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<front>
<journal-meta>
<journal-id journal-id-type="pmc">CMC</journal-id>
<journal-id journal-id-type="nlm-ta">CMC</journal-id>
<journal-id journal-id-type="publisher-id">CMC</journal-id>
<journal-title-group>
<journal-title>Computers, Materials &#x0026; Continua</journal-title>
</journal-title-group>
<issn pub-type="epub">1546-2226</issn>
<issn pub-type="ppub">1546-2218</issn>
<publisher>
<publisher-name>Tech Science Press</publisher-name>
<publisher-loc>USA</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">14413</article-id>
<article-id pub-id-type="doi">10.32604/cmc.2021.014413</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Article</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>A Secure NDN Framework for Internet of Things Enabled Healthcare</article-title>
<alt-title alt-title-type="left-running-head">A Secure NDN Framework for Internet of Things Enabled Healthcare</alt-title>
<alt-title alt-title-type="right-running-head">A Secure NDN Framework for Internet of Things Enabled Healthcare</alt-title>
</title-group>
<contrib-group content-type="authors">
<contrib id="author-1" contrib-type="author">
<name name-style="western">
<surname>Ullah</surname>
<given-names>Syed Sajid</given-names>
</name>
<xref ref-type="aff" rid="aff-1">1</xref>
</contrib>
<contrib id="author-2" contrib-type="author" corresp="yes">
<name name-style="western">
<surname>Hussain</surname>
<given-names>Saddam</given-names>
</name>
<xref ref-type="aff" rid="aff-1">1</xref>
<email>saddamicup1993@gmail.com</email>
</contrib>
<contrib id="author-3" contrib-type="author">
<name name-style="western">
<surname>Gumaei</surname>
<given-names>Abdu</given-names>
</name>
<xref ref-type="aff" rid="aff-2">2</xref><xref ref-type="aff" rid="aff-3">3</xref></contrib>
<contrib id="author-4" contrib-type="author">
<name name-style="western">
<surname>AlSalman</surname>
<given-names>Hussain</given-names>
</name>
<xref ref-type="aff" rid="aff-2">2</xref><xref ref-type="aff" rid="aff-4">4</xref></contrib>
<aff id="aff-1"><label>1</label><institution>IT Department, Hazara University</institution>, <addr-line>Mansehra, 21120</addr-line>, <country>Pakistan</country></aff>
<aff id="aff-2"><label>2</label><institution>Research Chair of Pervasive and Mobile Computing, Department of Information Systems, College of Computer and Information Sciences, King Saud University</institution>, <addr-line>Riyadh, 11543</addr-line>, <country>Saudi Arabia</country></aff>
<aff id="aff-3"><label>3</label><institution>Department of Computer Science, Taiz University</institution>, <addr-line>Taiz</addr-line>, <country>Yemen</country></aff>
<aff id="aff-4"><label>4</label><institution>Department of Computer Science, College of Computer and Information Sciences, King Saud University</institution>, <addr-line>Riyadh, 11543</addr-line>, <country>Saudi Arabia</country></aff>
</contrib-group>
<author-notes><corresp id="cor1">&#x002A;Corresponding Author: Saddam Hussain. Email: <email>saddamicup1993@gmail.com</email></corresp></author-notes>
<pub-date pub-type="epub" date-type="pub" iso-8601-date="2020-10-30">
<day>30</day>
<month>10</month>
<year>2020</year>
</pub-date>
<volume>67</volume>
<issue>1</issue>
<fpage>223</fpage>
<lpage>240</lpage>
<history>
<date date-type="received">
<day>18</day>
<month>09</month>
<year>2020</year>
</date>
<date date-type="accepted">
<day>18</day>
<month>10</month>
<year>2020</year>
</date>
</history>
<permissions>
<copyright-statement>&#x00A9; 2021 Ullah et al.</copyright-statement>
<copyright-year>2021</copyright-year>
<copyright-holder>Ullah et al.</copyright-holder>
<license xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>This work is licensed under a <ext-link ext-link-type="uri" xlink:type="simple" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</ext-link>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</license-p>
</license>
</permissions>
<self-uri content-type="pdf" xlink:href="TSP_CMC_14413.pdf"></self-uri>
<abstract>
<p>Healthcare is a binding domain for the Internet of Things (IoT) to automate healthcare services for sharing and accumulation patient records at anytime from anywhere through the Internet. The current IP-based Internet architecture suffers from latency, mobility, location dependency, and security. The Named Data Networking (NDN) has been projected as a future internet architecture to cope with the limitations of IP-based Internet. However, the NDN infrastructure does not have a secure framework for IoT healthcare information. In this paper, we proposed a secure NDN framework for IoT-enabled Healthcare (IoTEH). In the proposed work, we adopt the services of Identity-Based Signcryption (IBS) cryptography under the security hardness Hyperelliptic Curve Cryptosystem (HCC) to secure the IoTEH information in NDN. The HCC provides the corresponding level of security using minimal computational and communicational resources as compared to bilinear pairing and Elliptic Curve Cryptosystem (ECC). For the efficiency of the proposed scheme, we simulated the security of the proposed solution using Automated Validation of Internet Security Protocols and Applications (AVISPA). Besides, we deployed the proposed scheme on the IoTEH in NDN infrastructure and compared it with the recent IBS schemes in terms of computation and communication overheads. The simulation results showed the superiority and improvement of the proposed framework against contemporary related works.</p>
</abstract>
<kwd-group kwd-group-type="author">
<kwd>Named data networking</kwd>
<kwd>healthcare</kwd>
<kwd>identity-based signcryption</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="s1">
<label>1</label>
<title>Introduction</title>
<p>The IoTEH has recently been introduced to alleviate the issue of scarce resources due to the growing aging population [<xref ref-type="bibr" rid="ref-1">1</xref>,<xref ref-type="bibr" rid="ref-2">2</xref>]. The IoTEH system with all available resources to perform healthcare activities such as diagnosis, monitoring, and remote surgery [<xref ref-type="bibr" rid="ref-3">3</xref>]. The whole framework is devoted to extending the healthcare amenities from hospitals and communities to homes. Throughout wireless technology has been applied widely to integrate monitoring devices, including front-end network manager [<xref ref-type="bibr" rid="ref-4">4</xref>]. The system connects patients with all healthcare resources available in the community such as hospitals, physicians, rehabilitation centers, nurses, paramedics, and ambulances. All the content is networked together to the Internet, supported by programs based on Radio-Frequency Identification (RFID) technology [<xref ref-type="bibr" rid="ref-5">5</xref>,<xref ref-type="bibr" rid="ref-6">6</xref>]. Automated resource allocation has been developed to identify rehabilitation solutions to meet the specific needs of individual patients. However, IoTEH exchanges data/information over IP-based Internet with the risks related to security, privacy and mobility.</p>
<p>To overcome the aforementioned limitations of the IP-based Internet paradigm, a new Internet paradigm called Named Data Networking (NDN) has been introduced [<xref ref-type="bibr" rid="ref-7">7</xref>]. NDN aimed to offer in-network caching, built-in mobility support, and named-based routing that can provide scalable connectivity to the IoT devices with efficient information access to the end-users [<xref ref-type="bibr" rid="ref-8">8</xref>,<xref ref-type="bibr" rid="ref-9">9</xref>]. By keeping the positive aspects of NDN, a few schemes have been suggested for NDN based healthcare [<xref ref-type="bibr" rid="ref-10">10</xref>&#x2013;<xref ref-type="bibr" rid="ref-12">12</xref>].</p>
<p>However, until now, there is no concrete security plan suggested that can protect the NDN based healthcare information. As the IoTEH in NDN requires the essential properties of authentication and confidentiality, which can easily be achieved by implementing a secure digital signature and encryption (sign-then-encrypt) scheme [<xref ref-type="bibr" rid="ref-13">13</xref>,<xref ref-type="bibr" rid="ref-14">14</xref>]. Unfortunately, the trivial combination of sign-then-encrypt is costly and subject to some subtle attacks [<xref ref-type="bibr" rid="ref-15">15</xref>]. For this purpose, in 1997, Zheng [<xref ref-type="bibr" rid="ref-16">16</xref>], tossed the concept of a new cryptographic primitive toned as Signcryption, which provides the services of confidentiality and authenticity at a reasonable cost than the traditional sign-then-encrypt approach. Since then plenty of practical and innovative signcryption schemes have been suggested in recent years [<xref ref-type="bibr" rid="ref-17">17</xref>&#x2013;<xref ref-type="bibr" rid="ref-20">20</xref>]. However, the idea Zheng was primarily based on the old concept of Public Key Infrastructure (PKI) and therefore suffers from the certificate-related overheads.</p>
<p>In 1984, in a seminar, Shamir coined the concept of Identity-Based Cryptography (IBC), which is aimed to provide a viable alternative to traditional PKI in terms of convenience and efficiency [<xref ref-type="bibr" rid="ref-21">21</xref>]. An interesting feature of this type of cryptosystem is that any binary string that identifies the user, such as an email address, can be the public key of the users. Using identities as a public key eliminates the need for public-key certificates [<xref ref-type="bibr" rid="ref-22">22</xref>]. The first identity-based signature was mentioned in the Shamir proposal; however, the Identity-Based Encryption (IDBE) scheme was not established until 2001, when a practical IDBE scheme was proposed from bilinear pairing [<xref ref-type="bibr" rid="ref-23">23</xref>]. Since then, IBC and its applications have been the talk of the town for the past decade.</p>
<p>To provide efficient and robust security with minimal computation overheads, the common approaches used are Bilinear Pairing (BRPG), RSA, ECC, and HCC [<xref ref-type="bibr" rid="ref-24">24</xref>&#x2013;<xref ref-type="bibr" rid="ref-29">29</xref>]. However, HCC provides the same level of security in contrast with ECC, RSA, and BRPG [<xref ref-type="bibr" rid="ref-30">30</xref>&#x2013;<xref ref-type="bibr" rid="ref-32">32</xref>] using small key sizes. Therefore, HCC is considered as the most compact and efficient cryptographic mechanism that provides better performance than ECC, BRPG, and RSA with high efficiency and lower-key length [<xref ref-type="bibr" rid="ref-15">15</xref>]. The HCC uses 80-bit keys with strong security that will better suit the IoTEH in NDN infrastructure.</p>
<sec id="s1_1">
<label>1.1</label>
<title>NDN Overview</title>
<p>NDN is a new data-centric architecture that defines three different roles, such as Routers, Clients/Customer, and Providers with two types of packets (i.e., Interest Packet and Data Packet). Moreover, each router maintains three kinds of data structures, such as Pending Interest Table (PIT), Forwarding Information Base (FIB), and Content Store (CS) [<xref ref-type="bibr" rid="ref-33">33</xref>]. The data attainment process begins by sending an Interest with a particular name from the client&#x2019;s side. The routers rely on FIBs for transferring the interest to a potential provider and generate a PIT entry list on each router to establish an opposite path. Based on the opposite path, the provider of any interest returns the data to the client with the target data. The CS then stores the targeted data that pass through it for future use [<xref ref-type="bibr" rid="ref-34">34</xref>].</p>
<p>Suppose Client A begins the data attainment process by sending an Interest with a particular name, as shown in <xref ref-type="fig" rid="fig-1">Fig. 1</xref>. Initially, the Interest of Client A will be transferred using the services of FIB to the potential provider of the content/data. The feedback to that particular interest will be stored inside the CS of Router 4, Router 3, and Router 1 for future reuse. Later on, if another Client B needs the same content/data, then the interest will be satisfied locally from the CS of Router 3, instead of transferring the Interest of Client B to the original provider of the content/data [<xref ref-type="bibr" rid="ref-35">35</xref>].</p>
<fig id="fig-1">
<label>Figure 1</label>
<caption>
<title>Generic illustration of information distribution in NDN</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-1.png"/>
</fig>
</sec>
<sec id="s1_2">
<label>1.2</label>
<title>Contributions</title>
<p>Inspired by the above-mentioned discussion, we propose an IBS scheme for IoTEH in NDN networks. The proposed scheme is based on the concept of HCC, which provides the same level of security in contrast with ECC, RSA, and BRPG using small key sizes. The key research finding is mentioned below:
<list list-type="bullet">
<list-item><p>We proposed a secure NDN framework for Internet of Things Enabled Healthcare (IoTEH) using Identity-Based Signacryption (IBS) cryptography.</p></list-item>
<list-item><p>We used a lightweight Hyperelliptic Curve Cryptosystem (HCC) for the efficiency in terms of computation and communication overheads.</p></list-item>
<list-item><p>We also validate our scheme using the simulation tool &#x201C;AVISPA.&#x201D;</p></list-item>
<list-item><p>We deployed the newly proposed scheme on IoT enabled healthcare in NDN networks.</p></list-item>
<list-item><p>To conclude, we also compared our proposed scheme with relevant existing IBS schemes and the results show that the given scheme is more efficient in terms of computation and communication overheads than the previous.</p></list-item>
</list></p>
</sec>
<sec id="s1_3">
<label>1.3</label>
<title>Paper Organization</title>
<p>In Section 2, we discuss the related work about NDN based healthcare and IBS schemes. Section 3 comprises the preliminaries, threat model, and syntax of the proposed scheme. Section 4 includes the proposed network model and the proposed algorithm. Section 5 describes the security analysis for the proposed scheme. Section 6 includes a comparative analysis. In Section 7, we deployed our scheme on IoTBH in NDN networks, and Section 8 concludes our research.</p>
</sec>
</sec>
<sec id="s2">
<label>2</label>
<title>Related Work</title>
<p>In this section, we divide the given literature into two portions, such as NDN based schemes for healthcare and IBS schemes.</p>
<sec id="s2_1">
<label>2.1</label>
<title>NDN Based Schemes for Healthcare</title>
<p>In 2015, Saxena et al. [<xref ref-type="bibr" rid="ref-10">10</xref>] proposed an NDN based solution for healthcare. The proposed scheme can locate a network-based healthcare service. Later in 2017, Saxena et al. [<xref ref-type="bibr" rid="ref-11">11</xref>] tossed another NDN based scheme for emergency healthcare services. The author&#x2019;s aims to verify the authenticity of emergency messages in NDN based healthcare. However, in both the schemes [<xref ref-type="bibr" rid="ref-10">10</xref>,<xref ref-type="bibr" rid="ref-11">11</xref>], the authors did not provide a concrete security plan for the proposed scheme.</p>
<p>Recently, Wang et al. [<xref ref-type="bibr" rid="ref-12">12</xref>] proposed a monitoring framework to secure NDN-based healthcare infrastructure using the services of edge cloud. The authors, for the first time, introduce a security framework for NDN-based healthcare. In the given framework, the author exploits the advantages of NDN to enhance the efficiency of medical data. Unfortunately, the author used heavy attribute-based encryption using bilinear pairing.</p>
</sec>
<sec id="s2_2">
<label>2.2</label>
<title>Identity-Based Signcryption (IBS) Schemes</title>
<p>Signcryption and IBC [<xref ref-type="bibr" rid="ref-36">36</xref>] is an exciting research topic to develop a secure and effective IBS scheme. In 2002, Malone-Lee [<xref ref-type="bibr" rid="ref-23">23</xref>] provided the first IBS scheme using BRPG. Later in 2006, Duan et al. [<xref ref-type="bibr" rid="ref-37">37</xref>], proposed a multi-receiver IBS scheme for multiple receivers. However, the given scheme is subject to massive pairing operation due to BRPG. In 2008, Li et al. [<xref ref-type="bibr" rid="ref-38">38</xref>] coined an identity-based broadcast signcryption scheme for application to transmit a message securely and authentically. However, the given scheme is subject to massive pairing operation due to BRPG.</p>
<p>Later in 2013, Libert et al. [<xref ref-type="bibr" rid="ref-39">39</xref>] showed that the scheme of Malone-Lee&#x2019;s did not provide the semantic security because the signature of the signed message appears in the ultimate ciphertext. The authors also proposed three new IBS schemes, but they did not provide the essential security properties of public verifiability and forward secrecy. Similarly, the concept of IBS was further expanded to cater to further applications. In 2017, Nayak [<xref ref-type="bibr" rid="ref-40">40</xref>] constructed a new IBS scheme based on ECC. Unlike the previous schemes, the given approach reduces the computational and communicational resources. Besides, the given scheme provides the security assets of authentication, integrity, confidentiality, and unforgeability. However, there is still a need for improvement in the communication and computation cost because the cost of the scalar point multiplication on the elliptic curve is still not affordable for the resource-constrained environment. Later, in the same year, Reddi et al. [<xref ref-type="bibr" rid="ref-41">41</xref>] presented an IBS that is used to authenticate and verify both parties involved in the communication. In the proposed work, the author incorporated the idea of IBS into the Key Agreement Protocol. However, the given scheme is subject to massive pairing operation due to BRPG. Later, in 2017, Karati et al. [<xref ref-type="bibr" rid="ref-42">42</xref>] proposed an IBS scheme for the Industrial Internet of Things (IIoT).</p>
<p>Conversely, the proposed scheme suffers from a massive pairing operation due to the use of BRPG. Later in 2017, Swapna et al. [<xref ref-type="bibr" rid="ref-43">43</xref>] presented an IBS scheme to secure the communication between end-users and smart homes. The given scheme can provide the security assets of integrity, authentication, and confidentiality to protect the communication between end-users and smart homes from different types of possible security attacks. Unfortunately, the given scheme was constructed on bilinear pairing.</p>
<p>In 2020, Dharminder et al. [<xref ref-type="bibr" rid="ref-44">44</xref>] presented an IBS scheme for IIoT crowdsourcing under the standard model. In the proposed framework, the user adds a pairing free computation signing, making it efficient for the user. According to the authors, the proposed scheme is efficient in terms of computational and communicational costs. However, the given scheme suffers from high bandwidth usage and heavy computation costs due to the utilization of BRPG.</p>
</sec>
</sec>
<sec id="s3">
<label>3</label>
<title>Preliminaries</title>
<sec id="s3_1">
<label>3.1</label>
<title>Complexity Assumptions</title>
<p>For conducting the security analysis, we performed the following complexity assumptions:
<list list-type="bullet">
<list-item><p>The <inline-formula id="ieqn-1"><alternatives><inline-graphic xlink:href="ieqn-1.png"/><tex-math id="tex-ieqn-1"><![CDATA[$f_{\mathfrak{q}}$]]></tex-math><mml:math id="mml-ieqn-1"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D52E;</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> is a finite field with the order <inline-formula id="ieqn-2"><alternatives><inline-graphic xlink:href="ieqn-2.png"/><tex-math id="tex-ieqn-2"><![CDATA[$\mathfrak{q}$]]></tex-math><mml:math id="mml-ieqn-2"><mml:mi>&#x1D52E;</mml:mi></mml:math></alternatives></inline-formula>, where <inline-formula id="ieqn-3"><alternatives><inline-graphic xlink:href="ieqn-3.png"/><tex-math id="tex-ieqn-3"><![CDATA[$ \left(\mathfrak{q}\right)\approx 2^{160}$]]></tex-math><mml:math id="mml-ieqn-3"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D52E;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2248;</mml:mo><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>160</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>D is the divisor of the hyperelliptic curve (hec), which is the finite sum of the points; <inline-formula id="ieqn-4"><alternatives><inline-graphic xlink:href="ieqn-4.png"/><tex-math id="tex-ieqn-4"><![CDATA[$\mathrm{D}= \sum p_{i}\;\boldsymbol{\varepsilon}\;\mathrm{hec}\;m_{i}p_{i}$]]></tex-math><mml:math id="mml-ieqn-4"><mml:mstyle mathvariant="normal"><mml:mi>D</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mo>&#x2211;</mml:mo> <mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>&#x03B5;</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mstyle mathvariant="normal"><mml:mi>h</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi></mml:mstyle><mml:mspace width="2.77626pt" class="tmspace"/><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, where <inline-formula id="ieqn-5"><alternatives><inline-graphic xlink:href="ieqn-5.png"/><tex-math id="tex-ieqn-5"><![CDATA[$m_{i}\;\varepsilon\;f_{\mathfrak{q}}$]]></tex-math><mml:math id="mml-ieqn-5"><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>&#x03B5;</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D52E;</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>.</p></list-item>
</list></p>
<sec id="s3_1_1">
<label>3.1.1</label>
<title>Hyperelliptic Curve Discrete Logarithm Problem (HDLP)</title>
<p>The following supposition has been made for HDLP.</p>
<list list-type="bullet">
<list-item><p><inline-formula id="ieqn-6"><alternatives><inline-graphic xlink:href="ieqn-6.png"/><tex-math id="tex-ieqn-6"><![CDATA[$\Omega$]]></tex-math><mml:math id="mml-ieqn-6"><mml:mi>&#x03A9;</mml:mi></mml:math></alternatives></inline-formula> belongs to <inline-formula id="ieqn-7"><alternatives><inline-graphic xlink:href="ieqn-7.png"/><tex-math id="tex-ieqn-7"><![CDATA[$\{0, 1, 2, 3, 4, 5, \ldots, n-1\}$]]></tex-math><mml:math id="mml-ieqn-7"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Probability computation <inline-formula id="ieqn-8"><alternatives><inline-graphic xlink:href="ieqn-8.png"/><tex-math id="tex-ieqn-8"><![CDATA[$\Omega$]]></tex-math><mml:math id="mml-ieqn-8"><mml:mi>&#x03A9;</mml:mi></mml:math></alternatives></inline-formula> from <inline-formula id="ieqn-9"><alternatives><inline-graphic xlink:href="ieqn-9.png"/><tex-math id="tex-ieqn-9"><![CDATA[$\mathcal{M}=\Omega\cdot D$]]></tex-math><mml:math id="mml-ieqn-9"><mml:mi mathvariant="script">M</mml:mi><mml:mo>=</mml:mo><mml:mi>&#x03A9;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mi>D</mml:mi></mml:math></alternatives></inline-formula> is negligible.</p></list-item>
</list>
</sec>
<sec id="s3_1_2">
<label>3.1.2</label>
<title>Hyperelliptic Curve Computational Diffie&#x2013;Hellman (HCDH)</title>
<p>We also make the subsequent suppositions for <italic>HCDH</italic>.</p>
<list list-type="bullet">
<list-item><p>The <inline-formula id="ieqn-10"><alternatives><inline-graphic xlink:href="ieqn-10.png"/><tex-math id="tex-ieqn-10"><![CDATA[$\Omega$]]></tex-math><mml:math id="mml-ieqn-10"><mml:mi>&#x03A9;</mml:mi></mml:math></alternatives></inline-formula> and <inline-formula id="ieqn-11"><alternatives><inline-graphic xlink:href="ieqn-11.png"/><tex-math id="tex-ieqn-11"><![CDATA[$\mathcal{R}$]]></tex-math><mml:math id="mml-ieqn-11"><mml:mi mathvariant="script">R</mml:mi></mml:math></alternatives></inline-formula> belongs to <inline-formula id="ieqn-12"><alternatives><inline-graphic xlink:href="ieqn-12.png"/><tex-math id="tex-ieqn-12"><![CDATA[$\{0, 1, 2, 3, 4, 5, \ldots, \mathrm{n}-1\}$]]></tex-math><mml:math id="mml-ieqn-12"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>n</mml:mi></mml:mstyle><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Probability computation of <inline-formula id="ieqn-13"><alternatives><inline-graphic xlink:href="ieqn-13.png"/><tex-math id="tex-ieqn-13"><![CDATA[$\Omega$]]></tex-math><mml:math id="mml-ieqn-13"><mml:mi>&#x03A9;</mml:mi></mml:math></alternatives></inline-formula> and <inline-formula id="ieqn-14"><alternatives><inline-graphic xlink:href="ieqn-14.png"/><tex-math id="tex-ieqn-14"><![CDATA[$\mathcal{R}$]]></tex-math><mml:math id="mml-ieqn-14"><mml:mi mathvariant="script">R</mml:mi></mml:math></alternatives></inline-formula> from <inline-formula id="ieqn-15"><alternatives><inline-graphic xlink:href="ieqn-15.png"/><tex-math id="tex-ieqn-15"><![CDATA[$\Upsilon =\Omega \cdot \mathcal{R}\cdot \mathrm{D}$]]></tex-math><mml:math id="mml-ieqn-15"><mml:mi>&#x03D2;</mml:mi><mml:mo>=</mml:mo><mml:mi>&#x03A9;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mi mathvariant="script">R</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula> is negligible.</p></list-item>
</list>
</sec>
</sec>
<sec id="s3_2">
<label>3.2</label>
<title>Threat Model</title>
<p>In our scheme, we examine and consider the Doley&#x2013;Yao [<xref ref-type="bibr" rid="ref-45">45</xref>,<xref ref-type="bibr" rid="ref-46">46</xref>] threat model. According to Doley&#x2013;Yao, communication between two or more entities are not trusted and secure, as attackers have full command to expose the contents of the ciphertext and inject false encryption/signature text into the network. As NDN-based healthcare is posed to various types of security threats, this means that the user&#x2019;s sensitive information can be easily forged or delete by any adversaries. To maintain the security and authentication of IoTEH in NDN networks, authentication and secure communication between entities are required.</p>
</sec>
<sec id="s3_3">
<label>3.3</label>
<title>Syntax of the Proposed Scheme</title>
<p>The syntax of our newly proposed scheme consists of the following phases:
<list list-type="bullet">
<list-item><p>Setup Phase: In this phase, the Private Key Generation (PKG) produces its master secret key (<inline-formula id="ieqn-16"><alternatives><inline-graphic xlink:href="ieqn-16.png"/><tex-math id="tex-ieqn-16"><![CDATA[$\nu$]]></tex-math><mml:math id="mml-ieqn-16"><mml:mi>&#x03BD;</mml:mi></mml:math></alternatives></inline-formula>) and computes the master public key (<inline-formula id="ieqn-17"><alternatives><inline-graphic xlink:href="ieqn-17.png"/><tex-math id="tex-ieqn-17"><![CDATA[$\lambda$]]></tex-math><mml:math id="mml-ieqn-17"><mml:mi>&#x03BB;</mml:mi></mml:math></alternatives></inline-formula>) and the security parameter set <inline-formula id="ieqn-18"><alternatives><inline-graphic xlink:href="ieqn-18.png"/><tex-math id="tex-ieqn-18"><![CDATA[$\rho$]]></tex-math><mml:math id="mml-ieqn-18"><mml:mi>&#x03C1;</mml:mi></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Key Extraction Phase: In this phase, PKG makes a public key and private key for the consumer (<inline-formula id="ieqn-19"><alternatives><inline-graphic xlink:href="ieqn-19.png"/><tex-math id="tex-ieqn-19"><![CDATA[$\varsigma_{c}$]]></tex-math><mml:math id="mml-ieqn-19"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-20"><alternatives><inline-graphic xlink:href="ieqn-20.png"/><tex-math id="tex-ieqn-20"><![CDATA[$\varrho_{c}$]]></tex-math><mml:math id="mml-ieqn-20"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) and producer (<inline-formula id="ieqn-21"><alternatives><inline-graphic xlink:href="ieqn-21.png"/><tex-math id="tex-ieqn-21"><![CDATA[$\varsigma_{p}$]]></tex-math><mml:math id="mml-ieqn-21"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-22"><alternatives><inline-graphic xlink:href="ieqn-22.png"/><tex-math id="tex-ieqn-22"><![CDATA[$\varrho_{p}$]]></tex-math><mml:math id="mml-ieqn-22"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) on behalf of both consumer and producer identities (<inline-formula id="ieqn-23"><alternatives><inline-graphic xlink:href="ieqn-23.png"/><tex-math id="tex-ieqn-23"><![CDATA[$\mathrm{I}\mathtt{D}_{c}$]]></tex-math><mml:math id="mml-ieqn-23"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-24"><alternatives><inline-graphic xlink:href="ieqn-24.png"/><tex-math id="tex-ieqn-24"><![CDATA[$\mathrm{I}\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-24"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>). The PKG then send the keys to the consumer and producer by using a secure channel.</p></list-item>
<list-item><p>Signcryption Phase: In this phase, the producer generates signcrypted message ( &#x1d4c8;) by taking the consumer and its own identities (<inline-formula id="ieqn-25"><alternatives><inline-graphic xlink:href="ieqn-25.png"/><tex-math id="tex-ieqn-25"><![CDATA[$\mathrm{I}\mathtt{D}_{c}$]]></tex-math><mml:math id="mml-ieqn-25"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-26"><alternatives><inline-graphic xlink:href="ieqn-26.png"/><tex-math id="tex-ieqn-26"><![CDATA[$\mathrm{I}\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-26"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>), consumer public key (<inline-formula id="ieqn-27"><alternatives><inline-graphic xlink:href="ieqn-27.png"/><tex-math id="tex-ieqn-27"><![CDATA[$\varsigma_{c, }$]]></tex-math><mml:math id="mml-ieqn-27"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>), its private key (<inline-formula id="ieqn-28"><alternatives><inline-graphic xlink:href="ieqn-28.png"/><tex-math id="tex-ieqn-28"><![CDATA[$\varrho_{p}$]]></tex-math><mml:math id="mml-ieqn-28"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) as an input. Then send signcrypted message (&#x1d4c8;) to the consumer.</p></list-item>
<list-item><p>Unsigncryption Phase: In this phase, the consumer unsigncrypt the signcrypted message (<inline-formula id="ieqn-29"><alternatives><inline-graphic xlink:href="ieqn-29.png"/><tex-math id="tex-ieqn-29"><![CDATA[${\mathcal s}$]]></tex-math><mml:math id="mml-ieqn-29"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>). For this purpose, the consumer takes its private key (<inline-formula id="ieqn-30"><alternatives><inline-graphic xlink:href="ieqn-30.png"/><tex-math id="tex-ieqn-30"><![CDATA[$\varrho_{c}$]]></tex-math><mml:math id="mml-ieqn-30"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>), the public key of producer (<inline-formula id="ieqn-31"><alternatives><inline-graphic xlink:href="ieqn-31.png"/><tex-math id="tex-ieqn-31"><![CDATA[$\varsigma_{p}$]]></tex-math><mml:math id="mml-ieqn-31"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) its own and producer identity (<inline-formula id="ieqn-32"><alternatives><inline-graphic xlink:href="ieqn-32.png"/><tex-math id="tex-ieqn-32"><![CDATA[$\mathrm{I}\mathtt{D}_{c}$]]></tex-math><mml:math id="mml-ieqn-32"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-33"><alternatives><inline-graphic xlink:href="ieqn-33.png"/><tex-math id="tex-ieqn-33"><![CDATA[$\mathrm{I}\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-33"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) and signcrypted message (<inline-formula id="ieqn-34"><alternatives><inline-graphic xlink:href="ieqn-34.png"/><tex-math id="tex-ieqn-34"><![CDATA[${\mathcal s}$]]></tex-math><mml:math id="mml-ieqn-34"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>) as an input.</p></list-item>
</list></p>
</sec>
</sec>
<sec id="s4">
<label>4</label>
<title>Proposed NDN Framework for Internet of Things Enabled Healthcare</title>
<sec id="s4_1">
<label>4.1</label>
<title>Proposed Network Model</title>
<p>In <xref ref-type="fig" rid="fig-2">Fig. 2</xref>, we have shown the secure network modal for IoTEH in NDN networks. The proposed modal consists of the participants, such as consumer, producer, NDN routers, and PKG. The role of each participant is explained below:
<list list-type="bullet">
<list-item><p>Role of Consumer: The consumer can be a hospital, patient, doctor, or any IoT device (smartphone, smartwatch, sensor, etc.) that want secure healthcare-related information like (patient records, patient stats, online patient monitoring).</p></list-item>
<list-item><p>Role of Producer: Producer can be a hospital, patient, doctor, or any IoT device (smartphone, smartwatch, sensor, etc.) that provide healthcare-related information like (patient records, patient stats, online patient monitoring).</p></list-item>
<list-item><p>Role of NDN Routers: The NDN routers is responsible for providing a communication route among producer and consumer for sending healthcare-related information. Every NDN router maintains three types of Tables, such as CS, PIT, and FIB.</p></list-item>
<list-item><p>Private Key Generator (PKG): The PKG is a trustable authority that establishes and manage a secure communication between consumer and producer.</p></list-item>
</list></p>
<fig id="fig-2">
<label>Figure 2</label>
<caption>
<title>Proposed network modal</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-2.png"/>
</fig>
<p>In our proposed scheme, before starting a secure communication, the consumer and producer send their identities to PKG. After receiving the identities of both the consumer and producer, the PKG generates the private and public keys for both of them and delivers it using a secure connection.</p>
<p>Suppose a consumer sends an interest for healthcare-related information, the NDN routers will transfer that interest to the producer. The producer will signcrypt the information based on the interest of the consumer. The signcrypted information is then forwarded to the consumer through the NDN routers. The NDN router, after receiving the information from the producer, it will forward the information using the services of FIB by assigning a PIT interface without caching. The process of forwarding without caching will continue until the information is reached to the original consumer. Obviously, the caching of this information will not facilitate any consumer later because the information can only be designcrypted using the private key of the requested consumer.</p>
</sec>
<sec id="s4_2">
<label>4.2</label>
<title>Proposed Algorithm</title>
<p>The proposed algorithm consists of the following 4 phases, such as setup phase, key extraction phase, signcryption phase and unsigncryption phase [<xref ref-type="bibr" rid="ref-40">40</xref>].</p>
<p>The notation used in our algorithms is mentioned in <xref ref-type="table" rid="table-1">Tab. 1</xref>.</p>
<table-wrap id="table-1">
<label>Table 1</label>
<caption>
<title>Notation for the proposed algorithm</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Name</th>
<th>Notation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Security parameter</td>
<td><inline-formula id="ieqn-35"><alternatives><inline-graphic xlink:href="ieqn-35.png"/><tex-math id="tex-ieqn-35"><![CDATA[$\mu$]]></tex-math><mml:math id="mml-ieqn-35"><mml:mi>&#x03BC;</mml:mi></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Finite field</td>
<td><inline-formula id="ieqn-36"><alternatives><inline-graphic xlink:href="ieqn-36.png"/><tex-math id="tex-ieqn-36"><![CDATA[${\mathcal f}_{{\mathcal n}}$]]></tex-math><mml:math id="mml-ieqn-36"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BB;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Divisor</td>
<td><italic>D</italic></td>
</tr>
<tr>
<td>Master secret key</td>
<td><inline-formula id="ieqn-37"><alternatives><inline-graphic xlink:href="ieqn-37.png"/><tex-math id="tex-ieqn-37"><![CDATA[$ \nu $]]></tex-math><mml:math id="mml-ieqn-37"><mml:mi>&#x03BD;</mml:mi></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Master public key</td>
<td><inline-formula id="ieqn-38"><alternatives><inline-graphic xlink:href="ieqn-38.png"/><tex-math id="tex-ieqn-38"><![CDATA[$ \lambda $]]></tex-math><mml:math id="mml-ieqn-38"><mml:mi>&#x03BB;</mml:mi></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Public parameter set</td>
<td><inline-formula id="ieqn-39"><alternatives><inline-graphic xlink:href="ieqn-39.png"/><tex-math id="tex-ieqn-39"><![CDATA[$ \rho $]]></tex-math><mml:math id="mml-ieqn-39"><mml:mi>&#x03C1;</mml:mi></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Identity of users (consumer and producer)</td>
<td><inline-formula id="ieqn-40"><alternatives><inline-graphic xlink:href="ieqn-40.png"/><tex-math id="tex-ieqn-40"><![CDATA[$ \mathrm{I}\mathtt{D}_{u}$]]></tex-math><mml:math id="mml-ieqn-40"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Identity of producer &#x0026; consumer</td>
<td><inline-formula id="ieqn-41"><alternatives><inline-graphic xlink:href="ieqn-41.png"/><tex-math id="tex-ieqn-41"><![CDATA[$\mathrm{I}\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-41"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-42"><alternatives><inline-graphic xlink:href="ieqn-42.png"/><tex-math id="tex-ieqn-42"><![CDATA[$\mathrm{I}\mathtt{D}_{c}$]]></tex-math><mml:math id="mml-ieqn-42"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Producer private key and public key</td>
<td><inline-formula id="ieqn-43"><alternatives><inline-graphic xlink:href="ieqn-43.png"/><tex-math id="tex-ieqn-43"><![CDATA[$\varrho_{p}$]]></tex-math><mml:math id="mml-ieqn-43"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-44"><alternatives><inline-graphic xlink:href="ieqn-44.png"/><tex-math id="tex-ieqn-44"><![CDATA[$\varsigma _{p}$]]></tex-math><mml:math id="mml-ieqn-44"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Message</td>
<td><italic>m</italic></td>
</tr>
<tr>
<td>Fresh nonce</td>
<td><inline-formula id="ieqn-45"><alternatives><inline-graphic xlink:href="ieqn-45.png"/><tex-math id="tex-ieqn-45"><![CDATA[$ \Lambda $]]></tex-math><mml:math id="mml-ieqn-45"><mml:mtext>&#x039B;</mml:mtext></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Random and private number</td>
<td><inline-formula id="ieqn-46"><alternatives><inline-graphic xlink:href="ieqn-46.png"/><tex-math id="tex-ieqn-46"><![CDATA[${\mathcal r}$]]></tex-math><mml:math id="mml-ieqn-46"><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-47"><alternatives><inline-graphic xlink:href="ieqn-47.png"/><tex-math id="tex-ieqn-47"><![CDATA[${\mathcal b}$]]></tex-math><mml:math id="mml-ieqn-47"><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Message digest</td>
<td><inline-formula id="ieqn-48"><alternatives><inline-graphic xlink:href="ieqn-48.png"/><tex-math id="tex-ieqn-48"><![CDATA[$ \mathcal{Z}$]]></tex-math><mml:math id="mml-ieqn-48"><mml:mi mathvariant="script">Z</mml:mi></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Signature</td>
<td><inline-formula id="ieqn-49"><alternatives><inline-graphic xlink:href="ieqn-49.png"/><tex-math id="tex-ieqn-49"><![CDATA[$ \nabla $]]></tex-math><mml:math id="mml-ieqn-49"><mml:mo>&#x2207;</mml:mo></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Signed message</td>
<td><inline-formula id="ieqn-50"><alternatives><inline-graphic xlink:href="ieqn-50.png"/><tex-math id="tex-ieqn-50"><![CDATA[$ {\mathcal s}$]]></tex-math><mml:math id="mml-ieqn-50"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Consumer private key &#x0026; public key</td>
<td><inline-formula id="ieqn-51"><alternatives><inline-graphic xlink:href="ieqn-51.png"/><tex-math id="tex-ieqn-51"><![CDATA[$\varrho _{c}$]]></tex-math><mml:math id="mml-ieqn-51"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-52"><alternatives><inline-graphic xlink:href="ieqn-52.png"/><tex-math id="tex-ieqn-52"><![CDATA[$\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-52"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula></td>
</tr>
</tbody>
</table>
</table-wrap>
<p><bold>Setup:</bold></p>
<p>This algorithm is running by the PKG.</p>
<list list-type="bullet">
<list-item><p>It takes the security parameter (<inline-formula id="ieqn-53"><alternatives><inline-graphic xlink:href="ieqn-53.png"/><tex-math id="tex-ieqn-53"><![CDATA[$\mathrm{\mu}$]]></tex-math><mml:math id="mml-ieqn-53"><mml:mstyle mathvariant="normal"><mml:mi>&#x03BC;</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>)</p></list-item>
<list-item><p>Select a <inline-formula id="ieqn-54"><alternatives><inline-graphic xlink:href="ieqn-54.png"/><tex-math id="tex-ieqn-54"><![CDATA[$\mathtt{hec}$]]></tex-math><mml:math id="mml-ieqn-54"><mml:mstyle mathvariant="monospace"><mml:mi>h</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula> of the genus (<italic>g</italic> = 2)</p></list-item>
<list-item><p>Select a parameter (<inline-formula id="ieqn-55"><alternatives><inline-graphic xlink:href="ieqn-55.png"/><tex-math id="tex-ieqn-55"><![CDATA[${\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-55"><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>) of length 80 bits</p></list-item>
<list-item><p>Select a finite field (<inline-formula id="ieqn-56"><alternatives><inline-graphic xlink:href="ieqn-56.png"/><tex-math id="tex-ieqn-56"><![CDATA[${\mathcal f}_{{\mathcal n}}$]]></tex-math><mml:math id="mml-ieqn-56"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BB;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>)</p></list-item>
<list-item><p>Select a <inline-formula id="ieqn-57"><alternatives><inline-graphic xlink:href="ieqn-57.png"/><tex-math id="tex-ieqn-57"><![CDATA[$\mathtt{hec}$]]></tex-math><mml:math id="mml-ieqn-57"><mml:mstyle mathvariant="monospace"><mml:mi>h</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula> divisor (<inline-formula id="ieqn-58"><alternatives><inline-graphic xlink:href="ieqn-58.png"/><tex-math id="tex-ieqn-58"><![CDATA[$\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-58"><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>) of order <inline-formula id="ieqn-59"><alternatives><inline-graphic xlink:href="ieqn-59.png"/><tex-math id="tex-ieqn-59"><![CDATA[${\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-59"><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula></p></list-item>
<list-item><p>Select a master secret key (<inline-formula id="ieqn-60"><alternatives><inline-graphic xlink:href="ieqn-60.png"/><tex-math id="tex-ieqn-60"><![CDATA[$\nu$]]></tex-math><mml:math id="mml-ieqn-60"><mml:mi>&#x03BD;</mml:mi></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-61"><alternatives><inline-graphic xlink:href="ieqn-61.png"/><tex-math id="tex-ieqn-61"><![CDATA[$\nu \boldsymbol{\varepsilon} (0, 1, 2, 3, 4, \ldots, (n-1))$]]></tex-math><mml:math id="mml-ieqn-61"><mml:mi>&#x03BD;</mml:mi><mml:mi>&#x03B5;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></p></list-item>
<list-item><p>Compute master public key as <inline-formula id="ieqn-62"><alternatives><inline-graphic xlink:href="ieqn-62.png"/><tex-math id="tex-ieqn-62"><![CDATA[$\lambda=\nu \cdot \mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-62"><mml:mi>&#x03BB;</mml:mi><mml:mo>=</mml:mo><mml:mi>&#x03BD;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></p></list-item>
<list-item><p><inline-formula id="ieqn-63"><alternatives><inline-graphic xlink:href="ieqn-63.png"/><tex-math id="tex-ieqn-63"><![CDATA[${\mathcal h}_{0}$]]></tex-math><mml:math id="mml-ieqn-63"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-64"><alternatives><inline-graphic xlink:href="ieqn-64.png"/><tex-math id="tex-ieqn-64"><![CDATA[${\mathcal h}_{1}$]]></tex-math><mml:math id="mml-ieqn-64"><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> are one-way collision functions.</p></list-item>
<list-item><p>Then publish all the parameter set <inline-formula id="ieqn-65"><alternatives><inline-graphic xlink:href="ieqn-65.png"/><tex-math id="tex-ieqn-65"><![CDATA[$\rho =\{{\mathcal n}, {\mathcal f}_{{\mathcal n}}, {\mathcal h}_{0}, {\mathcal h}_{1}, \mu, \lambda, \mathtt{hec}, \mathsf{D}\}$]]></tex-math><mml:math id="mml-ieqn-65"><mml:mi>&#x03C1;</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BB;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>&#x03BC;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03BB;</mml:mi><mml:mo>,</mml:mo><mml:mstyle mathvariant="monospace"><mml:mi>h</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
</list>
<p><bold>Key Extraction:</bold></p>
<p>This algorithm is executed by the PKG that takes the identities of users (<inline-formula id="ieqn-66"><alternatives><inline-graphic xlink:href="ieqn-66.png"/><tex-math id="tex-ieqn-66"><![CDATA[$\mathrm{I}\mathtt{D}_{\mathtt{u}}$]]></tex-math><mml:math id="mml-ieqn-66"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>u</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) and compute the public and private keys for the users using the <inline-formula id="ieqn-67"><alternatives><inline-graphic xlink:href="ieqn-67.png"/><tex-math id="tex-ieqn-67"><![CDATA[$\mathrm{I}\mathtt{D}_{\mathtt{u}}$]]></tex-math><mml:math id="mml-ieqn-67"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>u</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> as:
<list list-type="bullet">
<list-item><p>Compute provider private key (<inline-formula id="ieqn-68"><alternatives><inline-graphic xlink:href="ieqn-68.png"/><tex-math id="tex-ieqn-68"><![CDATA[$\varrho _{p}$]]></tex-math><mml:math id="mml-ieqn-68"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-69"><alternatives><inline-graphic xlink:href="ieqn-69.png"/><tex-math id="tex-ieqn-69"><![CDATA[$\varrho _{p}= \nu \cdot {\mathcal h}_{0}(\mathrm{I}\mathtt{D}_{p})mod\;{\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-69"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x03BD;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute provider public key (<inline-formula id="ieqn-70"><alternatives><inline-graphic xlink:href="ieqn-70.png"/><tex-math id="tex-ieqn-70"><![CDATA[$\varsigma _{p}$]]></tex-math><mml:math id="mml-ieqn-70"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-71"><alternatives><inline-graphic xlink:href="ieqn-71.png"/><tex-math id="tex-ieqn-71"><![CDATA[$\varsigma _{p}=\varrho_{p}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-71"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute consumer private key (<inline-formula id="ieqn-72"><alternatives><inline-graphic xlink:href="ieqn-72.png"/><tex-math id="tex-ieqn-72"><![CDATA[$\varrho _{c}$]]></tex-math><mml:math id="mml-ieqn-72"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-73"><alternatives><inline-graphic xlink:href="ieqn-73.png"/><tex-math id="tex-ieqn-73"><![CDATA[$\varrho _{c}= \nu\cdot{\mathcal h}_{0}(\mathrm{I}\mathtt{D}_{c})mod\;{\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-73"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x03BD;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute consumer public key (<inline-formula id="ieqn-74"><alternatives><inline-graphic xlink:href="ieqn-74.png"/><tex-math id="tex-ieqn-74"><![CDATA[$\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-74"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-75"><alternatives><inline-graphic xlink:href="ieqn-75.png"/><tex-math id="tex-ieqn-75"><![CDATA[$\varsigma _{c}=\varrho_{c}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-75"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>.</p></list-item>
</list></p>
<p>After that, it sends the public and private key (<inline-formula id="ieqn-76"><alternatives><inline-graphic xlink:href="ieqn-76.png"/><tex-math id="tex-ieqn-76"><![CDATA[$\varrho_{\mathtt{u}}, \varsigma_{\mathtt{u}}$]]></tex-math><mml:math id="mml-ieqn-76"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>u</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>u</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) by using a secure network.</p>
<p><bold>Signcryption:</bold></p>
<p>This algorithm is run by the producer. It takes the message (<italic>m</italic>), fresh nonce (<inline-formula id="ieqn-77"><alternatives><inline-graphic xlink:href="ieqn-77.png"/><tex-math id="tex-ieqn-77"><![CDATA[$\Lambda$]]></tex-math><mml:math id="mml-ieqn-77"><mml:mtext>&#x039B;</mml:mtext></mml:math></alternatives></inline-formula>), <inline-formula id="ieqn-78"><alternatives><inline-graphic xlink:href="ieqn-78.png"/><tex-math id="tex-ieqn-78"><![CDATA[$\mathrm{I}\mathtt{D}_{\mathrm{c}}$]]></tex-math><mml:math id="mml-ieqn-78"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-79"><alternatives><inline-graphic xlink:href="ieqn-79.png"/><tex-math id="tex-ieqn-79"><![CDATA[$\mathrm{I}\mathtt{D}_{\mathrm{p}}$]]></tex-math><mml:math id="mml-ieqn-79"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-80"><alternatives><inline-graphic xlink:href="ieqn-80.png"/><tex-math id="tex-ieqn-80"><![CDATA[$\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-80"><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-81"><alternatives><inline-graphic xlink:href="ieqn-81.png"/><tex-math id="tex-ieqn-81"><![CDATA[$\varrho _{p}$]]></tex-math><mml:math id="mml-ieqn-81"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-82"><alternatives><inline-graphic xlink:href="ieqn-82.png"/><tex-math id="tex-ieqn-82"><![CDATA[$\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-82"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> as input and then perform the following computations.</p>
<list list-type="bullet">
<list-item><p>Take a random number <inline-formula id="ieqn-83"><alternatives><inline-graphic xlink:href="ieqn-83.png"/><tex-math id="tex-ieqn-83"><![CDATA[${\mathcal r}\;\boldsymbol{\varepsilon}\;(0, 1, 2, 3, 4, \ldots, (n-1))$]]></tex-math><mml:math id="mml-ieqn-83"><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>&#x03B5;</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute private number as: <inline-formula id="ieqn-84"><alternatives><inline-graphic xlink:href="ieqn-84.png"/><tex-math id="tex-ieqn-84"><![CDATA[${\mathcal b}={\mathcal r}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-84"><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute <inline-formula id="ieqn-85"><alternatives><inline-graphic xlink:href="ieqn-85.png"/><tex-math id="tex-ieqn-85"><![CDATA[$\mathfrak{J}= \mathrm{I}\mathtt{D}_{\mathrm{c}}\cdot{\mathcal r}\cdot\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-85"><mml:mi>&#x1D50D;</mml:mi><mml:mo>=</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute encrypted text <inline-formula id="ieqn-86"><alternatives><inline-graphic xlink:href="ieqn-86.png"/><tex-math id="tex-ieqn-86"><![CDATA[$\mathfrak{w}= \mathfrak{w}_{\mathfrak{J}}(\mathrm{m})$]]></tex-math><mml:math id="mml-ieqn-86"><mml:mi>&#x1D534;</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x1D534;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D50D;</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute message-digest <inline-formula id="ieqn-87"><alternatives><inline-graphic xlink:href="ieqn-87.png"/><tex-math id="tex-ieqn-87"><![CDATA[$\mathcal{Z}= {\mathcal h}_{1}(\mathrm{m}||\lambda | \left| \mathrm{I }\mathtt{D}_{\mathrm{p}}\right| |\mathrm{I }\mathtt{D}_{\mathrm{c}}||\varsigma _{c})$]]></tex-math><mml:math id="mml-ieqn-87"><mml:mi mathvariant="script">Z</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mtext>&#x039B;</mml:mtext><mml:mo>|</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Generate signature <inline-formula id="ieqn-88"><alternatives><inline-graphic xlink:href="ieqn-88.png"/><tex-math id="tex-ieqn-88"><![CDATA[$\nabla =\mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}-\mathrm{I }\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho_{p}\cdot{\mathcal b} mod {\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-88"><mml:mo>&#x2207;</mml:mo><mml:mo>=</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>-</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>.</p></list-item>
</list>
<p>Send the signcrypted text <inline-formula id="ieqn-89"><alternatives><inline-graphic xlink:href="ieqn-89.png"/><tex-math id="tex-ieqn-89"><![CDATA[${\mathcal s}= (\mathfrak{w}, \mathcal{Z}, \nabla, {\mathcal b})$]]></tex-math><mml:math id="mml-ieqn-89"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>,</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> to the consumer.</p>
<p><bold>Unsigncryption:</bold></p>
<p>This algorithm is run on the consumer side. It takes <inline-formula id="ieqn-90"><alternatives><inline-graphic xlink:href="ieqn-90.png"/><tex-math id="tex-ieqn-90"><![CDATA[${\mathcal s}= (\mathfrak{w}, \mathcal{Z}, \nabla, {\mathcal b})$]]></tex-math><mml:math id="mml-ieqn-90"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>,</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-91"><alternatives><inline-graphic xlink:href="ieqn-91.png"/><tex-math id="tex-ieqn-91"><![CDATA[$\varrho _{{\mathcal c}}$]]></tex-math><mml:math id="mml-ieqn-91"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4B8;</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-92"><alternatives><inline-graphic xlink:href="ieqn-92.png"/><tex-math id="tex-ieqn-92"><![CDATA[$\varsigma _{p}$]]></tex-math><mml:math id="mml-ieqn-92"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>.</p>
<list list-type="bullet">
<list-item><p>Compute <inline-formula id="ieqn-93"><alternatives><inline-graphic xlink:href="ieqn-93.png"/><tex-math id="tex-ieqn-93"><![CDATA[$\mathfrak{J}=\nabla \cdot\varsigma _{c}+\mathrm{I }\mathtt{D}_{p}\cdot \mathcal{Z}\cdot {\mathcal b}\cdot\varsigma _{p}\cdot\varrho _{{\mathcal c}}$]]></tex-math><mml:math id="mml-ieqn-93"><mml:mi>&#x1D50D;</mml:mi><mml:mo>=</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4B8;</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Decrypt text <inline-formula id="ieqn-94"><alternatives><inline-graphic xlink:href="ieqn-94.png"/><tex-math id="tex-ieqn-94"><![CDATA[$\mathrm{m}^{\prime}= d_{\mathfrak{J}}(\mathfrak{w})$]]></tex-math><mml:math id="mml-ieqn-94"><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D50D;</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>.</p></list-item>
<list-item><p>Compute <inline-formula id="ieqn-95"><alternatives><inline-graphic xlink:href="ieqn-95.png"/><tex-math id="tex-ieqn-95"><![CDATA[$\mathcal{Z}^{\prime}= {\mathcal h}_{1}(\mathrm{m}^{\prime}||\lambda | \left| \mathrm{I }\mathtt{D}_{\mathrm{p}}\right| |\mathrm{I }\mathtt{D}_{\mathrm{c}}||\varsigma _{c})$]]></tex-math><mml:math id="mml-ieqn-95"><mml:msup><mml:mrow><mml:mi mathvariant="script">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mtext>&#x039B;</mml:mtext><mml:mo>|</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. If <inline-formula id="ieqn-96"><alternatives><inline-graphic xlink:href="ieqn-96.png"/><tex-math id="tex-ieqn-96"><![CDATA[$\mathcal{Z}=\mathcal{Z}^{\prime}$]]></tex-math><mml:math id="mml-ieqn-96"><mml:mi mathvariant="script">Z</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> valid otherwise invalid.</p></list-item>
</list>
</sec>
<sec id="s4_3">
<label>4.3</label>
<title>Correctness</title>
<p>
<disp-formula id="eqn-1">
<alternatives>
<graphic mimetype="image" mime-subtype="png" xlink:href="eqn-1.png"/>
<tex-math id="tex-eqn-1"><![CDATA[$$\begin{align*}
\mathfrak{J}&=\nabla \cdot\varsigma_{c}+\mathrm{I }\mathtt{D}_{p}\cdot\mathcal{Z}\cdot{\mathcal b}\cdot\varsigma _{p}\cdot\varrho_{{\mathcal c}}\nonumber \\
&= (\mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}-\mathrm{I}\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho _{p}\cdot{\mathcal b})\cdot\varsigma _{c}+\mathrm{I }\mathtt{D}_{p}\cdot\mathcal{Z}\cdot{\mathcal b}\cdot\varsigma _{p}\cdot\varrho_{{\mathcal c}}\nonumber \\
&= \mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}\cdot \varsigma _{c}-\mathrm{I }\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho _{p}\cdot{\mathcal b}\cdot\varsigma _{c}+\mathrm{I }\mathtt{D}_{p}\cdot\mathcal{Z}\cdot{\mathcal b}\cdot\varsigma _{p}\cdot\varrho _{{\mathcal c}}\nonumber \\
&= \mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}\cdot \varsigma _{c}-\mathrm{I}\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho _{p}\cdot{\mathcal b}. \varsigma _{c}+\mathrm{I }\mathtt{D}_{p}\cdot\mathcal{Z}\cdot{\mathcal b}\cdot\varrho _{p}\cdot\mathsf{D}\varrho _{\mathcal{c}}\nonumber \\
&= \mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}\cdot \varsigma _{c}-\mathrm{I}\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho_{p}\cdot{\mathcal b}\cdot \varsigma_{c}+\mathrm{I}\mathtt{D}_{p}\cdot\mathcal{Z}\cdot{\mathcal b}\cdot\varrho _{p}\cdot\varsigma _{c}\nonumber \\
&= \mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}\cdot \varsigma_{c}=\mathfrak{J}
\end{align*}$$]]></tex-math>
<mml:math id="mml-eqn-1" display="block"><mml:mtable columnalign="right left" columnspacing="1pt"><mml:mtr><mml:mtd><mml:mi>&#x1D50D;</mml:mi></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mi 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</p>
</sec>
</sec>
<sec id="s5">
<label>5</label>
<title>Security Analysis</title>
<p>In this section, we discuss the proposed scheme to maintain the basic security assets, including confidentiality, authentication, unforgeability, integrity and Non-Repudiation. Each of the mentioned features is briefly analyzed in the following sections.</p>
<sec id="s5_1">
<label>5.1</label>
<title>Confidentiality</title>
<p>An IBS scheme is supposed to succeed in the property of confidentiality if no adversary can compromise the encryption key of the sender.</p>
<p><bold>Proof:</bold> The proposed plan ensures the property of confidentiality. If an intruder wants to steal the original content or secret key of the message, he/she must have information about the key in advance as <inline-formula id="ieqn-97"><alternatives><inline-graphic xlink:href="ieqn-97.png"/><tex-math id="tex-ieqn-97"><![CDATA[$\mathfrak{J}= \mathrm{I}\mathtt{D}_{\mathrm{c}}\cdot{\mathcal r}\cdot\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-97"><mml:mi>&#x1D50D;</mml:mi><mml:mo>=</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>. To determine <inline-formula id="ieqn-98"><alternatives><inline-graphic xlink:href="ieqn-98.png"/><tex-math id="tex-ieqn-98"><![CDATA[$\mathfrak{J}$]]></tex-math><mml:math id="mml-ieqn-98"><mml:mi>&#x1D50D;</mml:mi></mml:math></alternatives></inline-formula>, the intruder needs to compute <inline-formula id="ieqn-99"><alternatives><inline-graphic xlink:href="ieqn-99.png"/><tex-math id="tex-ieqn-99"><![CDATA[${\mathcal r}$]]></tex-math><mml:math id="mml-ieqn-99"><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula> from <inline-formula id="ieqn-100"><alternatives><inline-graphic xlink:href="ieqn-100.png"/><tex-math id="tex-ieqn-100"><![CDATA[${\mathcal b}= {\mathcal r}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-100"><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>, which infeasible due to the properties of HDLP.</p>
</sec>
<sec id="s5_2">
<label>5.2</label>
<title>Authentication</title>
<p>An IBS is considered to achieve the security asset of authentication if the consumer can verify the source of the message.</p>
<p><bold>Proof:</bold> The consumer can use his public key <inline-formula id="ieqn-101"><alternatives><inline-graphic xlink:href="ieqn-101.png"/><tex-math id="tex-ieqn-101"><![CDATA[$\varsigma_{c}$]]></tex-math><mml:math id="mml-ieqn-101"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> and signature <inline-formula id="ieqn-102"><alternatives><inline-graphic xlink:href="ieqn-102.png"/><tex-math id="tex-ieqn-102"><![CDATA[$\nabla$]]></tex-math><mml:math id="mml-ieqn-102"><mml:mo>&#x2207;</mml:mo></mml:math></alternatives></inline-formula> to verify the authenticity of the producer. As the message is signed with the private key <inline-formula id="ieqn-103"><alternatives><inline-graphic xlink:href="ieqn-103.png"/><tex-math id="tex-ieqn-103"><![CDATA[$\varrho_{p}$]]></tex-math><mml:math id="mml-ieqn-103"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> of the producer. In our scheme, the consumer can authenticate the identity of the producer.</p>
</sec>
<sec id="s5_3">
<label>5.3</label>
<title>Integrity</title>
<p>An IBS scheme is likely to achieve the security asset of integrity if no adversary can generate the same hash value for two different sizes/nature messages.</p>
<p><bold>Proof:</bold> The provider takes the &#x201C;hash value&#x201D; &#x201C;<inline-formula id="ieqn-104"><alternatives><inline-graphic xlink:href="ieqn-104.png"/><tex-math id="tex-ieqn-104"><![CDATA[$\mathcal{Z}= {\mathcal h}_{1}(\mathrm{m}||\Lambda | \left|\mathrm{I}\mathtt{D}_{\mathrm{p}}\right| |\mathrm{I}\mathtt{D}_{\mathrm{c}}||\varsigma _{c})$]]></tex-math><mml:math id="mml-ieqn-104"><mml:mi mathvariant="script">Z</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mtext>&#x039B;</mml:mtext><mml:mo>|</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>&#x201D; of the message before sending the message. If the attacker changes the ciphertext of the message, then the consumer can perform the following operation for verification of the ciphertext. The consumer take <inline-formula id="ieqn-105"><alternatives><inline-graphic xlink:href="ieqn-105.png"/><tex-math id="tex-ieqn-105"><![CDATA[$\mathrm{m}^{\prime}= d_{\mathfrak{J}}(\mathfrak{w})$]]></tex-math><mml:math id="mml-ieqn-105"><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D50D;</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and compute the <inline-formula id="ieqn-106"><alternatives><inline-graphic xlink:href="ieqn-106.png"/><tex-math id="tex-ieqn-106"><![CDATA[$Z^{\prime}= {\mathcal h}_{1}(\mathrm{m}||\Lambda | \left| \mathrm{I}\mathtt{D}_{\mathrm{p}}\right| |\mathrm{I}\mathtt{D}_{\mathrm{c}}||\varsigma_{c})$]]></tex-math><mml:math id="mml-ieqn-106"><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mtext>&#x039B;</mml:mtext><mml:mo>|</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. After that, the consumer compares the <inline-formula id="ieqn-107"><alternatives><inline-graphic xlink:href="ieqn-107.png"/><tex-math id="tex-ieqn-107"><![CDATA[$\mathcal{Z}=Z^{\prime}$]]></tex-math><mml:math id="mml-ieqn-107"><mml:mi mathvariant="script">Z</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> if they are equal, then the integrity of the message holds; otherwise, the message has been altered.</p>
</sec>
<sec id="s5_4">
<label>5.4</label>
<title>Unforgeability</title>
<p>An IBS scheme is considered to achieve the security assets of unforgeability if there exists no intruder which can compromise the private key of the producer.</p>
<p><bold>Proof:</bold> In our scheme, if an intruder tries to generate a valid signature, he/she must need to calculate <inline-formula id="ieqn-108"><alternatives><inline-graphic xlink:href="ieqn-108.png"/><tex-math id="tex-ieqn-108"><![CDATA[$\nabla$]]></tex-math><mml:math id="mml-ieqn-108"><mml:mo>&#x2207;</mml:mo></mml:math></alternatives></inline-formula> from (<inline-formula id="ieqn-109"><alternatives><inline-graphic xlink:href="ieqn-109.png"/><tex-math id="tex-ieqn-109"><![CDATA[$\mathrm{I}\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}-\mathrm{I}\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho _{p}\cdot{\mathcal b}$]]></tex-math><mml:math id="mml-ieqn-109"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>-</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>) and to do so, the attacker needs to find <inline-formula id="ieqn-110"><alternatives><inline-graphic xlink:href="ieqn-110.png"/><tex-math id="tex-ieqn-110"><![CDATA[${\mathcal r}$]]></tex-math><mml:math id="mml-ieqn-110"><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula> from the <inline-formula id="ieqn-111"><alternatives><inline-graphic xlink:href="ieqn-111.png"/><tex-math id="tex-ieqn-111"><![CDATA[${\mathcal b}={\mathcal r}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-111"><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>. Further, the attacker also needs to find <inline-formula id="ieqn-112"><alternatives><inline-graphic xlink:href="ieqn-112.png"/><tex-math id="tex-ieqn-112"><![CDATA[$\varrho _{p}$]]></tex-math><mml:math id="mml-ieqn-112"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> from <inline-formula id="ieqn-113"><alternatives><inline-graphic xlink:href="ieqn-113.png"/><tex-math id="tex-ieqn-113"><![CDATA[$\varsigma _{p}=\varrho _{p}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-113"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>. So, it is computationally infeasible for the attacker to solve a two-time HDLP.</p>
</sec>
<sec id="s5_5">
<label>5.5</label>
<title>Non-Repudiation</title>
<p>An IBS scheme is supposed to succeed in the security service of non-repudiation if a sender cannot repudiate his signcrypted text.</p>
<p><bold>Proof:</bold> As the message is signed with the private key <inline-formula id="ieqn-114"><alternatives><inline-graphic xlink:href="ieqn-114.png"/><tex-math id="tex-ieqn-114"><![CDATA[$\varrho _{p}$]]></tex-math><mml:math id="mml-ieqn-114"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> of the producer. In our scheme, the consumer can authenticate the provider identity <inline-formula id="ieqn-115"><alternatives><inline-graphic xlink:href="ieqn-115.png"/><tex-math id="tex-ieqn-115"><![CDATA[$\mathrm{I}\mathtt{D}_{\mathrm{p}}$]]></tex-math><mml:math id="mml-ieqn-115"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>. So, the provider later can&#x2019;t deny from his signature.</p>
</sec>
</sec>
<sec id="s6">
<label>6</label>
<title>Cost Analysis</title>
<p>In this section, we will analyze the performance of our newly proposed scheme in relation to computation cost and communication cost. First, we compared our scheme with four related schemes of Yosef et al. [<xref ref-type="bibr" rid="ref-43">43</xref>], Nayak [<xref ref-type="bibr" rid="ref-40">40</xref>], Karate et al. [<xref ref-type="bibr" rid="ref-42">42</xref>] and Dharminder et al. [<xref ref-type="bibr" rid="ref-44">44</xref>], to show the computational and communicational efficiency. The computational efficiency is determined by the computational cost of the algorithm, and the communication efficiency is determined by the length of the ciphertext. The symbol (<italic>P</italic>) indicates the pairing operation, the symbol (<inline-formula id="ieqn-116"><alternatives><inline-graphic xlink:href="ieqn-116.png"/><tex-math id="tex-ieqn-116"><![CDATA[$\Sigma$]]></tex-math><mml:math id="mml-ieqn-116"><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula>) represents an exponential operation, and the symbol (<inline-formula id="ieqn-117"><alternatives><inline-graphic xlink:href="ieqn-117.png"/><tex-math id="tex-ieqn-117"><![CDATA[$\mathcal{P}\mathrm{BM }$]]></tex-math><mml:math id="mml-ieqn-117"><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>) indicates a pairing based point multiplication operation, the symbol (<inline-formula id="ieqn-118"><alternatives><inline-graphic xlink:href="ieqn-118.png"/><tex-math id="tex-ieqn-118"><![CDATA[$\boldsymbol{\mathsf{SBPM}}$]]></tex-math><mml:math id="mml-ieqn-118"><mml:mstyle mathvariant="bold-sans-serif"><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:mi>P</mml:mi><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>) represent scalar point multiplication of elliptic curve and the symbol (<inline-formula id="ieqn-119"><alternatives><inline-graphic xlink:href="ieqn-119.png"/><tex-math id="tex-ieqn-119"><![CDATA[$\mathrm{H }\mathrm{E }\boldsymbol{\mathsf{D}}\mathrm{M }$]]></tex-math><mml:math id="mml-ieqn-119"><mml:mstyle mathvariant="normal"><mml:mi>H</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>E</mml:mi></mml:mstyle><mml:mstyle mathvariant="bold-sans-serif"><mml:mi>D</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>) represent the hyperelliptic curve divisor multiplication. Here, we ignore the cost of other operations like hashing, addition, and subtraction because they take a much shorter time than the other operations mentioned above.</p>
<p>According to [<xref ref-type="bibr" rid="ref-27">27</xref>], the operation cost and their timing are listed in <xref ref-type="table" rid="table-3">Tab. 3</xref> below. The hardware and software specifications used for the simulation results are Intel Core i74510UCPU, Processor 2.0 and 8 GB RAM, Operating system of Windows 7, and C Library (MIRACL) [<xref ref-type="bibr" rid="ref-32">32</xref>]. Similarly, the <inline-formula id="ieqn-120"><alternatives><inline-graphic xlink:href="ieqn-120.png"/><tex-math id="tex-ieqn-120"><![CDATA[$\mathrm{HE}\boldsymbol{\mathsf{D}}\mathrm{M}$]]></tex-math><mml:math id="mml-ieqn-120"><mml:mstyle mathvariant="normal"><mml:mi>H</mml:mi><mml:mi>E</mml:mi></mml:mstyle><mml:mstyle mathvariant="bold-sans-serif"><mml:mi>D</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula> will consume 0.48 ms [<xref ref-type="bibr" rid="ref-15">15</xref>,<xref ref-type="bibr" rid="ref-47">47</xref>].</p>
<p>The symbols represent the length of the element. For example, <inline-formula id="ieqn-121"><alternatives><inline-graphic xlink:href="ieqn-121.png"/><tex-math id="tex-ieqn-121"><![CDATA[$|\mathrm{G}| = 1024$]]></tex-math><mml:math id="mml-ieqn-121"><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>G</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mn>1024</mml:mn></mml:math></alternatives></inline-formula> bits denote the length of the element in the group, <inline-formula id="ieqn-122"><alternatives><inline-graphic xlink:href="ieqn-122.png"/><tex-math id="tex-ieqn-122"><![CDATA[$|\mathrm{m}| = 512$]]></tex-math><mml:math id="mml-ieqn-122"><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mn>512</mml:mn></mml:math></alternatives></inline-formula> bits represent the length of the message space. Similarly, <inline-formula id="ieqn-123"><alternatives><inline-graphic xlink:href="ieqn-123.png"/><tex-math id="tex-ieqn-123"><![CDATA[$|{\mathcal q}| = 160$]]></tex-math><mml:math id="mml-ieqn-123"><mml:mo>|</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c6;</mml:mtext></mml:mstyle><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mn>160</mml:mn></mml:math></alternatives></inline-formula> bits and <inline-formula id="ieqn-124"><alternatives><inline-graphic xlink:href="ieqn-124.png"/><tex-math id="tex-ieqn-124"><![CDATA[$|{\mathcal n}| = 80$]]></tex-math><mml:math id="mml-ieqn-124"><mml:mo>|</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mn>80</mml:mn></mml:math></alternatives></inline-formula> bits represent the length of elements in the elliptic curve and hyperelliptic curve cryptosystem, as shown in <xref ref-type="table" rid="table-2">Tab. 2</xref>. Our scheme has a lower communication overhead cost as compared to Yosef et al. [<xref ref-type="bibr" rid="ref-43">43</xref>], Nayak [<xref ref-type="bibr" rid="ref-40">40</xref>], Karate et al. [<xref ref-type="bibr" rid="ref-42">42</xref>], and Dharminder et al. [<xref ref-type="bibr" rid="ref-44">44</xref>].</p>
<table-wrap id="table-2">
<label>Table 2</label>
<caption>
<title>Comparative analysis based on major operations</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Schemes</th>
<th>[<xref ref-type="bibr" rid="ref-43">43</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-40">40</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-42">42</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-44">44</xref>]</th>
<th>Proposed</th>
</tr>
</thead>
<tbody>
<tr>
<td>Signcryption</td>
<td><inline-formula id="ieqn-125"><alternatives><inline-graphic xlink:href="ieqn-125.png"/><tex-math id="tex-ieqn-125"><![CDATA[$ 3\mathcal{P}\mathrm{BM }+1P $]]></tex-math><mml:math id="mml-ieqn-125"><mml:mn>3</mml:mn><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi><mml:mi>M</mml:mi></mml:mstyle><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mi>P</mml:mi></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-126"><alternatives><inline-graphic xlink:href="ieqn-126.png"/><tex-math id="tex-ieqn-126"><![CDATA[$ 8 {\mathsf{SBPM}}$]]></tex-math><mml:math id="mml-ieqn-126"><mml:mn>8</mml:mn><mml:mstyle mathvariant="sans-serif"><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:mi>P</mml:mi><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-127"><alternatives><inline-graphic xlink:href="ieqn-127.png"/><tex-math id="tex-ieqn-127"><![CDATA[$ 4\mathcal{P}\mathrm{B }\mathrm{M }+4\Sigma $]]></tex-math><mml:math id="mml-ieqn-127"><mml:mn>4</mml:mn><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-128"><alternatives><inline-graphic xlink:href="ieqn-128.png"/><tex-math id="tex-ieqn-128"><![CDATA[$ 3\Sigma $]]></tex-math><mml:math id="mml-ieqn-128"><mml:mn>3</mml:mn><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-129"><alternatives><inline-graphic xlink:href="ieqn-129.png"/><tex-math id="tex-ieqn-129"><![CDATA[$ 6\mathrm{H }\mathrm{E }{\mathsf{D}}\mathrm{M }$]]></tex-math><mml:math id="mml-ieqn-129"><mml:mn>6</mml:mn><mml:mstyle mathvariant="normal"><mml:mi>H</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>E</mml:mi></mml:mstyle><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Unsigncryption</td>
<td>3<italic>P</italic></td>
<td><inline-formula id="ieqn-130"><alternatives><inline-graphic xlink:href="ieqn-130.png"/><tex-math id="tex-ieqn-130"><![CDATA[$ 5 {\mathsf{SBPM}}$]]></tex-math><mml:math id="mml-ieqn-130"><mml:mn>5</mml:mn><mml:mstyle mathvariant="sans-serif"><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:mi>P</mml:mi><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-131"><alternatives><inline-graphic xlink:href="ieqn-131.png"/><tex-math id="tex-ieqn-131"><![CDATA[$ 2P +2\mathcal{P}\mathrm{B }\mathrm{M }+2\Sigma $]]></tex-math><mml:math id="mml-ieqn-131"><mml:mn>2</mml:mn><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-132"><alternatives><inline-graphic xlink:href="ieqn-132.png"/><tex-math id="tex-ieqn-132"><![CDATA[$ 2P +1\Sigma $]]></tex-math><mml:math id="mml-ieqn-132"><mml:mn>2</mml:mn><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-133"><alternatives><inline-graphic xlink:href="ieqn-133.png"/><tex-math id="tex-ieqn-133"><![CDATA[$ 5\mathrm{H }\mathrm{E }{\mathsf{D}}\mathrm{M }$]]></tex-math><mml:math id="mml-ieqn-133"><mml:mn>5</mml:mn><mml:mstyle mathvariant="normal"><mml:mi>H</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>E</mml:mi></mml:mstyle><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Total</td>
<td><inline-formula id="ieqn-134"><alternatives><inline-graphic xlink:href="ieqn-134.png"/><tex-math id="tex-ieqn-134"><![CDATA[$ 3\mathcal{P}\mathrm{B }\mathrm{M }+4P $]]></tex-math><mml:math id="mml-ieqn-134"><mml:mn>3</mml:mn><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mi>P</mml:mi></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-135"><alternatives><inline-graphic xlink:href="ieqn-135.png"/><tex-math id="tex-ieqn-135"><![CDATA[$ 13 {\mathsf{SBPM}}$]]></tex-math><mml:math id="mml-ieqn-135"><mml:mn>13</mml:mn><mml:mstyle mathvariant="sans-serif"><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:mi>P</mml:mi><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-136"><alternatives><inline-graphic xlink:href="ieqn-136.png"/><tex-math id="tex-ieqn-136"><![CDATA[$ 2P +6\mathcal{P}\mathrm{B }\mathrm{M }+6\Sigma $]]></tex-math><mml:math id="mml-ieqn-136"><mml:mn>2</mml:mn><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mn>6</mml:mn><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle><mml:mo>+</mml:mo><mml:mn>6</mml:mn><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-137"><alternatives><inline-graphic xlink:href="ieqn-137.png"/><tex-math id="tex-ieqn-137"><![CDATA[$ 2P +4\Sigma $]]></tex-math><mml:math id="mml-ieqn-137"><mml:mn>2</mml:mn><mml:mi>P</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-138"><alternatives><inline-graphic xlink:href="ieqn-138.png"/><tex-math id="tex-ieqn-138"><![CDATA[$ 11\mathrm{H }\mathrm{E }{\mathsf{D}}\mathrm{M }$]]></tex-math><mml:math id="mml-ieqn-138"><mml:mn>11</mml:mn><mml:mstyle mathvariant="normal"><mml:mi>H</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>E</mml:mi></mml:mstyle><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></td>
</tr>
</tbody>
</table>
</table-wrap>
<table-wrap id="table-3">
<label>Table 3</label>
<caption>
<title>Operations cost in milliseconds</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Operation</th>
<th><italic>P</italic></th>
<th><inline-formula id="ieqn-139"><alternatives><inline-graphic xlink:href="ieqn-139.png"/><tex-math id="tex-ieqn-139"><![CDATA[${\Sigma }$]]></tex-math><mml:math id="mml-ieqn-139"><mml:mtext>&#x03A3;</mml:mtext></mml:math></alternatives></inline-formula></th>
<th><inline-formula id="ieqn-140"><alternatives><inline-graphic xlink:href="ieqn-140.png"/><tex-math id="tex-ieqn-140"><![CDATA[${\mathcal{P}}\mathrm{B}\mathrm{M }$]]></tex-math><mml:math id="mml-ieqn-140"><mml:mi mathvariant="script">P</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>B</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></th>
<th><inline-formula id="ieqn-141"><alternatives><inline-graphic xlink:href="ieqn-141.png"/><tex-math id="tex-ieqn-141"><![CDATA[$ {\mathsf{SBPM}}$]]></tex-math><mml:math id="mml-ieqn-141"><mml:mstyle mathvariant="sans-serif"><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:mi>P</mml:mi><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></th>
<th><inline-formula id="ieqn-142"><alternatives><inline-graphic xlink:href="ieqn-142.png"/><tex-math id="tex-ieqn-142"><![CDATA[$ \mathrm{H E }{\mathsf{D}}\mathrm{M }$]]></tex-math><mml:math id="mml-ieqn-142"><mml:mstyle mathvariant="normal"><mml:mi>H</mml:mi><mml:mi>E</mml:mi></mml:mstyle><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle><mml:mstyle mathvariant="normal"><mml:mi>M</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula></th>
</tr>
</thead>
<tbody>
<tr>
<td>Cost in millisecond</td>
<td>14.31</td>
<td>1.25</td>
<td>4.32</td>
<td>0.97</td>
<td>0.48</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>In accordance, <xref ref-type="table" rid="table-4">Tabs. 4</xref> and <xref ref-type="table" rid="table-6">6</xref> show a comparative illustration of our proposed work with Yosef et al. [<xref ref-type="bibr" rid="ref-43">43</xref>], Nayak [<xref ref-type="bibr" rid="ref-40">40</xref>], Karate et al. [<xref ref-type="bibr" rid="ref-42">42</xref>], and Dharminder et al. [<xref ref-type="bibr" rid="ref-44">44</xref>], in term of computation and communication overheads. According to our comparative analysis, our scheme shows efficiency in terms of computation and communication overheads, as shown in <xref ref-type="fig" rid="fig-3">Figs. 3</xref> and <xref ref-type="fig" rid="fig-4">4</xref>. Furthermore, an exact computation and communication cost reduction are shown in <xref ref-type="table" rid="table-5">Tabs. 5</xref> and <xref ref-type="table" rid="table-7">7</xref>.</p>
<table-wrap id="table-4">
<label>Table 4</label>
<caption>
<title>Comparative analysis based in milliseconds</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Schemes</th>
<th>[<xref ref-type="bibr" rid="ref-43">43</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-40">40</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-42">42</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-44">44</xref>]</th>
<th>Proposed</th>
</tr>
</thead>
<tbody>
<tr>
<td>Signcryption</td>
<td>27.86</td>
<td>7.76</td>
<td>22.28</td>
<td>3.75</td>
<td>2.88</td>
</tr>
<tr>
<td>Unsigncryption</td>
<td>44.7</td>
<td>4.85</td>
<td>40.94</td>
<td>29.87</td>
<td>2.4</td>
</tr>
<tr>
<td>Total</td>
<td>72.56</td>
<td>12.61</td>
<td>63.22</td>
<td>33.64</td>
<td>5.28</td>
</tr>
</tbody>
</table>
</table-wrap>
<table-wrap id="table-5">
<label>Table 5</label>
<caption>
<title>Percentage computation cost reduction</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Schemes</th>
<th>[<xref ref-type="bibr" rid="ref-43">43</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-40">40</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-42">42</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-44">44</xref>]</th>
</tr>
</thead>
<tbody>
<tr>
<td>Total computation cost of <inline-formula id="ieqn-143"><alternatives><inline-graphic xlink:href="ieqn-143.png"/><tex-math id="tex-ieqn-143"><![CDATA[$(\boldsymbol{\alpha })$]]></tex-math><mml:math id="mml-ieqn-143"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03B1;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></td>
<td>72.56</td>
<td>12.61</td>
<td>63.22</td>
<td>33.64</td>
</tr>
<tr>
<td>Total computation cost of the proposed <inline-formula id="ieqn-144"><alternatives><inline-graphic xlink:href="ieqn-144.png"/><tex-math id="tex-ieqn-144"><![CDATA[$(\boldsymbol{\beta })$]]></tex-math><mml:math id="mml-ieqn-144"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03B2;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></td>
<td>5.28</td>
<td>5.28</td>
<td>5.28</td>
<td>5.28</td>
</tr>
<tr>
<td>Cost reduction in % <inline-formula id="ieqn-145"><alternatives><inline-graphic xlink:href="ieqn-145.png"/><tex-math id="tex-ieqn-145"><![CDATA[$(\boldsymbol{\gamma })$]]></tex-math><mml:math id="mml-ieqn-145"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03B3;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></td>
<td>92.72</td>
<td>58.12</td>
<td>91.64</td>
<td>84.30</td>
</tr>
</tbody>
</table>
<table-wrap-foot><p>Improvement in percentage (<inline-formula id="ieqn-146"><alternatives><inline-graphic xlink:href="ieqn-146.png"/><tex-math id="tex-ieqn-146"><![CDATA[$\gamma$]]></tex-math><mml:math id="mml-ieqn-146"><mml:mi>&#x03B3;</mml:mi></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-147"><alternatives><inline-graphic xlink:href="ieqn-147.png"/><tex-math id="tex-ieqn-147"><![CDATA[$ \left(\displaystyle\frac{\alpha -\beta }{\beta }\right)\ast100$]]></tex-math><mml:math id="mml-ieqn-147"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>&#x03B1;</mml:mi><mml:mo>-</mml:mo><mml:mi>&#x03B2;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x03B2;</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>*</mml:mo><mml:mn>100</mml:mn></mml:math></alternatives></inline-formula></p></table-wrap-foot></table-wrap>
<table-wrap id="table-6">
<label>Table 6</label>
<caption>
<title>Comparative analysis in ciphertext in bits</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Schemes</th>
<th>[<xref ref-type="bibr" rid="ref-43">43</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-40">40</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-42">42</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-44">44</xref>]</th>
<th>Proposed</th>
</tr>
</thead>
<tbody>
<tr>
<td>Communication cost</td>
<td><inline-formula id="ieqn-148"><alternatives><inline-graphic xlink:href="ieqn-148.png"/><tex-math id="tex-ieqn-148"><![CDATA[$2|\mathrm{G}| + | \mathrm{m}|$]]></tex-math><mml:math id="mml-ieqn-148"><mml:mn>2</mml:mn><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>G</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>+</mml:mo><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-149"><alternatives><inline-graphic xlink:href="ieqn-149.png"/><tex-math id="tex-ieqn-149"><![CDATA[$3| {\mathcal q}| + | m |$]]></tex-math><mml:math id="mml-ieqn-149"><mml:mn>3</mml:mn><mml:mo>|</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c6;</mml:mtext></mml:mstyle><mml:mo>|</mml:mo><mml:mo>+</mml:mo><mml:mo>|</mml:mo><mml:mi>m</mml:mi><mml:mo>|</mml:mo></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-150"><alternatives><inline-graphic xlink:href="ieqn-150.png"/><tex-math id="tex-ieqn-150"><![CDATA[$4| \mathrm{G}| + | m | $]]></tex-math><mml:math id="mml-ieqn-150"><mml:mn>4</mml:mn><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>G</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>+</mml:mo><mml:mo>|</mml:mo><mml:mi>m</mml:mi><mml:mo>|</mml:mo></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-151"><alternatives><inline-graphic xlink:href="ieqn-151.png"/><tex-math id="tex-ieqn-151"><![CDATA[$3| \mathrm{G}| + | m |$]]></tex-math><mml:math id="mml-ieqn-151"><mml:mn>3</mml:mn><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>G</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>+</mml:mo><mml:mo>|</mml:mo><mml:mi>m</mml:mi><mml:mo>|</mml:mo></mml:math></alternatives></inline-formula></td>
<td><inline-formula id="ieqn-152"><alternatives><inline-graphic xlink:href="ieqn-152.png"/><tex-math id="tex-ieqn-152"><![CDATA[$ 3 \left| {\mathcal n}\right| +|\mathrm{m}| $]]></tex-math><mml:math id="mml-ieqn-152"><mml:mn>3</mml:mn><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo></mml:math></alternatives></inline-formula></td>
</tr>
<tr>
<td>Ciphertext size</td>
<td>2560</td>
<td>992</td>
<td>4608</td>
<td>3584</td>
<td>752</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="fig-3">
<label>Figure 3</label>
<caption>
<title>Computation cost analysis in terms of milliseconds</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-3.png"/>
</fig>
<fig id="fig-4">
<label>Figure 4</label>
<caption>
<title>Communication cost analysis in terms of bits</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-4.png"/>
</fig>
<table-wrap id="table-7">
<label>Table 7</label>
<caption>
<title>Percentage communication cost reduction</title>
</caption>
<table>
<colgroup>
<col/>
<col/>
<col/>
<col/>
<col/>
</colgroup>
<thead>
<tr>
<th>Schemes</th>
<th>[<xref ref-type="bibr" rid="ref-43">43</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-40">40</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-42">42</xref>]</th>
<th>[<xref ref-type="bibr" rid="ref-44">44</xref>]</th>
</tr>
</thead>
<tbody>
<tr>
<td>Total communication cost <inline-formula id="ieqn-153"><alternatives><inline-graphic xlink:href="ieqn-153.png"/><tex-math id="tex-ieqn-153"><![CDATA[$(\mathbf{\alpha })$]]></tex-math><mml:math id="mml-ieqn-153"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold"><mml:mi>&#x03B1;</mml:mi></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></td>
<td>2560</td>
<td>992</td>
<td>4608</td>
<td>3584</td>
</tr>
<tr>
<td>Total communication cost <inline-formula id="ieqn-154"><alternatives><inline-graphic xlink:href="ieqn-154.png"/><tex-math id="tex-ieqn-154"><![CDATA[$(\mathbf{\beta })$]]></tex-math><mml:math id="mml-ieqn-154"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold"><mml:mi>&#x03B2;</mml:mi></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></td>
<td>752</td>
<td>752</td>
<td>752</td>
<td>752</td>
</tr>
<tr>
<td>Cost reduction in % <inline-formula id="ieqn-155"><alternatives><inline-graphic xlink:href="ieqn-155.png"/><tex-math id="tex-ieqn-155"><![CDATA[$(\mathbf{\gamma })$]]></tex-math><mml:math id="mml-ieqn-155"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="bold"><mml:mi>&#x03B3;</mml:mi></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula></td>
<td>70.62</td>
<td>24.19</td>
<td>83.68</td>
<td>79.01</td>
</tr>
</tbody>
</table>
<table-wrap-foot><p>Improvement in percentage (<inline-formula id="ieqn-156"><alternatives><inline-graphic xlink:href="ieqn-156.png"/><tex-math id="tex-ieqn-156"><![CDATA[$\mathbf{\gamma }$]]></tex-math><mml:math id="mml-ieqn-156"><mml:mstyle mathvariant="bold"><mml:mi>&#x03B3;</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-157"><alternatives><inline-graphic xlink:href="ieqn-157.png"/><tex-math id="tex-ieqn-157"><![CDATA[$ \left(\displaystyle\frac{\alpha -\mathbf{\beta }}{\mathbf{\beta }}\right)\ast100$]]></tex-math><mml:math id="mml-ieqn-157"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>&#x03B1;</mml:mi><mml:mo>-</mml:mo><mml:mstyle mathvariant="bold"><mml:mi>&#x03B2;</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mi>&#x03B2;</mml:mi></mml:mstyle></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>*</mml:mo><mml:mn>100</mml:mn></mml:math></alternatives></inline-formula></p></table-wrap-foot></table-wrap>
</sec>
<sec id="s7">
<label>7</label>
<title>Deployment of Proposed Scheme</title>
<p>In this section, we deployed our scheme on IoTEH in the NDN network. We consider several IoT devices that can sense and share healthcare-related information among the hospital, patient, doctor, and IoT devices through the NDN router. The information can be shared from the city to the city as well as from country to country. Moreover, the devices in healthcare are connected based on the NDN policy.</p>
<p>Here, every NDN router maintains three kinds of data structures, such as Pending Interest Table (PIT), Forwarding Information Base (FIB), and Content Store (CS) [<xref ref-type="bibr" rid="ref-33">33</xref>]. The data attainment process begins by sending an Interest with a particular name from the client&#x2019;s side. The routers rely on FIBs for transferring the interest to a potential provider and generate a PIT entry list on each router to establish an opposite path. Based on the opposite path, the provider of any interest returns the data to the client with the target data. The CS stores the targeted data are passing through it for future use [<xref ref-type="bibr" rid="ref-48">48</xref>].</p>
<p>Assume consumers require healthcare-related information from the producer, and the communication includes the participants such as client, private key generator (PKG), NDN routers, and provider. The consumers are those who need the information. The providers are those who distribute the information to the consumers, and the PKG is a trusted authority that is responsible for establishing secure communication between the consumer and producer. Communication among consumers and producers is discussed below.</p>
<p><bold><italic>Registration Phase:</italic></bold> In this stage, the consumer and producer registered themselves with the PKG by providing their Identities (<inline-formula id="ieqn-158"><alternatives><inline-graphic xlink:href="ieqn-158.png"/><tex-math id="tex-ieqn-158"><![CDATA[$\mathrm{I }\mathtt{D}_{c}, \mathrm{I }\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-158"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) to PKG. The PKG gets their Identities (<inline-formula id="ieqn-159"><alternatives><inline-graphic xlink:href="ieqn-159.png"/><tex-math id="tex-ieqn-159"><![CDATA[$\mathcal{Id}_{\mathrm{u}}$]]></tex-math><mml:math id="mml-ieqn-159"><mml:mi mathvariant="script">I</mml:mi><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>u</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>), and generate consumer private key <inline-formula id="ieqn-160"><alternatives><inline-graphic xlink:href="ieqn-160.png"/><tex-math id="tex-ieqn-160"><![CDATA[$(\varrho _{c})\colon\varrho _{c}= \nu\cdot{\mathcal h}_{0}(\mathrm{I}\mathtt{D}_{c})\;mod\;{\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-160"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x03BD;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>, consumer public key (<inline-formula id="ieqn-161"><alternatives><inline-graphic xlink:href="ieqn-161.png"/><tex-math id="tex-ieqn-161"><![CDATA[$\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-161"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-162"><alternatives><inline-graphic xlink:href="ieqn-162.png"/><tex-math id="tex-ieqn-162"><![CDATA[$\varsigma _{c}= \varrho _{c}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-162"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>, private producer key (<inline-formula id="ieqn-163"><alternatives><inline-graphic xlink:href="ieqn-163.png"/><tex-math id="tex-ieqn-163"><![CDATA[$\varrho _{p}$]]></tex-math><mml:math id="mml-ieqn-163"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-164"><alternatives><inline-graphic xlink:href="ieqn-164.png"/><tex-math id="tex-ieqn-164"><![CDATA[$\varrho _{p}= \nu\cdot{\mathcal h}_{0}(\mathrm{I}\mathtt{D}_{p})\;mod\;{\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-164"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x03BD;</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula> and producer public key (<inline-formula id="ieqn-165"><alternatives><inline-graphic xlink:href="ieqn-165.png"/><tex-math id="tex-ieqn-165"><![CDATA[$\varsigma _{p}$]]></tex-math><mml:math id="mml-ieqn-165"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>): <inline-formula id="ieqn-166"><alternatives><inline-graphic xlink:href="ieqn-166.png"/><tex-math id="tex-ieqn-166"><![CDATA[$\varsigma _{p}= \varrho _{p}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-166"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula> by using their identity (<inline-formula id="ieqn-167"><alternatives><inline-graphic xlink:href="ieqn-167.png"/><tex-math id="tex-ieqn-167"><![CDATA[$\mathrm{I }\mathtt{D}_{\mathrm{c}}, \mathrm{I }\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-167"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>) Then the PKG sends the <inline-formula id="ieqn-168"><alternatives><inline-graphic xlink:href="ieqn-168.png"/><tex-math id="tex-ieqn-168"><![CDATA[$(\varrho _{c}, \varsigma _{c}, \varrho _{p}, \varsigma _{p})$]]></tex-math><mml:math id="mml-ieqn-168"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> to consumer and producer, as shown in <xref ref-type="fig" rid="fig-5">Fig. 5</xref>.</p>
<fig id="fig-5">
<label>Figure 5</label>
<caption>
<title>Registration of consumer and producer with PKG</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-5.png"/>
</fig>
<p><bold><italic>Signcryption Phase:</italic></bold> When the consumer shows interest in information <italic>m</italic>, upon receiving interest, the producer will signcrypt <italic>m</italic>. For this purpose, the producer takes information (<italic>m</italic>), fresh nonce (<inline-formula id="ieqn-169"><alternatives><inline-graphic xlink:href="ieqn-169.png"/><tex-math id="tex-ieqn-169"><![CDATA[$\lambda$]]></tex-math><mml:math id="mml-ieqn-169"><mml:mtext>&#x039B;</mml:mtext></mml:math></alternatives></inline-formula>), <inline-formula id="ieqn-170"><alternatives><inline-graphic xlink:href="ieqn-170.png"/><tex-math id="tex-ieqn-170"><![CDATA[$\mathrm{I }\mathtt{D}_{c}, \mathrm{I}\mathtt{D}_{p}$]]></tex-math><mml:math id="mml-ieqn-170"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> <inline-formula id="ieqn-171"><alternatives><inline-graphic xlink:href="ieqn-171.png"/><tex-math id="tex-ieqn-171"><![CDATA[$\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-171"><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-172"><alternatives><inline-graphic xlink:href="ieqn-172.png"/><tex-math id="tex-ieqn-172"><![CDATA[$\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-172"><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-173"><alternatives><inline-graphic xlink:href="ieqn-173.png"/><tex-math id="tex-ieqn-173"><![CDATA[$\varrho _{p}$]]></tex-math><mml:math id="mml-ieqn-173"><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> as input. First, take a random number <inline-formula id="ieqn-174"><alternatives><inline-graphic xlink:href="ieqn-174.png"/><tex-math id="tex-ieqn-174"><![CDATA[${\mathcal r}\;\varepsilon\; (0, 1, 2, 3, 4\ldots (n-1))$]]></tex-math><mml:math id="mml-ieqn-174"><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>&#x03B5;</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>&#x2026;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, compute a private number (<inline-formula id="ieqn-175"><alternatives><inline-graphic xlink:href="ieqn-175.png"/><tex-math id="tex-ieqn-175"><![CDATA[${\mathcal b}$]]></tex-math><mml:math id="mml-ieqn-175"><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-176"><alternatives><inline-graphic xlink:href="ieqn-176.png"/><tex-math id="tex-ieqn-176"><![CDATA[${\mathcal b}={\mathcal r}\cdot\mathsf{D}$]]></tex-math><mml:math id="mml-ieqn-176"><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:mstyle mathvariant="sans-serif"><mml:mi>D</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>. After that, the producer computes an encryption key (<inline-formula id="ieqn-177"><alternatives><inline-graphic xlink:href="ieqn-177.png"/><tex-math id="tex-ieqn-177"><![CDATA[$\mathfrak{J}$]]></tex-math><mml:math id="mml-ieqn-177"><mml:mi>&#x1D50D;</mml:mi></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-178"><alternatives><inline-graphic xlink:href="ieqn-178.png"/><tex-math id="tex-ieqn-178"><![CDATA[$\mathfrak{J}= \mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot{\mathcal r}\cdot\varsigma _{c}$]]></tex-math><mml:math id="mml-ieqn-178"><mml:mi>&#x1D50D;</mml:mi><mml:mo>=</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>. Then perform the encryption process on information (<inline-formula id="ieqn-179"><alternatives><inline-graphic xlink:href="ieqn-179.png"/><tex-math id="tex-ieqn-179"><![CDATA[$\mathfrak{w}$]]></tex-math><mml:math id="mml-ieqn-179"><mml:mi>&#x1D534;</mml:mi></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-180"><alternatives><inline-graphic xlink:href="ieqn-180.png"/><tex-math id="tex-ieqn-180"><![CDATA[$\mathfrak{w}= \mathfrak{w}_{\mathfrak{J}}(\mathrm{m})$]]></tex-math><mml:math id="mml-ieqn-180"><mml:mi>&#x1D534;</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x1D534;</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D50D;</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. The producer then computes information digest (<inline-formula id="ieqn-181"><alternatives><inline-graphic xlink:href="ieqn-181.png"/><tex-math id="tex-ieqn-181"><![CDATA[$\mathcal{Z}$]]></tex-math><mml:math id="mml-ieqn-181"><mml:mi mathvariant="script">Z</mml:mi></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-182"><alternatives><inline-graphic xlink:href="ieqn-182.png"/><tex-math id="tex-ieqn-182"><![CDATA[$\mathcal{Z}= {\mathcal h}_{1}(\mathrm{m}||\lambda | \left| \mathrm{I }\mathtt{D}_{\mathrm{p}}\right| |\mathrm{I }\mathtt{D}_{\mathrm{c}}||\varsigma _{c})$]]></tex-math><mml:math id="mml-ieqn-182"><mml:mi mathvariant="script">Z</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mtext>&#x039B;</mml:mtext><mml:mo>|</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. After that, the producer generates signature (<inline-formula id="ieqn-183"><alternatives><inline-graphic xlink:href="ieqn-183.png"/><tex-math id="tex-ieqn-183"><![CDATA[$\nabla$]]></tex-math><mml:math id="mml-ieqn-183"><mml:mo>&#x2207;</mml:mo></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-184"><alternatives><inline-graphic xlink:href="ieqn-184.png"/><tex-math id="tex-ieqn-184"><![CDATA[$\nabla =\mathrm{I }\mathtt{D}_{\mathrm{c}}\cdot {\mathcal r}-\mathrm{I }\mathtt{D}_{\mathrm{p}}\cdot\mathcal{Z}\cdot\varrho _{p}\cdot{\mathcal b }\;mod\;{\mathcal n}$]]></tex-math><mml:math id="mml-ieqn-184"><mml:mo>&#x2207;</mml:mo><mml:mo>=</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1d4c7;</mml:mtext></mml:mstyle><mml:mo>-</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mspace width="2.77626pt" class="tmspace"/><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mspace width="2.77626pt" class="tmspace"/><mml:mstyle class="text"><mml:mtext>&#x1D4C3;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>. Finally, producers generate the signcrypted information (<inline-formula id="ieqn-185"><alternatives><inline-graphic xlink:href="ieqn-185.png"/><tex-math id="tex-ieqn-185"><![CDATA[${\mathcal s}$]]></tex-math><mml:math id="mml-ieqn-185"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-186"><alternatives><inline-graphic xlink:href="ieqn-186.png"/><tex-math id="tex-ieqn-186"><![CDATA[${\mathcal s}= ( \mathfrak{w}, \mathcal{Z}, \nabla, {\mathcal b})$]]></tex-math><mml:math id="mml-ieqn-186"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>,</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and send it to the consumer, as shown in <xref ref-type="fig" rid="fig-6">Fig. 6</xref>.</p>
<fig id="fig-6">
<label>Figure 6</label>
<caption>
<title>Communication between consumer and producer</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-6.png"/>
</fig>
<p><bold><italic>Unsigncryption Phase:</italic></bold> After receiving the signcrypted information <inline-formula id="ieqn-187"><alternatives><inline-graphic xlink:href="ieqn-187.png"/><tex-math id="tex-ieqn-187"><![CDATA[${\mathcal s}= ( \mathfrak{w}, \mathcal{Z}, \nabla , {\mathcal b})$]]></tex-math><mml:math id="mml-ieqn-187"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>,</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, the consumer unsigncrypt the <inline-formula id="ieqn-188"><alternatives><inline-graphic xlink:href="ieqn-188.png"/><tex-math id="tex-ieqn-188"><![CDATA[${\mathcal s}$]]></tex-math><mml:math id="mml-ieqn-188"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle></mml:math></alternatives></inline-formula>. For unsigncryption, the consumer takes <inline-formula id="ieqn-189"><alternatives><inline-graphic xlink:href="ieqn-189.png"/><tex-math id="tex-ieqn-189"><![CDATA[${\mathcal s}= (\mathfrak{w}, \mathcal{Z}, \nabla, {\mathcal b})$]]></tex-math><mml:math id="mml-ieqn-189"><mml:mstyle class="text"><mml:mtext>&#x1d4c8;</mml:mtext></mml:mstyle><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>,</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>,</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, <inline-formula id="ieqn-190"><alternatives><inline-graphic xlink:href="ieqn-190.png"/><tex-math id="tex-ieqn-190"><![CDATA[$\mathrm{I }\mathtt{D}_{\mathrm{c}}, \mathrm{I }\mathtt{D}_{\mathrm{p}}\varrho _{{\mathcal c}},  \text{a}\text{n}\text{d}\;\varsigma _{p}$]]></tex-math><mml:math id="mml-ieqn-190"><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4B8;</mml:mtext></mml:mstyle></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mstyle class="text"><mml:mtext>a</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>n</mml:mtext></mml:mstyle><mml:mstyle class="text"><mml:mtext>d</mml:mtext></mml:mstyle><mml:mspace width="2.77626pt" class="tmspace"/><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula> as input. First, the consumer computes the decryption key (<inline-formula id="ieqn-191"><alternatives><inline-graphic xlink:href="ieqn-191.png"/><tex-math id="tex-ieqn-191"><![CDATA[$\mathfrak{J}$]]></tex-math><mml:math id="mml-ieqn-191"><mml:mi>&#x1D50D;</mml:mi></mml:math></alternatives></inline-formula>) from <inline-formula id="ieqn-192"><alternatives><inline-graphic xlink:href="ieqn-192.png"/><tex-math id="tex-ieqn-192"><![CDATA[${\mathfrak{J}=\nabla \cdot\varsigma _{c}+\mathrm{I }\mathtt{D}_{p}\cdot\mathcal{Z}\cdot{\mathcal b}\cdot\varsigma _{p}\cdot\varrho _{{\mathcal c}}}$]]></tex-math><mml:math id="mml-ieqn-192"><mml:mi>&#x1D50D;</mml:mi><mml:mo>=</mml:mo><mml:mo>&#x2207;</mml:mo><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:mi mathvariant="script">Z</mml:mi><mml:mo>&#x22C5;</mml:mo><mml:mstyle class="text"><mml:mtext>&#x1D4B7;</mml:mtext></mml:mstyle><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>&#x22C5;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03F1;</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4B8;</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>. Then decrypt the information <inline-formula id="ieqn-193"><alternatives><inline-graphic xlink:href="ieqn-193.png"/><tex-math id="tex-ieqn-193"><![CDATA[$(\mathrm{m}^{\prime})$]]></tex-math><mml:math id="mml-ieqn-193"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> where <inline-formula id="ieqn-194"><alternatives><inline-graphic xlink:href="ieqn-194.png"/><tex-math id="tex-ieqn-194"><![CDATA[$\mathrm{m}^{\prime}= d_{\mathfrak{J}}(\mathfrak{w})$]]></tex-math><mml:math id="mml-ieqn-194"><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x1D50D;</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x1D534;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Finally, compute information digest (<inline-formula id="ieqn-195"><alternatives><inline-graphic xlink:href="ieqn-195.png"/><tex-math id="tex-ieqn-195"><![CDATA[$\mathcal{Z}^{\prime}$]]></tex-math><mml:math id="mml-ieqn-195"><mml:msup><mml:mrow><mml:mi mathvariant="script">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>) where <inline-formula id="ieqn-196"><alternatives><inline-graphic xlink:href="ieqn-196.png"/><tex-math id="tex-ieqn-196"><![CDATA[$\mathcal{Z}^{\prime}= {\mathcal h}_{1}(\mathrm{m}^{\prime}||\lambda | \left| \mathrm{I }\mathtt{D}_{\mathrm{p}}\right| |\mathrm{I }\mathtt{D}_{\mathrm{c}}||\varsigma _{c})$]]></tex-math><mml:math id="mml-ieqn-196"><mml:msup><mml:mrow><mml:mi mathvariant="script">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle class="text"><mml:mtext>&#x1D4BD;</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>m</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:mtext>&#x039B;</mml:mtext><mml:mo>|</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>p</mml:mi></mml:mstyle></mml:mrow></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>I</mml:mi></mml:mstyle><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>D</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>c</mml:mi></mml:mstyle></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>&#x03C2;</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. If <inline-formula id="ieqn-197"><alternatives><inline-graphic xlink:href="ieqn-197.png"/><tex-math id="tex-ieqn-197"><![CDATA[$\mathcal{Z}=\mathcal{Z}^{\prime}$]]></tex-math><mml:math id="mml-ieqn-197"><mml:mi mathvariant="script">Z</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="script">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x2032;</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> valid otherwise invalid, as shown in <xref ref-type="fig" rid="fig-6">Fig. 6</xref>.</p>
</sec>
<sec id="s8">
<label>8</label>
<title>Conclusion</title>
<p>In this paper, we proposed a secure NDN framework for the Internet of Things Enabled Healthcare (IoTEH) using a lightweight Identity-Based Signcryption (IBS) cryptography to secure the information of IoT enabled healthcare in NDN infrastructure. To minimize the cost consumption, we used a Hyperelliptic Curve Cryptosystem (HCC) which provides the corresponding level of security as compared to bilinear pairing and Elliptic Curve Cryptosystem (ECC). To show the efficiency of our newly proposed scheme we compared the proposed scheme with recently presented identity-based signcryption schemes in terms of computation and communication overheads. The final results show the superiority of our scheme in terms of computation and communication costs. For further security, we simulate the security of our scheme using Automated Validation of Internet Security Protocols and Applications (AVISPA). Finally, we deployed our proposed scheme on NDN enabled healthcare.</p>
</sec>
</body>
<back>
<ack><p>The authors are grateful to the Deanship of Scientific Research, King Saud University for funding through Vice Deanship of Scientific Research Chairs.</p></ack>
<fn-group><fn fn-type="other"><p><bold>Funding Statement:</bold> The authors received no financial support for the research, authorship, and/or publication of this article.</p></fn>
<fn fn-type="conflict"><p><bold>Conflicts of Interest:</bold> The authors declare that they have no conflicts of interest to report regarding the present study.</p></fn></fn-group>
<ref-list content-type="authoryear">
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</ref-list>
<app-group>
<app>
<title>Appendix A.</title> 
<sec>
<title>Simulation and Validation</title>
<p>There are several formal security verification tools, such as ProVerif [<xref ref-type="bibr" rid="ref-49">49</xref>], AVISPA (Internet Security Protocol and Automatic Verification of Application) [<xref ref-type="bibr" rid="ref-50">50</xref>], and Scyther [<xref ref-type="bibr" rid="ref-51">51</xref>]. In our proposed work, we use AVISPA as it is popular in the security community. Specifically, we encode our proposed scheme using the &#x201C;role-oriented language&#x201D; of High-Level Protocol Specification Language (HLPSL) [<xref ref-type="bibr" rid="ref-50">50</xref>], which has a variety of basic roles (the roles for consumer, the role for the producer). Defined the proposed scheme two mandatory roles (session, goal and environment). The HLPSL2IF Translator helps to convert the HLPSL code to &#x201C;Intermediate Format (IF),&#x201D; and the IF is then sent to one of AVISPA&#x2019;s four available backends: &#x201C;On-the-Fly Model-Checker (OFMC),&#x201D; &#x201C;Automatic Approval for Analysis of Tree Automata, Security Protocol (TA4SP),&#x201D; &#x201C;Structure Logic Based Attack Search (CL-AtSe)&#x201D; and &#x201C;SAT-based Model-Checker (SATMC)&#x201D;  [<xref ref-type="bibr" rid="ref-52">52</xref>]. The basic top-down illustration of AVISPA is shown in <xref ref-type="fig" rid="fig-7">Fig. 7</xref>.</p>
<fig id="fig-7">
<label>Figure 7</label> 
<caption>
<title>Top-down illustration of AVISPA </title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-7.png"/>
</fig>
<sec>
<title>A.1 Simulation and Validation Results</title>
<p>Here we validate our scheme according to the backend tools of AVISPA tool such as ATSE and OFMC.</p>
<p><italic>A.1.1 Results of OFMC Protocol</italic></p>
<p>In <xref ref-type="fig" rid="fig-8">Fig. 8</xref> below, we provide the simulation result of the proposed scheme under the back-end checkers of AVISPA called OFMC. The simulation result under OFMC shows that our scheme is perfectly safe.</p>
<fig id="fig-8">
<label>Figure 8</label>
<caption>
<title>OFMC protocol result of our scheme</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-8.png"/>
</fig>
</sec>
<sec>
<title>A.1.2 Results of ATSE Protocol</title>
<p>In <xref ref-type="fig" rid="fig-9">Fig. 9</xref>, we also provide the simulation result of our scheme under the function of the another AVISPA back end checker called CL-AtSe. The result shows that the given scheme is safe under CL-AtSe. The CL-AtSe supports a type-imperfection discovery that is responsible for the associativity message concatenation.</p>
<fig id="fig-9">
<label>Figure 9</label>
<caption>
<title>ATSE protocol result of our scheme</title>
</caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="fig-9.png"/>
</fig>
</sec>
</sec>
</app>
</app-group>
</back>
</article>