Using easy coefficients conjecture for rotation symmetric Boolean functions

Abstract

A Boolean function in n variables is rotation symmetric (RS) if it is invariant under powers of ρ(x1,…,xn)=(x2,…,xn,x1). An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial. Completing earlier research on special cases, the author showed in 2018 that for any RS function fn in n variables, the sequence of Hamming weights wt(fn) for all values of n satisfies a linear recurrence relation. It was also proved that the associated recursion polynomial could be explicitly calculated as the minimal polynomial of a rules matrix and an algorithm for computing the rules matrix was explained. Examples showed that the usual formula which gives the values of wt(fn)−2n−1 as a linear combination of powers of the irrational roots of the minimal polynomial has simple coefficients which are all 1/2 if the multiset made up of the roots of the characteristic polynomial of the rules matrix is used instead, no matter what the degree of the Boolean function is. The conjecture that this is always true is called the Easy Coefficients Conjecture (ECC). The special case of MRS quadratics was proved in an earlier paper. The present paper gives some ECC applications valid for any function for which the ECC is true, even though there is a proof only for the quadratic MRS case so far.