On the Coset Weight Divisibility and Nonlinearity of Resilient and Correlation-Immune Functions

Sarkar and Maitra have recently shown that, given any m-resilient function f on F 2 n , the Hamming distance between f and any affine function on F 2 n is divisible by 2m+1. We show that their result is a simple consequence of a recent characterization of resilient functions by means of their numerical normal forms. This characterization allows us to obtain a better divisibility bound, involving n, m and the algebraic degree d of the function. Smaller is d and/or m, stronger is our improvement. We show that our divisibility bound is tight for every positive n, every non-negative m ≤ n − 2 and every positive d ≤ n − m − 1. We deduce a bound on the nonlinearity of resilient functions involving n, m and d. This bound improves upon those given recently and independently by Sarkar and Maitra and by Tarannikov. We finally show that the same bound stands in the more general framework of m-th order correlation-immune functions, for sufficiently large m.