A Lower Bound of Fast Algebraic Immunity of a Class of 1-Resilient Boolean Functions

Abstract

Boolean functions should possess high fast algebraic immunity when used in stream ciphers in order to stand up to fast algebraic attacks. However, in previous research, the fast algebraic immunity of Boolean functions was usually calculated by the computer. In 2017, Tang, Carlet, and Tang first mathematically proved that every function belonging to a class of 1-resilient Boolean functions has the fast algebraic immunity no less than n − 6. Inspired by the Tang's method, we also demonstrate that the fast algebraic immunity of another class of the 1-resilient Boolean functions is no less than n − 6. Meanwhile, we also prove some combinator facts originated from the Tu-Deng Conjecture. INDEX TERMS Fast algebraic immunity, algebraic immunity, resiliency, Boolean functions.

A Bound of Fast Algebraic Immunity of Constructions With BPR Using a Four-Disjoint-Part Support

“…More importantly, they deduced that the FAI of a class of 1-resilent Boolean functions is greater than or equal to n − 6 by means of mathematical derivation, which is also a groundbreaking demonstration that a class of Boolean functions can effectively resist FAA by theoretical proof instead of computeraided calculation. Later, Chen also proved that a lower bound of a class of 1-resilent Boolean functions is n − 6 [12]. Inspired by Tang and Chen' idea, we demonstrate that a bound of FAI of constructions with bivariate polynomial representation (BPR) using a four-disjoint-part support.…”

“…Later, Chen also proved that a lower bound of the FAI of a class of 1-resilent Boolean functions is n − 6 [12]. The Boolean functions in [11], [12] belongs to the case v = 1.…”

“…In 2017, Tang et al proved that the FAI of a class of 1-resilient Boolean functions modified from TCT functions is no less than n − 6 by theoretical proof instead of computer-aided calculation [11]. Later, Chen also proved that a lower bound of the FAI of a class of 1-resilent Boolean functions is n − 6 [12]. The Boolean functions in [11], [12] belongs to the case v = 1.…”

SAO 1-Resilient Functions With Lower Absolute Indicator in Even Variables

“…However, no matter the balanced Boolean functions given in [13], [24] or in [25], they cannot be transformed into m-resilient (m ≥ 1) functions, and there is no evidence to show the existence of such functions. Many works on resilient functions are devoted to estimating the nonlinearity or other cryptographic criteria of resilient functions, but seldom considering their absolute indicators (see [4], [5], [16], [23], [27]- [30] and the references therein). Until now, there are only a few works (see [10], [17]) on this topic and the best known upper bound of the minimum absolute indicator of 1-resilient functions on n-variables (n even) is 5 • 2 n/2 − 2 n/4+2 + 4, which was obtained by Ge et al [10] for the calculation of the absolute indicator of 1-resilient functions designed by Zhang et al in [30], and it turned out that those 1-resilient functions possess the currently highest nonlinearity 2 n−1 − 2 n/2−1 − 2 n/4 and lowest absolute indicator 5 • 2 n/2 − 2 n/4+2 + 4.…”

Fast Algebraic Immunity of $2^m+2$ & $2^m+3$ Variables Majority Function

“…In recent years, more efforts have been made to investigate the FAI of Boolean functions [2], [5], [15]- [20]. In 2006, Armknecht et al proposed some efficient methods for evaluating the ability of resisting algebraic attacks and fast algebraic attacks for Boolean functions, with which they also gave a proof of bad resistance of the majority function resist fast algebraic attacks [15].…”