On solving systems of diagonal polynomial equations over finite fields

TL;DR

This paper introduces a polynomial-time algorithm for solving systems of diagonal polynomial equations over finite fields with a fixed degree, leading to efficient quantum algorithms for specific algebraic hidden structure problems.

Contribution

It presents a novel polynomial-time algorithm for diagonal polynomial systems over finite fields and applies it to develop quantum algorithms for hidden subgroup and hidden polynomial graph problems.

Findings

Algorithm solves systems with fixed polynomial degree efficiently

Quantum algorithms for certain hidden subgroup problems

Quantum algorithms for hidden polynomial graph problems

Abstract

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant. As a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems: for the hidden subgroup problem in certain semidirect product p-groups of constant nilpotency class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant.