Project: 18.783 Spring 2021

Algorithm to compute binary quadratic form corresponding to a prime ideal in an imaginary quadratic field.

Algebraic proof of the associativity of the elliptic curve group law on curves defined by a short Weierstrass equation, as presented in Lecture 2 of 18.783

Helper functions for Problem Set 12 of 18.783

Schoof's algorithm for counting points on elliptic curves, as presented in Lecture 8 of 18.783

Code to compute and verify division polynomials of elliptic curves, as presented in Lecture 5 of 18.783

Demonstration of Rabin's root-finding algorithm and the Cantor-Zassenhaus algorithm for factoring polynomials in finite fields, as presented in Lecture 3 of 18.783

Implementation of the Elliptic curve factorization method (ECM) using the Montgomery ladder for scalar multiplication.

Simple implementation of Pollard p-1 algorithm for factoring integers, as presented in Lecture 10 of 18.783

Simple index calculus algorithm for computing discrete logarithms, as presented in Lecture 10 of 18.783

Simple algorithm to enumerate neighbors in the ell-isogeny graph of an elliptic curve over a finite field, used in Problem Set 11 of 18.783

Algorithm to compute solutions (t,v) to the norm equation 4p = t^2-v^D, where D is the discriminant of an order in an imaginary quadratic field, used in Problem Set 11 of 18.783

Implementation of Schoof's algorithm as described in Lecture 9

Code to symbolically compute division polynomials, both directly from the group law, and via recurrences.