Asymptotic enumeration of correlation-immune boolean functions

TL;DR

This paper provides an improved asymptotic estimate for counting correlation-immune boolean functions of n variables, especially as the order k increases with n, correcting previous misconceptions and including functions with specific weights like resilient functions.

Contribution

It offers a new asymptotic estimate for the number of correlation-immune boolean functions that extends previous results to larger k and specific weights, including resilient functions.

Findings

Corrects Denisov's repudiation of earlier estimates

Estimates valid for k increasing with n within limits

Includes functions with specified weights, such as resilient functions

Abstract

A boolean function of $n$ boolean variables is {correlation-immune} of order $k$ if the function value is uncorrelated with the values of any $k$ of the arguments. Such functions are of considerable interest due to their cryptographic properties, and are also related to the orthogonal arrays of statistics and the balanced hypercube colourings of combinatorics. The {weight} of a boolean function is the number of argument values that produce a function value of 1. If this is exactly half the argument values, that is, $2^{n-1}$ values, a correlation-immune function is called {resilient}. An asymptotic estimate of the number $N(n,k)$ of $n$-variable correlation-immune boolean functions of order $k$ was obtained in 1992 by Denisov for constant $k$. Denisov repudiated that estimate in 2000, but we will show that the repudiation was a mistake. The main contribution of this paper is an asymptotic estimate of $N(n,k)$ which holds if $k$ increases with $n$ within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of $k=1$, our estimates are valid for all weights.