Construction of balanced even-variable Boolean functions with optimal algebraic immunity

Abstract

Given a positive integer n, let k = n/2 − 1 and s = k i=0 n i . The generator matrix G(k, n) s×2 n of the kth-order Reed-Muller code RM(k, n) is an important tool in the study of Boolean functions' algebraic immunity. In this paper, choosing the last s column vectors in G(k, n) as a basis of the vector space F s 2 , we study the values of the coefficients in the linear expressions of G(k, n)'s column vectors over this basis. As an application, we present a new construction of balanced even-variable Boolean functions on F n 2 with optimal algebraic immunity by modifying the outputs of majority function. The nonlinearities of these constructed functions are also determined.

Simpler proof for nonlinearity of majority function

“…A useful recent survey of the applications of majority functions to algebraic immunity is given in [2]. The papers [1,6,9,11,12,13] and others referenced in those papers all contain recent work on cryptographic applications of these functions. It is necessary to know the nonlinearity of the majority function to begin such studies.…”

“…Now (9) follows from (11), (12) and (13), and (10) follows from (9) by elementary properties of binomial coefficients. Simple estimates using (10) show that wt(C(B(2n)) * ) < 2 2n−2 .…”