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We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero.

- Andrew Shallue
- ANTS
- 2008

Given sets L1, . . . , Lk of elements from Z/mZ, the k-set birthday problem is to find an element from each list such that their sum is 0 modulo m. We give a new analysis of the algorithm in [16], proving that it returns a solution with high probability. By the work of Lyubashevsky [10], we get as an immediate corollary an improved algorithm for the randomâ€¦ (More)

- Andrew Shallue
- 2007

This thesis contains work on two problems in algorithmic number theory. The first problem is to give an algorithm that constructs a rational point on an elliptic curve over a finite field. A fast and easy randomized algorithm has existed for some time. We prove that in the case where the finite field has characteristic 2, there is a deterministic algorithmâ€¦ (More)

- W. R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue
- Math. Comput.
- 2014

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the propertyâ€¦ (More)

- Eric Bach, Andrew Shallue
- Math. Comput.
- 2015

- Andrew Shallue
- ACM Trans. Algorithms
- 2016

This article explores the asymptotic complexity of two problems related to the Miller-Rabin-Selfridge primality test. The first problem is to tabulate strong pseudoprimes to a single fixed base <i>a</i>. It is now proven that tabulating up to <i>x</i> requires <i>O</i>(<i>x</i>) arithmetic operations and <i>O</i>(<i>x</i>log <i>x</i>) bits of space.â€¦ (More)

- Andrew Shallue
- ArXiv
- 2012

Given positive integers a1, . . . , an, t, the fixed weight subset sum problem is to find a subset of the ai that sum to t, where the subset has a prescribed number of elements. It is this problem that underlies the security of modern knapsack cryptosystems, and solving the problem results directly in a message attack. We present new exponential algorithmsâ€¦ (More)

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