Paper 2026/701

Boolean Arithmetic over $\mathbb{F}_2$ from Group Commutators

Abstract

This paper studies efficient realizations of arithmetic over the binary field $\mathbb{F}_2$ in nonabelian groups using only intrinsic group operations, namely multiplication and inversion. The constructions rely on commutators to implement Boolean computation within the group structure. Two complementary approaches are presented: a realization of a universal Boolean gate (NAND) and direct realizations of the field operations XOR and AND. These approaches apply to finite nonabelian simple groups and can be implemented using a small number of group operations. Explicit realizations are provided in the alternating groups $A_5$ and $A_6$. For the smallest nonabelian simple group $A_5$, these constructions achieve state-of-the-art efficiency in the number of group operations.

Note: Added an appendix with hand-optimized realizations of standard Boolean gates over $A_5$ and $S_5$.