Enric Florit Zacarías

I studied Mathematics and Computer Engineering at Universitat de Barcelona (2014 - 2020). I did my Master's degree in Advanced Mathematics at Universitat de Barcelona, which I finished in July 2021. During my master's degree I had support from a Màster+UB scholarship at IMUB. I started my PhD at UB in October 2021, funded by an FPU2020 grant from the Spanish Ministry of Universities. I am currently working on abelian varieties of GL4-type and their conjectural connection to Siegel paramodular forms.

Here you will find a growing (and at times, incomplete) list of my projects. In the materials section you'll find some slides for talks I have given. This website is updated once every few months, mostly when I have some new material to share.

A short list of places to go from here:

• The SIMBa Seminar: informal mathematics seminar in Barcelona
• My giving-back-to-the-community, have-fun-with-some-friends project: Hacking Lliure
• My github and gitlab accounts

Publications

• March 2022. Seven small simple groups not previously known to be Galois over Q, with Luis Dieulefait and Núria Vila. Preprint.
• January 2021. An atlas of the Richelot isogeny graph, with Ben Smith. To appear in the RIMS Kôkyûro Bessatsu volume on supersingular curves and abelian varieties. [arXiv, hal]
• December 2020. Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph, with Ben Smith. To appear in the AGC2T 2021 proceedings. [arXiv, hal]

Materials

• October 2021. Random walks on supersingular isogeny graphs ([email protected]).

Abstract: We survey several aspects of supersingular elliptic curves and their isogeny graphs. Isogeny graphs have obtained attention for the last fifteen years due to their uses in quantum-resistant cryptographic protocols. Studying them involves looking at elliptic curves, quaternion algebras, and random walks on (almost) regular graphs, among other topics. In particular, we give the tools necessary to state the Ramanujan property, connecting supersingular curves in characteristic p with modular forms of level p. We also explain the hash function of Charles, Lauter and Goren as an example of application.

• August 2021. Towards a database of isogeny graphs, with Gerard Finol.

  The extended abstract for our poster in the 2020 BYMAT conference has been published in the conference proceedings.

• July 2021. Abelian surfaces, Siegel modular forms, and the Paramodularity Conjecture. Master's Thesis.

  I studied abelian varieties and their associated representations and L-functions, in order to connect them to modular forms. The construction of Eichler and Shimura is treated and the statement of modularity for elliptic curves and GL2-type abelian varieties is given. Then I moved on to studying Siegel paramodular forms and stated the paramodularity conjecture for abelian surfaces with trivial endomorphism ring. The code written to check the first known case of the conjecture has lots of room for improvement, and I want to rewrite it in the following months to improve efficiency (and possibly allow for a full check of paramodularity in level 277). The document can be found here, you can also see my slides.

• November 2020. A database of isogeny graphs, with Gerard Finol.

  We computed genus 1 supersingular isogeny graphs for p up to 30,000 and isogeny degrees 2,3,5,7,11, complementing some of the work done in Adventures in Supersingularland (Arpin et al., 2019). Some tables with the relevant properties are available from isogenies.enricflorit.com, and the full database (including adjacency matrices) is available from Zenodo (DOI 10.5281/zenodo.4304044). We also presented the main ideas as a poster at the third BYMAT conference.

• July 2020. Random walks in genus 2 isogeny graphs, with Ben Smith. ANTS XIV Rump Session (slides).
• June 2020. Superspecial isogeny graphs in genus 2. Talk given at the SIMBa Seminar.

  Abstract: Supersingular isogeny graphs allow us to formulate supposedly post-quantum protocols, based on the fact that we don't have efficient algorithms to find isogenies between curves. These graphs are expanders and Ramanujan; in particular, the stationary distribution of the random walk is (essentially) uniform. After defining these notions, we will talk about the generalization of these graphs with hyperelliptic genus 2 curves, and we will explain some properties appearing in this case and the differences with elliptic curve graphs. You can find the slides here (pdf).

• February 2020. Isogeny-based cryptography. Talk given at the STNB2020 (Barcelona Number Theory Seminar).

  Abstract: The SIDH protocol was presented in 2011 by Jao and De Feo, giving an alternative to the key exchange of Diffie-Hellman which is resistent to quantum cryptanalysis. Its security is based in the difficulty of finding isogenies between two supersingular elliptic curves. The main concept used is quotienting a curve by a finite subgroup, so that private keys are cyclic subgroups of a certain initial curve, and public keys are the respective quotient curves. These are computed with the Vélu formulas, although the computation of high-degree isogenies must be optimized to make the protocol effective. After explaining the protocol and justifying the choice of parameters, we will see two attacks with which we can try to break keys. You can find the slides here (pdf).

• January 2020. Criptografia basada en isogènies. Treball Final de Grau (Bachelor's thesis).

  The topic of my Bachelor's thesis was isogeny-based cryptography. More specifically, I studied (supersingular) elliptic curves, the SIDH protocol and its basic cryptanalysis.

  • The written part (in catalan): http://hdl.handle.net/2445/164891 or here.
  • The presentation slides (also in catalan).
  • Source code (written and documented in english) for the curstom implementation of SIDH and its attacks in Sage. Warning: do not use in production!
  • Isogeny Explorer: Understanding isogeny-based cryptography through visualization. Visualizations of supersingular isogeny graphs. To better understand isogeny graphs and to be able to explain them, I made a way to save isogeny graphs in JSON files for visualization with D3.js.

• November 2019. Postquantum Cryptography: what, why, and how? Talk given at the SIMBa Seminar.

  Abstract: The key agreement scheme proposed by Diffie and Hellman in 1976 relies on the problem of finding discrete logarithms. One can choose appropriate groups where the best algorithms for solving this problem are too slow, such as certain elliptic curves over finite fields. There are already proposed quantum algorithms that break discrete logarithms in polynomial time. For this reason multiple "postquantum" cryptography primitives have appeared in the last years, while trying to find harder computational problems. One of the proposed protocols using elliptic curves is SIDH/SIKE, candidate to the NIST Post-Quantum Cryptography Competition. You can find the slides here (pdf) and a recording of the talk here.

• July 2019. Elliptic curves, Pairings, and the ECDLP.

  I had the opportunity to attend to a Summer School named "Cryptography meets Graph Theory" (more info here) where we were encouraged to present a poster. Mine was about the MOV attack, an algorithm that tries to simplify Discrete Logarithms over Elliptic Curves. You can find the poster here. I also wrote some study notes, although they are very incomplete. The material is fairly technical, but accessible once one has some basic knowledge on groups, arithmetic and elliptic curves.

• April 2019. p-adic attacks on elliptic curves.

  The people at the Overdrive Hacking Conference asked me to do some talk on cryptograpy, and so I started studying about elliptic curve theory and ECC. During the actual talk I didn't have much time to explain the attack, but it is partially detailed in the slides. The main reference for study was Lawrence Washington's "Elliptic Curves: Number theory and Cryptography". You can find the slides here (pdf).