The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Sajjad Shaukat Jamal,Dawood Shah,Tariq Shah,Abdulaziz Deajim

doi:10.1155/2020/8883884

Abstract

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree 8 primitive irreducible polynomial over ℤ2. In this manuscript, we study affine power affine S-boxes for all the 16 distinct degree 8 primitive irreducible polynomials over ℤ2 to propose 16 different 8×8 substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

Highlights

The exchange of digital data through the Internet has revolutionized the communication parameters over the years
To counter the emerging challenges of security, cryptography and steganography are used to hide the secret information whereas watermarking is used for copyright protection
Cryptography is divided into two types named symmetric key cryptography and asymmetric key cryptography

Summary

Introduction

The exchange of digital data through the Internet has revolutionized the communication parameters over the years. We have constructed 16 different robust 8 × 8 S-boxes over the elements of these 16 irreducible polynomials. We define 16 affine power affine transformations on these different Galois fields which can be given as z ⟶ (az + b)o(cz + d)− 1 ; here, for a, b, c, d values, we would be able to get 16 distinct S-boxes. The motivation behind this work is to study all primitive irreducible polynomials and their role in the construction of S-boxes. In this manuscript, we studied all binary degree 8 primitive irreducible polynomials for the construction of S-boxes. (1) We constructed S-boxes associated with the 16 binary degree 8 primitive irreducible polynomials. We construct 16 S-boxes out of the Galois fields corresponding to the aforementioned sixteen primitive irreducible polynomials. S-boxes have strong cryptographic properties certified with the help of analyses such as nonlinearity, strict avalanche criterion (SAC), bit independent criterion (BIC), linear approximation probability (LP), and differential approximation probability (DP) [20]

Security Analysis

Balanced Boolean Function