Equivalence classes of boolean functions for first-order correlation

Abstract

Boolean functions are very important cryptographic primitives in stream or block ciphers. In this context, these functions need to satisfy good properties like high algebraic degree, nonlinearity and correlation immunity. We present here an original and efficient method to enumerate all the correlation-immune functions of a fixed Hamming weight, in particular the class of 1-resilient functions. The key idea consists in defining equivalent classes to split boolean functions along their distance from correlation-immune boolean functions. These classes, called first-order correlation classes, are built using a recursive decomposition of smaller classes. We derive from this method several algorithms to enumerate their elements and to count their cardinality. We first show that the exact number of 1-resilient boolean functions with 7 variables is 23478015754788854439497622689296 and we obtain a tight estimation of their number with 8 variables, between 4 1067 and 5.6 1068. We then present a general lower bound for the number of 1-resilient boolean functions and improve Schneider's upper bound. We also propose a general lower bound for the number of k-resilient functions. Most of the bounds presented in this paper, substantially improve the best known bounds in the literature. We finally establish that the probability of a Boolean function being 1-resilient is asymptotically between (npi)n/2/2n2-3/2n-1en-1/2.