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

Abstract

Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions.

Implementing Symmetric Cryptography Using Sequence of Semi-Bent Functions

“…Siegenthaler [9] proved that m þ d ≤ n þ 1ifm ≤ n À 2. The exact nature of trade-offs among order of correlation immunity, nonlinearity, and algebraic degree has also been investigated, for instance, ( [12,13]. Using the above bounds, one may naturally try to provide the construction of an n; m; d; N f ÀÁ function for any given n and m while at the same time attempting to optimize d and N f .…”