The computational efficacy of finite-field arithmetic

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 ƒ p on the prime fields with two properties: first, ƒ p has a polynomial-size Boolean circuit in any good representation, i.e. ƒ 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 ƒ p has polynomial-size arithmetic circuits. Informally, ƒ p is the hardest function to compute with arithmetic that has small Boolean circuits. We reduce the function ƒ p to the pair of functions g p = ∑ k=1 p−1 x k k on the field F p , and m p on Z p 2 . Here m p is the “modulo p” function defined in the natural way. We show that ƒ p has polynomial-size arithmetic circuits if and only if g p and m p have polynomial-size arithmetic circuits, the latter being arithmetic circuits over the ring Z p2 . Finally, we establish a connection of ƒ p and m p with the Bernoulli polynomials and determine the coefficients of the unique degree p − 1 polynomial over F p that computes ƒ p .