Generation of Nonlinear Substitutions by Simulated Annealing Algorithm

Abstract

The problem of nonlinear substitution generation (S-boxes) is investigated in many related works in symmetric key cryptography. In particular, the strength of symmetric ciphers to linear cryptanalysis is directly related to the nonlinearity of substitution. In addition to being highly nonlinear, S-boxes must be random, i.e., must not contain hidden mathematical constructs that facilitate algebraic cryptanalysis. The generation of such substitutions is a complex combinatorial optimization problem. Probabilistic algorithms are used to solve it, for instance the simulated annealing algorithm, which is well-fitted to a discrete search space. We propose a new cost function based on Walsh–Hadamard spectrum computation, and investigate the search efficiency of S-boxes using a simulated annealing algorithm. For this purpose, we conduct numerous experiments with different input parameters: initial temperature, cooling coefficient, number of internal and external loops. As the results of the research show, applying the new cost function allows for the rapid generation of nonlinear substitutions. To find 8-bit bijective S-boxes with nonlinearity 104, we need about 83,000 iterations. At the same time, the probability of finding the target result is 100%.

Enhancing Smart Communication Security: A Novel Cost Function for Efficient S-Box Generation in Symmetric Key Cryptography

“…In our study in [26] delve into the optimization of the Hill Climbing algorithm for the efficient generation of S-boxes, crucial for modern symmetric ciphers. Another our pivotal contribution in [27], explores the simulated annealing algorithm enhanced by a novel cost function based on the Walsh-Hadamard spectrum. In [28] introduce an innovative cost function that revolutionizes the hill-climbing algorithm's efficiency in generating S-boxes.…”