Heuristic Design of Cryptographically Strong Balanced Boolean Functions

1 I n t r o d u c t i o n It is well known that the resistance of a product cipher to modern cryptanalytic attacks such as linear and differential cryptanalysis [10,1] depends critically upon the nonlinearity of the Boolean functions comprising the round function. Typically these functions must be balanced, so there is considerable interest in the design of highly nonlinear balanced Boolean functions. In addition we would like cipher functions to satisfy other cryptographic properties also, such as correlation immunity [20] and the strict avalanche criterion (SAC) [25]. Previous work on the design of balanced functions includes [6,7,17-19,21]. The existing body of research concentrates on specific constructions, supported by algebraic proofs that the resulting Boolean functions will be both balanced and satisfy one or more other properties. In contrast, some recent publications [13,12] address the issue of applying combinatorial optimisation methods to the design of Boolean functions. The methods of gradient descent (or hill climbing (HC)) and the use of a genetic algorithm have proven useful in the quasi-random generation of highly nonlinear Boolean functions. In this paper we present a modification of the genetic algorithm (GA) presented in [12] so that it is confined to balanced Boolean functions. When combined with the two-step hill climbing algorithm [13] a very effective means of generating highly nonlinear balanced Boolean functions is obtained. The results