Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

Abstract

This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler's upper bound of algebraic degree. Also a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

“…Compared with Carlet-Feng functions [9] and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables [19], ours show better immunity against fast algebraic attacks. We check that our constructions are almost perfect algebraic immune functions (see Definition 5).…”

“…We notice that Pan et al [19] presented a construction for a class of 1-resilient Boolean functions with optimal algebraic immunity on an even number of variables by dividing them into two correlation classes, that is, equivalence classes. However, the cryptographic properties of the resulting functions are highly related to those of the initial functions we choose, and in particular, one would not expect strong resistance against fast algebraic attack in the resulting Boolean functions.…”

“…Pan et al [19] presented a construction for a class of 1-resilient Boolean functions with optimal algebraic immunity on an even number of variables by dividing them into two correlation classes. More precisely, Pan et al proposed a secondary construction (i.e., Siegenthaler's [6] construction) by concatenating two balanced Boolean functions , with odd variables , where deg( ) = −1, AI( ) = ( +1)/2.…”

“…However, the cryptographic properties of the resulting functions are highly related to those of the initial functions they chose as building block, and in particular, this does not rule out the possibility that these functions are vulnerable to fast algebraic attack; that is, one would not expect strong resistance against fast algebraic attack in the resulting Boolean functions. More details on the rationale of their constructions can be found in [19] where two constructions are presented and security properties are analyzed mathematically step by step. In Section 6, we also compare the properties of fast algebraic immunity between our construction and the proposal of Pan et al [19].…”

“…More details on the rationale of their constructions can be found in [19] where two constructions are presented and security properties are analyzed mathematically step by step. In Section 6, we also compare the properties of fast algebraic immunity between our construction and the proposal of Pan et al [19].…”