The Structure and Degrees of Polynomials Computing Square Roots $\mod p$

TL;DR

This paper investigates the structure and minimal degrees of polynomials over finite fields that compute square roots, focusing on differences between primes congruent to 1 or 3 modulo 4.

Contribution

It characterizes the degrees and structures of polynomials computing square roots over finite fields, especially for primes where existing methods are not minimal.

Findings

For primes p ≡ 3 mod 4, the polynomial X^{(p+1)/4} is minimal.

The paper provides new insights into polynomial degrees for primes p ≡ 1 mod 4.

It explores the algebraic structure of polynomials computing square roots.

Abstract

For an odd prime $p$, we say a polynomial $f\in \mathbb F_p[X]$ computes square roots if $f(a)^2=a$ for all nonzero, perfect squares $a\in \mathbb F_p$. When $p\equiv 3 \mod 4$, it is easy to see that $f(X)=X^{\frac{p+1}{4}}$ is the smallest such polynomial. For $p\equiv 1 \mod 4$, the situation is less clear. Tonelli-Shanks offers an algorithm for constructing polynomials that compute square roots, but the question of whether their degree is minimal remains. In this paper, we study the various degrees and structures of polynomials computing square roots.