- A study of error floor behavior in QC-MDPC codes (with Sarah Arpin, Tyler Raven Billingsley, Jun Bo Lau, Ray Perlner, and Angela Robinson), in Post-Quantum Cryptography, PQCrypto 2022, Lecture Notes in Computer Science, vol. 13512. DOI: 10.1007/978-3-031-17234-2_5; IACR ePrint: 2022/1043.
We present experimental findings on the decoding failure rate (DFR) of BIKE, a fourth-round candidate in the NIST Post-Quantum Standardization process, at the 20-bit security level. We select parameters according to BIKE design principles and conduct a series of experiments. We directly compute the average DFR on a range of BIKE block sizes and identify both the waterfall and error floor regions of the DFR curve. We then study the influence on the average DFR of three sets C, N, and 2N of near-codewords — vectors of low weight that induce syndromes of low weight — defined by Vasseur in 2021. We find that error vectors leading to decoding failures have small maximum support intersection with elements of these sets; further, the distribution of intersections is quite similar to that of sampling random error vectors and counting the intersections with C, N, and 2N. Our results indicate that these three sets are not sufficient in classifying vectors expected to cause decoding failures. Finally, we study the role of syndrome weight on the decoding behavior and conclude that the set of error vectors that lead to decoding failures differ from random vectors by having low syndrome weight.
- Explicit two-cover descent for genus 2 curves (in collection ANTS XV), Research in Number Theory vol. 8 (2022), no. 67. DOI: 10.1007/s40993-022-00375-0; arXiv:2009.10313 [math.NT].
Given a genus 2 curve C with a rational Weierstrass point defined over a number field, we construct a family of genus 5 curves that realize descent by maximal unramified abelian two-covers of C, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus 2 quintic curves over Q of Mordell–Weil rank 2 or 3 whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases.
- Functional transcendence for the unipotent Albanese map, Algebra & Number Theory vol. 15 (2021), no. 6, pp. 1565–1580. DOI: 10.2140/ant.2021.15.1565; arXiv:1911.00587 [math.NT].
We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax–Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty–Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty–Kim theory to arbitrary number fields.
- Rational points on solvable curves over Q via non-abelian Chabauty (with Jordan S. Ellenberg), Int. Math. Res. Not. 2021. DOI: 10.1093/imrn/rnab141; arXiv:1706.00525 [math.NT].
We study the Selmer varieties of smooth projective curves of genus at least two defined over Q which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov–Tschinkel and Poonen on unramified correspondences, we deduce that any cover of P1 with solvable Galois group, and in particular any superelliptic curve over Q, has only finitely many rational points over Q.
- Higher moments of arithmetic functions in short intervals: a geometric perspective (with Vlad Matei), Int. Math. Res. Not. 2019, no. 21, pp. 6554–6584. DOI: 10.1093/imrn/rnx310; arXiv:1604.02067 [math.NT].
We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field Fq. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the l-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree n in the limit as q → ∞. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.
- Rational points and unipotent fundamental groups (Ph.D. thesis, University of Wisconsin–Madison, June 2018).
We investigate rational points on higher genus curves over number fields using Kim's non-abelian Chabauty method. We provide an exposition of this method, including a brief survey of the literature in the area. In joint work with Ellenberg, we then study the Selmer varieties of smooth projective curves of genus at least two defined over Q which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that the non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov–Tschinkel and Poonen on unramified correspondences, we deduce that any cover of P1 with solvable Galois group, and in particular any superelliptic curve over Q, has only finitely many rational points over Q.
We also present a strategy for generalizing the non-abelian Chabauty method to real number fields: A conjecture on certain transcendence properties of the unipotent Albanese map is formulated in the final two chapters of this thesis, together with a proof that this conjecture allows a generalization of several major results in the non-abelian Chabauty method to curves over a real number field.