The proposed chaotic map (Sine Ikeda modulated map) has been constructed by adding sine and Ikeda maps. The chaotic region of the map is stretched over a wide range. Thus, the keyspace of the designed map is expanded. Moreover, the total number of chaotic parameters and the sensitivity of the map to initial conditions increase compared to the Ikeda map. The proposed chaotic map can be defined mathematically as follows:

(4) xn+1=1+α(xncosφn−ynsinφn)+μsin(πxn)

(5) yn+1=α(xnsinφn+yncosφn)+μsin(πyn)

(6) φn=β−γ/(1+xn2+yn2)

The control parameters μ,α  and γ act chaotic when   0≤μ<∞ , 0≤α≤1.18 and 0≤γ<∞  and the parameter  β=0.4 . The state space plot helps us to visualize the dynamics of the chaotic map. Fig. 1 shows the state space plot of the Ikeda and the Sine Ikeda modulated map for  α=0.95,  γ=6 and  μ=10 .

The chaotic nature of the proposed map can be proved mathematically by using Schwarzian derivative, considering the Eqs. (4) and (5) separately.

Definition: The Schwarzian derivative [30] of the differentiable function f at point z=(xn,yn) is given by,(7) Sfz=U−32(V)2 where U=d3f(Z)/dZ3df(Z)/dZ and V=(d2f(Z)/dZ2df(Z)/dZ)2

Theorem: The map f(X,Y)∈C3  shows the period–doubling bifurcation route to chaos when the Schwarzian derivative  (Sfz) is negative for X,Y≥0.

The evidence of the concept is in [31]. The designed map in Eqs. (4) and (5) meets the condition  Sfz<0 , as (V)2 in Eqs. (4) and (5) are much larger than U for all values of  X,Y≥0 .

The most important property of the chaotic map is its sensitive dependence on the initial conditions. Fig. 2a shows a time series plot of two Sine Ikeda modulated maps with one-digit variation in their initial conditions. The first map (x₁) is plotted with the initial condition   x0=1.8888887 , and the second map (x₂) is plotted with the initial condition  x0=1.8888888 . Even though, the two maps initially occupy closer points, they take different paths from the second iteration. This outcome shows that the designed map is highly sensitive to the initial conditions. Fig. 2b exhibits the difference between maps x₁ and x₂. Figs. 2c and 2d show the time series plot of two Ikeda maps with one-digit variation in initial conditions and the difference between the two Ikeda maps. Comparing Figs. 2b and 2d, takes approximately thirty-five iterations, whereas Fig. 2b takes only two iterations for the two maps to diverge. This outcome shows that the designed map is extremely sensitive to initial conditions.

The bifurcation diagram illustrates the change in qualitative behavior of the map as the chaotic control parameters are varied. Fig. 3 shows the bifurcation structure of the Ikeda and the Sine Ikeda modulated maps. The bifurcation structure of the Ikeda map has blank windows, but the Sine Ikeda modulated map shows no sign of any blank windows. The blank windows do not exhibit chaos. Moreover, the proposed map exhibits bifurcations over a large chaotic parameter range compared to the state-of-the-art techniques. As a result, when compared to other chaotic maps, the Sine Ikeda modulated map is better suited for cryptographic systems.

The characteristics of discrete chaotic maps are determined using a tool termed the Lyapunov exponent. A positive Lyapunov exponent signifies the presence of chaos. The Lyapunov exponents are calculated using the Jacobian matrix-based method that is described in [32]. Using (4), (5) and (6), the derivatives and the Jacobian matrix (J) can be written as:

(8) ∂xn+1∂xn=α[costn−2xn2(β−tn)sintn−2xnyn(β−tn)costn]+μπcosπxn

(9) ∂xn+1∂yn=α[−sintn−2xnyn(β−tn)sintn−2yn2(β−tn)costn

(10) ∂yn+1∂xn=α[sintn+2xn2(β−tn)costn−2xnyn(β−tn)sintn

(11) ∂yn+1∂yn=α[costn+2xnyn(β−tn)costn−2yn2(β−tn)sintn]+μπcosπyn

(12) J=[∂xn+1∂xn∂xn+1∂yn∂yn+1∂xn∂yn+1∂yn]

Similarly, the Jacobian matrix of the Ikeda map is computed using Eqs. (1)–(3). Fig. 4 provides the Lyapunov exponent plot of the Ikeda, and the Sine Ikeda modulated map. The Lyapunov exponent of the Sine Ikeda modulated map is more positive compared to the Ikeda map. Therefore, the Sine Ikeda modulated map outperforms the basic Ikeda map in terms of the Lyapunov exponent. Furthermore, the Lyapunov exponent value is much larger than the existing chaotic map, which indicates that the proposed map is highly sensitive to initial conditions. Additionally, the two Lyapunov exponent values (LE1 and LE2) are positive in the proposed map, which shows that the Sine Ikeda modulated map exhibits hyperchaotic behavior. The Ikeda map exhibits chaotic behavior because only one Lyapunov exponent is positive. Also, existing 2D maps exhibit hyperchaotic behavior for very few values of the chaotic parameter. Therefore, compared to the Ikeda map and the existing 2D maps, it is very difficult to predict the output of the proposed map because of the hyperchaotic property. Hence, the proposed chaotic map exhibits the desirable characteristics of a cryptographic system.

The bifurcation entropy specifies the measure of uncertainty present in the bifurcation structure. Fig. 5 shows the plot of the bifurcation entropy of the Ikeda map and the Sine Ikeda modulated map for 256 samples. The bifurcation entropy of the Ikeda map is maximal for very few values of the chaotic parameter, but the bifurcation entropy of the Sine Ikeda modulated map is always at its peak after some initial iteration. Also, the bifurcation entropy is zero for a few values of the chaotic parameter in the case of the Ikeda map, but the bifurcation entropy is non-zero and is greater than four in the case of the proposed map. Therefore, the Sine Ikeda modulated map outperforms other maps when used in cryptographic systems.

The randomness of the keys generated using the proposed sine-modulated Ikeda is verified using the NIST statistical test suite SP 800-22 [33]. First, the keys generated were represented in binary form and tested using hundred samples of ten lakh bit length keys, and their significance level is 0.001. In Tab. 1, all the P-values are greater than the significant value and have passed the test. Fifteen test results demonstrate that the designed sine-modulated Ikeda map has enough randomness to be used in cryptographic systems.

In this section, the designed chaotic map has been compared with the existing techniques in terms of sensitive dependence on initial conditions, average cycle length, maximum Lyapunov exponent, and approximate entropy. Tab. 2 shows the comparison of the existing chaotic map-based key with that of the proposed key.

The first parameter from the table shows the number of iterations required for the chaotic maps to change their paths with a one-digit change in the initial condition. The initial conditions for all maps are assumed to be the same. The proposed chaotic map requires only two iterations compared to the existing map. This outcome shows that the sensitive dependence on the initial conditions is high compared to that of the existing 2D maps, which is the desirable characteristic of a chaotic key.

The second parameter is the cycle length. The cycle length of the chaotic map is important to decide on the dynamical degradation of the chaotic map. Also, Fig. 6 shows the comparison plot for different precisions. From Tab. 2 and Fig. 6, it could be inferred that the cycle length is larger compared to the existing technique. Therefore, the proposed map can be applied to the cryptographic system.

The last parameter listed in the table is the approximate entropy of the key generated. The entropy is used to measure the randomness of the key. The maximum entropy for an eight-bit key is 8. It can be inferred from the above table that the designed map has an entropy close to 8 compared to the existing technique. As a result, an attacker will find it difficult to decode and interpret the keys.