Connections among nonlinearity, avalanche and correlation immunity

Yuliang Zheng, Xian-Mo Zhang
2003 Theoretical Computer Science  
Nonlinear Boolean functions play an important role in the design of block ciphers, stream ciphers and one-way hash functions. Over the years researchers have identiÿed a number of indicators that forecast nonlinear properties of these functions. Studying the relationships among these indicators has been an area that has received extensive research. The focus of this paper is on the interplay of three notable nonlinear indicators, namely nonlinearity, avalanche and correlation immunity. We
more » ... ish, for the ÿrst time, an explicit and simple lower bound on the nonlinearity N f of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, N f ¿ 2 n−1 − 2 n−1−(1=2)p . We also identify all the functions whose nonlinearity attains the lower bound. As a further contribution of this paper, we prove that except for very few cases, the sum of the degree of avalanche and the order of correlation immunity of a Boolean function of n variables is at most n − 2. The new results obtained in this work further highlight the signiÿcance of the fact that while avalanche property is in harmony with nonlinearity, both go against correlation immunity. .au (X.-M. Zhang). 0304-3975/03/$ -see front matter c 2002 Published by Elsevier Science B.V. PII: S 0 3 0 4 -3 9 7 5 ( 0 2 ) 0 0 3 1 9 -5
doi:10.1016/s0304-3975(02)00319-5 fatcat:4goegm2qojcyrj7xke4oopdsyy