Improving the high order nonlinearity lower bound for Boolean functions with given algebraic immunity

Abstract

Algebraic immunity is a recently introduced cryptographic parameter for Boolean functions used in stream ciphers. If p A I ( f ) and p A I ( f ⊕ 1 ) are the minimum degree of all annihilators of f and f ⊕ 1 respectively, the algebraic immunity A I ( f ) is defined as the minimum of the two values. Several relations between the new parameter and old ones, like the degree, the r -th order nonlinearity and the weight of the Boolean function, have been proposed over the last few years. In this paper, we improve the existing lower bounds of the r -th order nonlinearity of a Boolean function f with given algebraic immunity. More precisely, we introduce the notion of complementary algebraic immunity A I ¯ ( f ) defined as the maximum of p A I ( f ) and p A I ( f ⊕ 1 ) . The value of A I ¯ ( f ) can be computed as part of the calculation of A I ( f ) , with no extra computational cost. We show that by taking advantage of all the available information from the computation of A I ( f ) , that is both A I ( f ) and A I ¯ ( f ) , the bound is tighter than all known lower bounds, where only the algebraic immunity A I ( f ) is used.