Fourteenth Algorithmic Number Theory Symposium, ANTS-XIV,
University of Auckland, New Zealand
June 29 - July 4, 2020
- Andrew Booker (Bristol, UK) 33 and all that
Abstract: I will speak about sums of three cubes, while trying to convey the surreal experience of seeing something that I did go viral. If time permits I will describe on-going follow-up work, joint with Andrew Sutherland.
- David Harvey (UNSW, Australia) Recent results on fast multiplication
Abstract: Joris van der Hoeven and I recently proposed an algorithm that multiplies two $n$-bit integers in $O(n \log n)$ bit operations. I will explain the main ideas behind this algorithm, and discuss the difficulties that arise when trying to adapt this algorithm to multiplication of polynomials in $F[x]$ where $F$ is a finite field.
- Isabel Vogt (Stanford, USA) Arithmetic and geometry of Brill--Noether loci of curves
Abstract: Given an abstract curve C, the explicit realizations of C in projective spaces are parameterized by the Brill--Noether loci of C. In this talk, we will explore some natural questions about the geometry and arithmetic of Brill--Noether loci.
- Rachel Pries (Colorado State University, USA) Principal polarizations and Shimura data for families of cyclic covers of the projective line
Abstract: Consider a family of degree m cyclic covers of the projective line, with any number of branch points and inertia type. The Jacobians of the curves in this family are abelian varieties having an automorphism of order m with a prescribed signature. For each such family, the signature determines a PEL-type Shimura variety. Under a condition on the class number of m, we determine the Hermitian form and Shimura datum of the component of the Shimura variety containing the Torelli locus. For the proof, we study the boundary of Hurwitz spaces, investigate narrow class numbers of real cyclotomic fields, and build on an algorithm of Van Wamelen about principal polarizations on abelian varieties with complex multiplication. This is joint work with Li, Mantovan, and Tang.
- David Jao (Waterloo, Canada) Isogeny-based cryptography: past, present, and future
Abstract: Isogeny-based cryptography is becoming an increasingly well-established subject within the post-quantum cryptography landscape, but widespread understanding remains elusive due to the technical and specialized nature of the topic. In this talk we present a clear comparison of modern-day isogeny-based cryptosystems, explain the known relationships between the various security assumptions used in isogeny-based cryptography today, and give algorithmic details of the known attacks against these assumptions. We also provide our thoughts regarding what kinds of isogeny-based cryptosystems and security assumptions may emerge in the future.
- Felipe Voloch (Canterbury, New Zealand)
Commitment Schemes and Diophantine Equations
Abstract: Motivated by questions in cryptography, we look for diophantine equations that are hard to solve but for which determining the number of solutions is easy.