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


Document Type


Student Type


Degree Name

Applied and Computational Mathematics (MS)

Department, Program, or Center

School of Mathematical Sciences (COS)


Manuel Lopez

Advisor/Committee Member

Anurag Agarwal

Advisor/Committee Member

James Marengo


RIT – Main Campus

Plan Codes