Constructing Two Classes of Boolean Functions With Good Cryptographic Properties

Abstract

Wu et al. proposed a generalized Tu-Deng conjecture over F 2 rm ×F 2 m , and constructed Boolean functions with good properties. However the proof of the generalized conjecture is still open. Based on Wu's work and assuming that the conjecture is true, we come up with a new class of balanced Boolean functions which has optimal algebraic degree, high nonlinearity and optimal algebraic immunity. The Boolean function also behaves well against fast algebraic attacks. Meanwhile we construct another class of Boolean functions by concatenation, which is 1-resilient and also has other good cryptographic properties.

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.…”