The computational efficacy of finite-field arithmetic

https://doi.org/10.1016/0304-3975(93)90022-LGet rights and content
Under an Elsevier user license
open archive

Abstract

We investigate the computational power of finite-field arithmetic operations as compared to Boolean operations. We pursue this goal in a representation-independent fashion. We define a good representation of the finite fields to be essentially one in which the field arithmetic operations have polynomial-size Boolean circuits. We exhibit a function ƒp on the prime fields with two properties: first, ƒp has a polynomial-size Boolean circuit in any good representation, i.e. ƒp is easy to compute with general operations; second, any function that has polynomial-size Boolean circuits in some good representation also has polynomial-size arithmetic circuits if and only if ƒp has polynomial-size arithmetic circuits. Informally, ƒp is the hardest function to compute with arithmetic that has small Boolean circuits.

We reduce the function ƒp to the pair of functions gp = k=1p−1xkk on the field Fp, and mp on Zp2. Here mp is the “modulo p” function defined in the natural way. We show that ƒp has polynomial-size arithmetic circuits if and only if gp and mp have polynomial-size arithmetic circuits, the latter being arithmetic circuits over the ring Zp2. Finally, we establish a connection of ƒp and mp with the Bernoulli polynomials and determine the coefficients of the unique degree p − 1 polynomial over Fp that computes ƒp.

Cited by (0)