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.