Construction for Balanced Boolean Function with Maximum Algebraic Immunity

Abstract

Algebraic immunity is a new cryptographic criterion. In order to resist algebraic attacks, Boolean functions used in cryptosystem should have high algebraic immunity. This paper presents a new recursive construction of Boolean function with maximum algebraic immunity. The constructed Boolean functions are pointed out to be balanced.

Heuristic Methods for the Design of Cryptographic Boolean Functions

“…The basic concepts and definitions of the mathematical apparatus of Boolean algebra used in evaluating the effectiveness of non-linear nodes to replace symmetric ciphers were introduced in [21][22][23][24][25][26][27][28].…”

“…The Boolean function f of n variables is the function [21][22][23][24][25][26][27][28], variables is the function that maps from the field GF(2 n ) of all binary vectors x = (x1, …, xn) of length n to the field GF (2). Usually Boolean functions are represented in algebraic normal form (ANF) and are considered as the sum of the products of the component coordinates.…”

“…) [21][22][23][24][25][26][27][28]. The truth table of a function f is (0,1) -a sequence, defined as (f(0), f(1), 2 1 ( ) n f   ) [21][22][23][24][25][26][27][28]. The sequence of the function f is balanced if its (0,1) -sequence ((1, -1) -sequence) contains the same number of zeros and ones (ones and minus ones).…”

“…The sequence of the function f is balanced if its (0,1) -sequence ((1, -1) -sequence) contains the same number of zeros and ones (ones and minus ones). The function f is balanced if its sequence is balanced [21][22][23][24][25][26][27][28].…”

A Proof of Constructions for Balanced Boolean Function with Optimum Algebraic Immunity

“…In [13], we proposed the upper construction, and checked that the constructed Boolean functions  n is balanced and has optimum AI for 1n6. In the following, we will prove in detail that  n is balanced and has optimum AI for all n.…”