New Results on Correlation Immunity

Abstract

The absolute indicator for GAC forecasts the overall avalanche characteristics of a cryptographic Boolean function. From a security point of view, it is desirable that the absolute indicator of a function takes as small a value as possible. The first contribution of this paper is to prove a tight lower bound on the absolute indicator of an mth- order correlation immune function with n variables, and to show that a function achieves the lower bound if and only if it is affine. The absolute indicator for GAC achieves the upper bound when the underlying function has a non-zero linear structure. Our second contribution is about a relationship between correlation immunity and non-zero linear structures. The third contribution of this paper is to address an open problem related to the upper bound on the nonlinearity of a correlation immune function. More specifically, we prove that given any odd mth-order correlation immune function f with n variables, the nonlinearity of f, denoted by N_f, must satisfy N_f ≤ 2^n-1 - 2^m+1 for 1/2n - 1 ≤ m < 0.6n - 0.4 or f has a non-zero linear structure. This extends a known result that is stated for 0.6n - 0.4 ≤ m ≤ n - 2.

Acknowledgments

The second author was supported by a Queen Elizabeth II Fellowship (227 23 1002). Both authors would like to thank Yuriy Tarannikov and Subhamoy Maitra for pointing out an error in an earlier version.