A Mathematical Approach for Generating a Highly Non-Linear Substitution Box Using Quadratic Fractional Transformation

Abstract

Nowadays, one of the most important difficulties is the protection and privacy of confidential data. To address these problems, numerous organizations rely on the use of cryptographic techniques to secure data from illegal activities and assaults. Modern cryptographic ciphers use the non-linear component of block cipher to ensure the robust encryption process and lawful decoding of plain data during the decryption phase. For the designing of a secure substitution box (S-box), non-linearity (NL) which is an algebraic property of the S-box has great importance. Consequently, the main focus of cryptographers is to achieve the S-box with a high value of non-linearity. In this suggested study, an algebraic approach for the construction of 16 × 16 S-boxes is provided which is based on the fractional transformation Q (z) = 1 α (z) m + β (mod257) and finite field. This technique is only applicable for the even number exponent in the range (2-254) that are not multiples of 4. Firstly, we choose a quadratic fractional transformation, swap each missing element with repeating elements, and acquire the initial S-box. In the second stage, a special permutation of the symmetric group S 256 is utilized to construct the final S-box, which has a higher NL score of 112.75 than the Advanced Encryption Standard (AES) S-box and a lower linear probability score of 0.1328. In addition, a tabular and graphical comparison of various algebraic features of the created S-box with many other S-boxes from the literature is provided which verifies that the created S-box has the ability and is good enough to withstand linear and differential attacks. From different analyses, it is ensured that the proposed S-boxes are better than as compared to the existing S-boxes. Further these S-boxes can be utilized in the security of the image data and the text data.