Formulas For The Square Root Modulo p

TL;DR

This paper presents explicit polynomial formulas over finite fields for computing square roots modulo primes of the form 2^k n + 1, providing new algebraic tools for number theory and cryptography.

Contribution

It introduces specific polynomial representations for the square root function modulo primes of special form, expanding algebraic methods in finite field computations.

Findings

Formulas for k=2, 3, 4 cases are provided.

Polynomial representations are constructed over finite fields.

Enhances algebraic techniques for square root computations modulo primes.

Abstract

A method of constructing specific polynomial representations $f(x)$ over the finite field $\mathbb{F}_p$ of the square roots function modulo a prime $p = 2^kn + 1$, $n$ odd, is presented. The formulas for the cases $k = 2$, $3$ and $4$ are given.