Improved upper bound on the nonlinearity of high order correlation immune functions

Abstract

It has recently been shown that when m > (Formula Presented)-1, the nonlinearity Nf of an mth-order correlation immune function f with n variables satisfies the condition of Nf ≤ 2n−1 − 2m, and that when m > 1(Formula Presented) − 2 and f is a balanced function, the nonlinearity satisfies Nf ≤ 2n−1 − 2m+1. In this work we prove that the general inequality, namely Nf ≤ 2n−1 − 2m, can be improved to Nf ≤ 2n−1 − 2m+1 for m ≥ 0.6n − 0.4, regardless of the balance of the function. We also show that correlation immune functions achieving the maximum nonlinearity for these functions have close relationships with plateaued functions. The latter have a number of cryptographically desirable properties.