Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented

Easy decision-Diffie-Hellman groups

The decision-Diffie-Hellman problem (DDH) is a central computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. Distortion maps are an important tool for solving DDH problems using pairings and it is known that distortion maps exist for all supersingular elliptic curves. We present an algorithm to construct suitable distortion maps. The algorithm is efficient on the curves usable in practice, and hence all DDH problems on these curves are easy. We also discuss the issue of which DDH problems on ordinary curves are easy

Cryptanalysis of an oblivious PRF from supersingular isogenies

We cryptanalyse the SIDH-based oblivious pseudorandom function from supersingular isogenies proposed at Asiacrypt’20 by Boneh, Kogan and Woo. To this end, we give an attack on an assumption, the auxiliary one-more assumption, that was introduced by Boneh et al. and we show that this leads to an attack on the oblivious PRF itself. The attack breaks the pseudorandomness as it allows adversaries to evaluate the OPRF without further interactions with the server after some initial OPRF evaluations and some offline computations. More specifically, we first propose a polynomial-time attack. Then, we argue it is easy to change the OPRF protocol to include some countermeasures, and present a second subexponential attack that succeeds in the presence of said countermeasures. Both attacks break the security parameters suggested by Boneh et al. Furthermore, we provide a proof of concept implementation as well as some timings of our attack. Finally, we examine the generation of one of the OPRF parameters and argue that a trusted third party is needed to guarantee provable security.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

Isogeny graphs with maximal real multiplication

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We fully describe the isogeny graphs in that case. Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2

Fast Arithmetics in Artin-Schreier Towers over Finite Fields

An Artin-Schreier tower over the finite field F_p is a tower of field extensions generated by polynomials of the form X^p - X - a. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.Comment: 28 pages, 4 figures, 3 tables, uses mathdots.sty, yjsco.sty Submitted to J. Symb. Compu