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 on the field p, and mp on p2. 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 p2. 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 p that computes ƒp.