The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography. This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize the AES-like cryptology on
AES (Advanced Encryption Standard) is a standard offered for encryption of electronic data. AES, adopted by the American government, is also used as a defacto encryption standard in the international arena. It replaces DES (Data Encryption Standard). The encryption algorithm defined by AES is a symmetric-key algorithm in which the keys used in both encryption and decryption of encrypted text are related to each other. The encryption and decryption keys are the same for AES.
The algorithm standardized with AES was created by making some changes to the Rijndael algorithm, which was mainly developed by Vincent Rijmen and Joan Daeman. Rijndael is a name obtain using the developers’ names: RIJmen and DAEmen.
AES is based on the design known as substitution-permutation. Its predecessor, DES, is an algorithm designed in Feistel structure. AES’ software and hardware performance is high. The 128-bit input block has a key length of 128, 192 and 256 bits. Rijndael, on which AES is based, supports input block lengths that are multiples of 32 between 128 and 256 bits and key lengths longer than 128 bits. Therefore, in the standardization process, key and input block lengths were restricted. AES works on a
The algorithm consists of identical rounds that transform a certain number of repeating input open text into output ciphertext. Each cycle consists of four steps, except for the last cycle. These cycles are applied in reserve order to decode the encrypted text. The number of repetitions of cycles is a function of the key length according to
Key lengths | Cycles |
---|---|
128 bits | 10 |
192 bits | 12 |
256 bits | 14 |
These cycles include key addition, byte substitution, ShiftRow and MixColumn. We can see these cycles in
A finite field, sometimes also called Galois field, is a set with a finite number of elements. Roughly speaking, a Galois field is a finite set of elements in which we can add, subtract, multiply and invert. Before we introduce the definition of a field, we first need the concept of a simple algebraic structure, a field.
A field F is a set of elements with the following properties:
All elements of F form an additive group with the group operation “+” and the neutral element 0.
All elements of F except 0 form a multiplicative group with the group operation “
When the two group operations are mixed, the distributivity law holds, i.e., for all
Galois field arithmetic is the most widely used field involving matrix operations. One can see detailed information about the Galois field and the operations performed on it in [
In extension fields
Note that there are exactly 256 = 2^{8} such polynomials. The set of these 256 polynomials is the finite field
In particular, we do not have to store the factors
Fibonacci numbers are defined by the recurrence relation of
Fibonacci polynomials that belong to the large polynomial classes are defined by a recurrence relation similar to Fibonacci numbers. The Belgian mathematician Eugene Charles Catalan and the German mathematician E. Jacobsthal were studied Fibonacci polynomials in 1983. The polynomials
In [
The sequence of polynomials
Kizilates et al. studied a new generalization of convolved
In [
In [
where
Diskaya et al. created a new encryption algorithm (known as AES-like) by using the AES algorithm in [
Fibonacci polynomials have many applications in algebra. In recent years, we see that these polynomials have many uses in the field of engineering. Also, Fibonacci polynomials are used in solving differential equations. These solutions are used in engineering and science, adding new approaches to the solution of engineering problems. Mirzae and Hoseini solved singularly perturbed differential-difference equations arising in science and engineering with Fibonacci polynomials in [
In this paper, we generalize the encryption algorithm given in [
In this chapter, we redefine the elements of k-order Fibonacci polynomial sequences using a certain irreducible polynomial in our coding algorithm. In extension fields
The AES encryption algorithm uses the
The irreducible polynomials of
In this paper, we consider the irreducible polynomials as
For later use the first few terms of the sequence Fibonacci polynomials can be seen in the following
0 | 1 | 2 | 3 | 4 | 5 | ||
---|---|---|---|---|---|---|---|
0 | 1 |
0 | 0 | |
1 | 1 | |
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 | ||
8 | ||
9 | ||
Polynomials of the Galois field are equivalent of each alphabet in
No. | Bit | Polynomial | Alphabet |
---|---|---|---|
0 | 00000 | 0 | A |
1 | 00001 | 1 | B |
2 | 00010 | C | |
3 | 00011 | Ç | |
4 | 00100 | D | |
5 | 00101 | E | |
6 | 00110 | F | |
7 | 00111 | G | |
8 | 01000 | Ğ | |
9 | 01001 | H | |
10 | 01010 | I | |
11 | 01011 | ||
12 | 01100 | J | |
13 | 01101 | K | |
14 | 01110 | L | |
15 | 01111 | M | |
16 | 10000 | N | |
17 | 10001 | O | |
18 | 10010 | Ö | |
19 | 10011 | P | |
20 | 10100 | R | |
21 | 10101 | S | |
22 | 10110 | Ş | |
23 | 10111 | T | |
24 | 11000 | U | |
25 | 11001 | Ü | |
26 | 11010 | V | |
27 | 11011 | W | |
28 | 11100 | X | |
29 | 11101 | Y | |
30 | 11110 | Z | |
31 | 11111 | Q |
Now, we obtain our encryption algorithm in line preliminary information we have given.
If there is an ascending 2 letters in the text, it letters is multiplied by 2. Key matrix in
If there is an ascending 2 letters in the text, it letters is multiplied by 2. Inverse key matrix in
“HELLO”
We can get as
It is known that
So, it is
Since the word “HELLO” has 5 letters, we divide it into blocks of
We can get in
So, it is
It results
Since we have 2 letters left, we can use our 2. Key matrix,
It results
where
It results
where
It results
Since we have 2 letters left, we can use our 2. Inverse key matrix.
It results
So, it is
Since we have 2 letters left, we can get
So, it is
It results
We have handled the example given in [
“PUBLIC”
It is known that
So, it is
Since the word “PUBLIC” has 6 letters, we divide it into blocks of
We encrypt the
We can get as
It results
Since we have 3 letters left, we can use our 1. Key matrix again.
It results
where
It results
where
It results
Since we have 3 letters left, we can use our 1. Inverse key matrix again.
It results
and for
It results
AES (Advanced Encryption Standard) is a standard offered for encryption of electronic data. The AES cipher is almost identical to the block cipher Rijndael. The Rijndael block and key size vary between 128, 192 and 256 bits. However, the AES standard only calls for a block size of 128 bits. Hence, only Rijndael with a block length of 128 bits is known as the AES algorithm. In the remainder of this page, we only discuss the standard version of Rijndael with a block length of 128 bits.
The Rijndael algorithm perform encryption with the help of polynomials in Galois fields. We have obtained a new encryption algorithm by generalizing the previous studies. In this paper, we generalized the encryption algorithm given in [
In this paper, we present the mathematical basis for understanding the design rationale and the features that follow the description itself. Then, we define AES-like encryption by giving the encryption method and its implementation.