Affine equivalence for quadratic rotation symmetric Boolean functions

Let fn( x0, x1,..., xn-1) denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree d in n = d variables and let wt( fn) denote the Hamming weight of this function. Let (0, a1,..., ad-1) n denote the function fn of degree d in n variables generated by themonomial x0xa1... xad-1. Such a function fn is called monomial rotation symmetric (MRS). It was proved in a 2012 paper that for any MRS fn with d = 3, the sequence of weights {wk = wt( fk) : k = 3, 4,...} satisfies a homogeneous linear recursion with integer coefficients. This result was gradually generalized in the following years, culminating around 2016 with the proof that such recursions exist for any rotation symmetric function fn. Recursions for quadratic RS functions were not explicitly considered, since a 2009 paper had already shown that the quadratic weights themselves could be given by an explicit formula. However, this formula is not easy to compute for a typical quadratic function. This paper shows that the weight recursions for the quadratic RS functions have an interesting special form which can be exploited to solve various problems about these functions, for example, deciding exactly which quadratic RS functions are balanced.