Computing the Number of Equivalent Classes on $\mathcal{R}(s,n)/\mathcal{R}(k,n)$

TL;DR

This paper introduces an efficient algorithm to compute the number of affine equivalence classes of Boolean functions within certain quotient spaces, extending previous limits up to 10 variables and improving computational feasibility.

Contribution

The authors develop a novel counting formula and an isomorphic permutation group approach to compute equivalence classes in quotient Reed-Muller spaces for up to 10 variables, surpassing prior limitations.

Findings

Number of classes for $ ext{R}(1,n)$ computed up to 10 variables.

Number of classes for $ ext{R}(s,n)/ ext{R}(k,n)$ computed for $n ext{ up to } 10$.

Significant reduction in computational complexity.

Abstract

Affine equivalent classes of Boolean functions have many applications in modern cryptography and circuit design. Previous publications have shown that affine equivalence on the entire space of Boolean functions can be computed up to 10 variables, but not on the quotient Boolean function space modulo functions of different degrees. Computing the number of equivalent classes of cosets of Reed-Muller code $\mathcal{R}(1,n)$ is equivalent to classifying Boolean functions modulo linear functions, which can be computed only when $n\leq 7$. Based on the linear representation of the affine group $\mathcal{AGL}(n,2)$ on $\mathcal{R}(s,n)/\mathcal{R}(k,n)$, we obtain a useful counting formula to compute the number of equivalent classes. Instead of computing the conjugate classes and representatives directly in $\mathcal{AGL}(n,2)$, we reduce the computation complexity by introducing an isomorphic permutation group $P_n$ and performing the computation in $P_n$. With the proposed algorithm, the number of equivalent classes of cosets of $R(1,n)$ can be computed up to 10 variables. Furthermore, the number of equivalent classes on $\mathcal{R}(s,n)/\mathcal{R}(k,n)$ can also be computed when $-1\leq k< s\leq n\leq 10$, which is a major improvement and advancement comparing to previous methods.