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

One class genera of ternary quadratic forms over number

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Verifiable delay functions from supersingular isogenies and pairings

International audienceWe present two new Verifiable Delay Functions (VDF) based on assumptions from elliptic curve cryptography. We discuss both the advantages and drawbacks of our constructions, we study their security and we demonstrate their practicality with a proof-of-concept implementation