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Square Root Computation In Finite Fields

Ebru Adiguzel-Goktas, Enver Ozdemir

TL;DR

A unifying framework where each of these well-known algorithms can be seen as a special case of it is described, offering a broad perspective to compare and further improve the existing methods in addition to offering a new avenue for square root computation algorithms in finite fields.

Abstract

In this paper, we present a review of three widely-used practical square root algorithms. We then describe a unifying framework where each of these well-known algorithms can be seen as a special case of it. The framework with singular curves offers a broad perspective to compare and further improve the existing methods in addition to offering a new avenue for square root computation algorithms in finite fields.

Square Root Computation In Finite Fields

TL;DR

A unifying framework where each of these well-known algorithms can be seen as a special case of it is described, offering a broad perspective to compare and further improve the existing methods in addition to offering a new avenue for square root computation algorithms in finite fields.

Abstract

In this paper, we present a review of three widely-used practical square root algorithms. We then describe a unifying framework where each of these well-known algorithms can be seen as a special case of it. The framework with singular curves offers a broad perspective to compare and further improve the existing methods in addition to offering a new avenue for square root computation algorithms in finite fields.
Paper Structure (12 sections, 6 theorems, 36 equations, 1 table, 4 algorithms)

This paper contains 12 sections, 6 theorems, 36 equations, 1 table, 4 algorithms.

Table of Contents

  1. Square Roots Algorithms
  2. Tonelli-Shanks Algorithm
  3. Cipolla's Algorithm
  4. Peralta's Algorithm
  5. Geometric Analogue of Algorithms
  6. Square Root Algorithm With Curves:
  7. Computing In An Elliptic Curve Group
  8. Affine Coordinates:
  9. Projective Coordinates:
  10. Singular Curves
  11. Tonelli-Shanks and Peralta's Algorithms with Singular Curves
  12. Cipolla's Algorithm with Singular Curves

Key Result

Theorem 1.1

The element $b=(t+w)^{p+1/2}$ lies in $\mathbb{F}_p$ and it is a square root of $a$.

Theorems & Definitions (15)

  • Theorem 1.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 5 more