Enumerative encoding of correlation immune Boolean functions

Abstract

Boolean functions are very important cryptographic primitives in stream or block ciphers. In order to be useful for cryptographic applications, these functions should satisfy some properties like high algebraic degree, high non linearity or being correlation immune. Since for most of the cryptographic criteria presented in the literature there is no complete characterization of the set of functions that optimally satisfy any of them, the possibility of finding an enumerative encoding of any such class of functions is extremely hard. In a recent paper Le Bars and Viola have presented an innovative recursive decomposition of the first order correlation immune Boolean functions. It is not a trivial task, however, to derive from this characterization an enumerative encoding. This paper presents an enumerative encoding for first order correlation immune functions. It provides the first enumerative encoding of a class of Boolean functions with cryptographic applications. The encoding naturally leads to efficient random generation algorithms. For example, we may construct, with uniform probability, any 1-resilient function (balanced first order correlation immune function) with 8 variables in less than 30 seconds, from a universe of around 1068 functions