Quaternary quadratic lattices over number fields

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q)

Algorithmic enumeration of ideal classes for quaternion orders

We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2.Comment: 39 pages, includes 2 tables; corrections made to Table 8.

Computing automorphic forms on Shimura curves over fields with arbitrary class number

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms.Comment: 15 pages; final submission to ANTS I

Supersingular isogeny graphs and endomorphism rings:reductions and solutions

In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the -isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism ring itself. We also give constructive versions of Deuring’s correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. The reductions are based on heuristics regarding the distribution of norms of elements in quaternion algebras. We show that conjugacy classes of maximal orders have a representative of polynomial size, and we define a way to represent endomorphism ring generators in a way that allows for efficient evaluation at points on the curve. We relate these problems to the security of the Charles-Goren-Lauter hash function. We provide a collision attack for special but natural parameters of the hash function and prove that for general parameters its preimage and collision resistance are also equivalent to the endomorphism ring computation problem.SCOPUS: cp.kinfo:eu-repo/semantics/published37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2018; Tel Aviv; Israel; 29 April 2018 through 3 May 2018ISBN: 978-331978371-0Volume Editors: Nielsen J.B.Rijmen V.Publisher: Springer Verla

The explicit membership problem for discrete free subgroups of PSL(2,R)

Non UBCUnreviewedAuthor affiliation: RWTH Aachen UniversityPostdoctora