# The computational efficacy of finite-field arithmetic

Carl Sturtivant, Gudmund Skovbjerg Frandsen

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

## 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 f{hook}p on the prime fields with two properties: first, f{hook}p has a polynomial-size Boolean circuit in any good representation, i.e. f{hook}p 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 f{hook}p has polynomial-size arithmetic circuits. Informally, f{hook}p is the hardest function to compute with arithmetic that has small Boolean circuits. We reduce the function f{hook}p to the pair of functions gp = ∑ k=1 p-1xk k on the field Fp, and mp on Zp2. Here mp is the "modulo p" function defined in the natural way. We show that f{hook}p 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 f{hook}p and mp with the Bernoulli polynomials and determine the coefficients of the unique degree p - 1 polynomial over Fp that computes f{hook}p.

Original language English (US) 291-309 19 Theoretical Computer Science 112 2 https://doi.org/10.1016/0304-3975(93)90022-L Published - May 10 1993