COBLAH: A chaotic OBL initialized hybrid algebraic-heuristic algorithm for optimal S-box construction

Abstract

The Substitution box (S-box) is the main nonlinear component responsible for the cryptographic strength of any Substitution-Permutation Network (SPN) based block cipher. Generating the S-box with optimal cryptographic properties is one of cryptography's most challenging combinatorial problems because of its enormous search space, lack of guidance, and conflicting performance criteria. This paper introduces a novel Chaotic Opposition-based Learning Initialized Hybrid Algebraic-Heuristic (COBLAH) algorithm, combining the favorable traits of Algebraic and heuristics methods based on Galois field inversion, affine mapping, and Genetic Algorithm (GA). The Galois field inversion and affine mapping are used to construct the S-box, while the GA guides the algebraic construction to find the best bit-matrix and additive vector based on any irreducible polynomial for GF(2⁸). GA initializes with a random population generated using a newly constructed cosine-cubic map incorporated with binarization and Opposition-based Learning (OBL). Further, Multi-Objective Optimization Ratio Analysis (MOORA) is utilized to identify the best S-box from the final optimized population. The performance of the proposed algorithm is evaluated by comparing the generated COBLAH S-box with more than twenty state-of-the-art S-boxes, including Advanced Encryption Standard (AES), Skipjack, Gray, and Affine Power Affine (APA). The COBLAH S-box has nonlinearity 112, Strict Avalanche Criterion (SAC) offset 0.0202, Distance to SAC (DSAC) 332, Differential Approximation Probability (DP) 0.0625, Linear Approximation Probability (LP) 0.0156, Bit Independence Criterion-Strict Avalanche Criterion (BIC-SAC) 0.50006, and Bit Independence Criterion-Nonlinearity (BIC-NL) 112, which stands as the optimal observed thus far. The absence of fixed and opposite fixed points and the fact that it adheres to a single cycle aligns the COBLAH S-box with an ideal S-box. In addition, an image encryption mechanism is utilized to encrypt and decrypt the different images sourced from the standard USC-SIPI image dataset using COBLAH S-box and compared against different state-of-the-art S-boxes based on various image characteristics.