#### Abstract

Finding square roots in the modular integers is a well known problem that is the basis for many modern cryptosystems. For primes of the form p=4n+3, given C\in\mathbb Z_p^\times , finding solutions to x^2\equiv C \pmod{p} is deterministic. For primes of the form p=4n+1, no known deterministic computation exists for determining x given C. Tonelli (later improved by Shanks,) Cipolla, and Pocklington, among others, found sophisticated algorithms to perform this task. Brute force is a transparent approach, but offers no insights into the problem. In this thesis, we produce a transparent approach to this problem, visualized using a model built on Symplectic Geometry. One of the insights from viewing the problem in this way is a conjecture on the distribution of quadratic residues, which we exploit in our algorithm. Even though the conjecture is not essential to the workings of the algorithm, it gives it an edge over brute force for large enough primes. Finally, we follow this with examples of the algorithm's execution.

#### Library of Congress Subject Headings

Square root; Algorithms

#### Publication Date

10-29-2018

#### Document Type

Thesis

#### Student Type

Graduate

#### Degree Name

Applied and Computational Mathematics (MS)

#### Department, Program, or Center

School of Mathematical Sciences (COS)

#### Advisor

Manuel Lopez

#### Advisor/Committee Member

Anurag Agarwal

#### Advisor/Committee Member

James Marengo

#### Recommended Citation

Spink, Michael R., "A Transparent Square Root Algorithm to Beat Brute Force for Sufficiently Large Primes of the Form p = 4n + 1" (2018). Thesis. Rochester Institute of Technology. Accessed from

https://scholarworks.rit.edu/theses/9905

#### Campus

RIT – Main Campus

#### Plan Codes

ACMTH-MS