On the correlation of symmetric functions

Abstract

The correlation between two Boolean functions of n inputs is de ned 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 t wo 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 a n d a n y circuit consisting of a M o d 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 ; (n) .In addition, we nd that for a large class of symmetric functions the correlation with parity i s identically zero for in nitely many n. W e c haracterize exactly those symmetric Boolean functions having this property.

Closed formulas for exponential sums of symmetric polynomials over Galois fields

“…In other words, in this paper we settle the problem of finding closed formulas for exponential sums of linear combinations of elementary symmetric polynomials over any Galois field. This extends the work of Cai, Green and Thierauf for the binary field [2] to every finite field. As far as we know, this is new.…”

“…In general, this problem is very hard to tackle, but imposing conditions on these functions may ease the problem. For instance, symmetric Boolean functions are good candidates for efficient implementations and today they are an active area research [2,6,7,8,10,11,12].…”

“…In general, to find closed formulas for exponential sums of symmetric Boolean functions was an open problem until Cai, Green and Thierauf found formulas for them in the 1990's [2]. Moreover, their formulas imply that exponential sums of symmetric Boolean functions have a recursive nature.…”

Value distribution of elementary symmetric polynomials and its perturbations over finite fields

“…Furthermore, in this article we use the term exponential sums to refer to both (1.1) and (1.2). In [11], closed formulas for exponential sums of type (1.1) of elementary symmetric polynomials were found (extending the results of [3] to every finite field). There is a natural connection between the formulas presented in [11] and the value distribution of elementary symmetric polynomials over F q .…”

“…The exponential sum over F q of F is defined as where ξ p = exp(2πi/p) and Tr = Tr Fq/Fp is the field trace function. These exponential sums have been extensively studied when the characteristic of the field is 2 because of their cryptographic applications, see [3,4,5,8,9,12,13,24]. Recently, some cryptographic applications when the characteristic of the field is different than 2 has been found.…”