TY - JOUR

T1 - The computational efficacy of finite-field arithmetic

AU - Sturtivant, Carl

AU - Skovbjerg Frandsen, Gudmund

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1993/5/10

Y1 - 1993/5/10

N2 - 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.

AB - 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.

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U2 - 10.1016/0304-3975(93)90022-L

DO - 10.1016/0304-3975(93)90022-L

M3 - Article

AN - SCOPUS:0027589536

VL - 112

SP - 291

EP - 309

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2

ER -