New Concepts in Evolutionary Search for Boolean Functions in Cryptology

Abstract

In symmetric cryptology the resistance to attacks depends critically on the nonlinearity properties of the Boolean functions describing cipher components like Substitution boxes (S-boxes). Some of the most effective methods known to generate functions that satisfy multiple criteria are based on evolutionary heuristics. In this paper, we improve on these algorithms by employing an adaptive strategy. Additionally, using recent improvements in the understanding of these combinatorial structures, we discover essential properties of the graph formed by affine equivalence classes of Boolean functions, which offers several advantages as a conceptual model for multiobjective seeking evolutionary heuristics. Finally, we propose the first major global cooperative effort to discover new bounds for cryptographic properties of Boolean functions.

Heuristic Methods for the Design of Cryptographic Boolean Functions

“…The basic concepts and definitions of the mathematical apparatus of Boolean algebra used in evaluating the effectiveness of non-linear nodes to replace symmetric ciphers were introduced in [21][22][23][24][25][26][27][28].…”

“…The Boolean function f of n variables is the function [21][22][23][24][25][26][27][28], variables is the function that maps from the field GF(2 n ) of all binary vectors x = (x1, …, xn) of length n to the field GF (2). Usually Boolean functions are represented in algebraic normal form (ANF) and are considered as the sum of the products of the component coordinates.…”

“…In particular, the algebraic properties of S-blocks of modern block ciphers are investigated in [5][6][7] and their influence on sustainability to algebraic cryptanalysis is shown. In [8][9][10][11], combinatorial properties of non-linear knots in the context of the security evaluation of various encryption modes and key schedules were properties are examined; in [24] the method of simulation of annealing is investigated; evolutionary methods are studied in [25,26]; papers [27,28] are dedicated to the heuristic methods of gradient search.…”

Classification of cubic (n-4)-resilient Boolean functions

“…The set of 2 n connection functions are invariant. Moreover, this concept can be generalized to connection functions of any distance 1 ≤ d ≤ 2 n − 1 (see [MFD04]).…”

Dragon: A Fast Word Based Stream Cipher

“…Existing methods [14] were adopted to generate the individual functions, then they were compared to the existing s-box functions to check the above-listed requirements for the XOR pairs. When the candidate function was acceptable it was appended to the s-box, else another function was tested.…”