Collaborative Calculation and Data Science

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AndrewVSutherland

18.783 Lecture 8: A simple implementation of Schoof's algorithm for counting points on an elliptic curve over a finite field.

Worksheet that accompanies the YouTube video https://www.youtube.com/watch?v=HIiotVxxi8c

Algebraic proof of the associativity of the group law on an elliptic curve using SageMath.

Example implementations of algorithms to search for representations of an integer as a sum of three cubes in the computer algebra systems Pari/GP, Magma, Oscar, and SageMath prepared as part of the PCMI/IAS summer program on Number Theory Informed by Computation.

Implementation of the group law on an Edwards curve

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

Simple SageMath implementation of Schoof's algorithm.

Data associated to Problem 2 of Problem Set 6 on computing the image of Galois

Demonstration of Pollard's p-1 factorization method

Code to compute the semismooth probability function and a table needed to solve Problem 6 on Problem Set 5.

SageMath code to compute and verify division polynomials of elliptic curves

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

18.783 Lecture 10: Simple implementation of the elliptic curve factorization method (ECM) using Montgomery curves

18.783 Lecture 10: Index calculus algorithm for computing discrete logarithms in the Fp*

18.783 Lecture 10: Simple implementation of Pollard's p-1 algorithm for factoring integers

18.783 Lecture 5: Explicit derivation of the n-division polynomial of an elliptic curve using projective Jacobian coordinates.

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.

18.783 Lecture 2 (proof of associativity)

Andrew Sutherland

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