A Modern Method for Constructing the S-Box of Advanced Encryption Standard

The substitution table (S-Box) of Advanced Encryption Standard (AES) and its properties are key elements in cryptanalysis ciphering. We aim here to propose a straightforward method for the non-linear transformation of AES S-Box construction. The method reduces the steps needed to compute the multiplicative inverse, and computes the matrices multiplication used in this transformation, without a need to use the characteristic matrix, and the result is a modern method constructing the S-Box.


Introduction
The S-Box table of AES is taken as a lookup table to substitute an input byte by another, this table is constructed using a non-linear transformation depends on the usual method taking more calculation steps to give the corresponding byte.
The S-Box plays a fundamental role in encryption and decryption processes, as byte substitution appears in many steps.At the first round of the encryption process, we add the plaintext matrix to the key matrix, then we substitute each byte by another byte according to S-Box, for example, to substitute the byte xy(say), we take the byte in the cell that has x as the column index and y as the row index, we do this substitute byte step in all rounds of the encryption process, and in all round of the decryption process, we do the inverse substitute byte step, to substitute the byte xy(say), we take the index of the column, and the index of the row of the cell that contains xy, as the left and the right character of the result byte, respectively.The S-Box ( The S-Box is constructed using the following operations [1]: 1) Finding the multiplicative inverse of an input byte in the finite field ( ) based on the irreducible polynomial ( ) 8 4 3 1 P x x x x x = + + + + .
2) Multiplying this multiplicative inverse by a specific matrix (matrix M).
3) Adding the multiplication result to a specific vector { } ( ) We convert the hexadecimal presentation of the input byte into binary presentation as ( ) a a a a a a a a and write it as a polynomial ( ) by the following characteristic matrix: Then, we add the result to (01100011).
We note that, for the input {00} the output is {63}.

Problem Statement
We search for an easier and straightforward method for constructing the AES S-Box.

Proposed Solution
The multiplicative inverse of an input byte can be computed in clear steps using an iterated formula.
Multiplying the multiplicative inverse matrix by the characteristic matrix can be determined directly from this multiplicative inverse using simple XOR operations, without a need to use the characteristic matrix.

Traditional Way
In cryptography, the extended Euclidean algorithm has wide uses especially for finding a multiplicative inverse (modular inverse).
Euclidean algorithm is used to find the greatest common divisor of two integers a and b, (denoted by for some integers r and q, we say and if ( ) With the polynomials ( ) A x and ( ) The algorithm below gives A x P x , where ( ) ( )
( ) The multiplicative inverse [2] of ( ) A x modulo ( ) and since ( ) ( ) ( ) ( ) So, the procedure of the extended Euclidean algorithm finds the greatest common divisor, also it finds the multiplicative inverse.
1 0 0 0 1 1 1 1 0 0 The output is 11101101 ED 6) Go to 2 Now, we want to multiply ( ) T x by the matrix M. First, write M as [7] 1 0 0 0 1 1 1 1 Let And write ( ) So, the multiplication of M and ( ) of the second matrix, so we don't need to use matrix M, as the traditional method.
In the last step, we add

Example
Using the modern way, we want to find the output of {53} { } First, finding the multiplicative inverse (Table 3).
Then, computing the matrices multiplication: Last, adding (01100011) So, the output is 11101101 ED = .

Conclusions
In this paper, a straightforward method for obtaining the Advanced Encryption Standard S-Box look-up table without the traditional use of the characteristic Matrix M is proposed.We have demonstrated that the two methods are equivalent.In addition, the multiplicative inverse of ( ) A x has been found more ele- gantly.
In future work, we will investigate the properties and the impact of this technique on cipher complexity analysis.

−
with its value from the above equation, continue doing this replacement, we obtain ( )

Table 1 )
, involves substitution bytes for all

Table 1 .
The AES S-Box.