## Abstract

The*correlation* between two Boolean functions of*n* inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2^{ n }. In this paper we compute, in closed form, the correlation between any two*symmetric* Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has an*exponentially small* correlation (in*n*) with the parity function. This improves a result of Smolensky [12] restricted to symmetric Boolean functions: the correlation between parity and any circuit consisting of a Mod_{ q } gate over AND gates of small fan-in, where*q* is odd and the function computed by the sum of the AND gates is symmetric, is bounded by 2^{−Ω(n)}.

In addition, we find that for a large class of symmetric functions the correlation with parity is*identically* zero for infinitely many*n*. We characterize exactly those symmetric Boolean functions having this property.

## Keywords

Boolean Function Symmetric Function Parity Function Random Oracle Symmetric Polynomial## Preview

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