The endomorphism ring problem and supersingular isogeny graphs

Supersingular isogeny graphs, which encode supersingular elliptic curves and their isogenies, have recently formed the basis for a number of post-quantum cryptographic protocols. The study of supersingular elliptic curves and their endomorphism rings has a long history and is intimately related to the study of quaternion algebras and their maximal orders. In this thesis, we give a treatment of the theory of quaternion algebras and elliptic curves over finite fields as these relate to supersingular isogeny graphs and computational problems on such graphs, in particular, consolidating and surveying results in the research literature. We also perform some numerical experiments on supersingular isogeny graphs and establish a number of refined upper bounds on supersingular elliptic curves with small non-integer endomorphisms

On Isogeny Graphs of Supersingular Elliptic Curves over Finite Fields

We study the isogeny graphs of supersingular elliptic curves over finite fields, with an emphasis on the vertices corresponding to elliptic curves of $j$-invariant 0 and 1728

Ramanujan graphs in cryptography

In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Pl\^{u}t proposed a cryptographic system based on Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.Comment: 33 page

Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos

We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields

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