The design of S-boxes by simulated annealing

Abstract

Abstract-Substitution boxes are important components in many modern day block and stream ciphers. Their study has attracted a great deal of attention over many years. The development of a variety of cryptosystem attacks over the years has lead to the development of criteria for resilience to such attacks. Some general criteria such as high non-linearity and low autocorrelation have been proposed as useful criteria (providing some protection against attacks such as linear cryptanalysis and differential cryptanalysis). There has been little application of evolutionary search to the development of S-boxes. In this paper we show how a cost function that has found excellent single output Boolean functions can be generalised to provide improved results for small S-boxes.

Reversed genetic algorithms for generation of bijective s-boxes with good cryptographic properties

“…NS deg(S) AC(S)max δ-uniformity Fixed points Linear Redundancy Pseudo-random S-BOX [20], [21] 98 NR NR NR NR NR (usually zero) Finite Field Inversion [24] 112 7 32 4 2 complete Hill Climbing [20] 100 NR NR NR NR NR GA/HC [21] 100 NR NR NR NR NR Simulated Annealing [8] 102 NR 80 NR NR NR GaT [32] 104 Note that algebraic immunity is also included in the target set of criteria for the Gradient descent method. The reported value for algebraic immunity is 3 [15].…”

“…Specific heuristic techniques include the hill climbing method, the simulated annealing method, the genetic algorithm or a combination of these. For example the highest nonlinearity achieved by: the hill climbing method is 100 [20], the simulated annealing method is 102 [8], and by a special genetic algorithm is 104 [32]. Recently in [15], values of 104 have been achieved for the nonlinearity by a method, referred as the modified gradient descent, which is based on swapping a number of values in a permutation.…”

“…Simulated annealing method provides an extension to the hill climbing technique in which the search process is able to move out of a local optimum in order to continue. For the case of (8 × 8) S-boxes by using this method S-boxes possessing nonlinearity 102 are generated [8]. The genetic algorithms work with a population of candidate solutions.…”

A Deterministic Optimization Approach for Generating Highly Nonlinear Balanced Boolean Functions in Cryptography

“…We are then faced with the minimization of a convex polyhedral function with binary variables under linear constraints which is in fact equivalent to a mixed zero-one linear program (with exactly one continuous variable). In our convenient for DC programming approach, for the moment, we consider the problem in the form (8).…”

Cartesian Genetic Programming Approach for Generating Substitution Boxes of Different Sizes

“…On the other side, 8×8 size S-boxes are used when speed and security of a cipher are of primary importance; as an example consider the AES algorithm [5]. For a detailed description of relevant S-box properties, we refer the reader to [3,4,[6][7][8] In this paper we investigate the efficiency of two algorithms previously unexplored when discussing the evolution of S-boxes. Those algorithms are Cartesian Genetic Programming (CGP) and Genetic Programming (GP).…”