On the Number Of Nonlinear Regular S-boxes

A. M. Youssef and S. E. Tavares Department Of Electrical and Computer Engineering Queen’s University Kingston, Ontario, Canada, K7L 3N6 Phone : (613) 545–2945 Fax : (613) 545–6615 email : [email protected] Indexing terms: Cryptography, Private-key cryptosystems, Substitution boxes, Combinatorics. Abstract Nonlinearity is a crucial requirement for the substitution boxes in secure block ciphers. In this letter, we calculate the probability of linearity in any nonzero linear combination of the output coordinates of a randomly selected regular substitution box. Introduction Gordon and Retkin [1] calculated the probability that any of the output coordinates of a random, reversible substitution box (i.e., a permutation) is an affine function. After both differential cryptanalysis [2], and linear cryptanalysis [3] were introduced, it was realized that the cryptographic strength of a multi-output function depends not only on the strength of its individual output coordinates but also on the strength of every nonzero linear combination of these coordinates [4]. One requirement in substitution-box (s-box) design is to have a regular s-box (also known as a balanced s-box). This means that each output symbol should appear an