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 GL(4)-type and their conjectural connection to Siegel paramodular forms.

Here you will find a growing (and at times, incomplete) list of my projects. You will also 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.

My email address is: enricflorit at ub dot edu.

Some links:

• The SIMBa Seminar: informal mathematics seminar in Barcelona
• My github and gitlab accounts

Publications and preprints

• Fité, F.; Florit, E.; Guitart, X. Abelian varieties genuinely of GL_n-type. Preprint [arXiv]
  Abstract.

  A simple abelian variety A defined over a number field k is called of GL_n-type if there exists a number field of degree 2dim(A)/n which is a subalgebra of End⁰(A). We say that A is genuinely of GL_n-type if its base change A_k contains no isogeny factor of GL_m-type for m<n. This generalizes the classical notion of abelian variety of GL₂-type without potential complex multiplication introduced by Ribet. We develop a theory of building blocks, inner twists and nebentypes for these varieties. When the center of End⁰(A) is totally real, Chi, Banaszak, Gajda, and Krasoń have attached to A a compatible system of Galois representations of degree n which is either symplectic or orthogonal. We extend their results under the weaker assumption that the center of End⁰(A_k) be totally real. We conclude the article by showing an explicit family of abelian fourfolds genuinely of GL₄-type. This involves the construction of a family of genus 2 curves defined over a quadratic field whose Jacobian has trivial endomorphism ring and is isogenous to its Galois conjugate.

• Florit, E.; Pacetti, A. K-varieties and Galois representations. Preprint [arXiv]
  Abstract.

  In a remarkable article Ribet showed how to attach rational 2-dimensional representations to elliptic ℚ-curves. An abelian variety A is a (weak) K-variety if it is isogenous to all of its GalK-conjugates. In this article we study the problem of attaching an absolutely irreducible ℓ-adic representation of GalK to an abelian K-variety, which sometimes has smaller dimension than expected. When possible, we also construct a Galois-equivariant pairing, which restricts the image of this representation. As an application of our construction, we prove modularity of abelian surfaces over ℚ with potential quaternionic multiplication.

• Florit, E. Abelian varieties that split modulo all but finitely many primes. Proc. Amer. Math. Soc. 153 (2025), 1491-1499 [arXiv, journal]
  Abstract.

  Let A be a simple abelian variety over a number field k such that End(A) is noncommutative. We show that A splits modulo all but finitely many primes of k. We prove this by considering the subalgebras of End(A_p)⊗Q which have prime Schur index. Our main tools are Tate's characterization of endomorphism algebras of abelian varieties over finite fields, and a Theorem of Chia-Fu Yu on embeddings of simple algebras.

• Gàmez-Montolio, A; Florit, E; Brain, M; Howe, J. M. Efficient Normalized Reduction and Generation of Equivalent Multivariate Binary Polynomials. Workshop on Binary Analysis Research (BAR 2024) [proceedings, eprint]
  Abstract.

  Polynomials over fixed-width binary numbers (bytes, Z/2^w Z, bit-vectors, etc.) appear widely in computer science including obfuscation and reverse engineering, program analysis, automated theorem proving, verification, error-correcting codes and cryptography. As some fixed-width binary numbers do not have reciprocals, these polynomials behave differently to those normally studied in mathematics. In particular, polynomial equality is harder to determine; polynomials having different coefficients is not sufficient to show they always compute different values. Determining polynomial equality is a fundamental building block for most symbolic algorithms. For larger widths or multivariate polynomials, checking all inputs is computationally infeasible. This paper presents a study of the mathematical structure of null polynomials (those that evaluate to 0 for all inputs) and uses this to develop efficient algorithms to reduce polynomials to a normalized form. Polynomials in such normalized form are equal if and only if their coefficients are equal. This is a key building block for more mathematically sophisticated approaches to a wide range of fundamental problems.

• Fité, F.; Florit, E.; Guitart, X. Endomorphism algebras of geometrically split genus 2 Jacobians over Q. Preprint [arXiv]
  Abstract.

  The main result of [FG20] classifies the 92 geometric endomorphism algebras of geometrically split abelian surfaces defined over Q. We show that 54 of them arise as geometric endomorphism algebras of Jacobians of genus 2 curves defined over Q, and that the remaining 38 do not. In particular, we exhibit 38 examples of Qbar-isogeny classes of abelian surfaces defined over Q that do not contain any Jacobian of a genus 2 curve defined over Q, and for each of the 54 algebras that do arise we exhibit a curve realizing them.

• Florit, E.; Smith, B. An atlas of the Richelot isogeny graph (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties). RIMS Kôkyûroku Bessatsu 2022, B90: 195-219. [proceedings, arXiv]
  Abstract.

  We describe and illustrate the local neighbourhoods of vertices and edges in the (2, 2)-isogeny graph of principally polarized abelian surfaces, considering the action of automorphisms. Our diagrams are intended to build intuition for number theorists and cryptographers investigating isogeny graphs in dimension/genus 2, and the superspecial isogeny graph in particular.

• Florit, E.; Smith, B. Automorphisms and Isogeny Graphs of Abelian Varieties, with Applications to the Superspecial Richelot Isogeny Graph. In Arithmetic, Geometry, Cryptography, and Coding Theory 2021, 779:103-32. Contemp. Math. Amer. Math. Soc., 2022. [proceedings, arXiv]
  Abstract.

  We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.

Talks

• July 2024. Abelian varieties that split modulo all but finitely many primes. Décimas Jornadas de Teoría de Números, Madrid (slides).
• February 2024. On simple reductions of abelian varieties. 37th Seminari de Teoria de Nombres de Barcelona.
• November 2023. Isogeny graphs and isogeny cryptography. Seminário do Grupo de Álgebra e Geometria, Universidade de Aveiro.
• September 2022. How to find a stationary distribution. Talk given at a workshop at the Facultat de Matemàtiques i Informàtica, where I explained the motivation behind the two papers with Ben Smith. You can see the slides here (pdf).
• June 2022. Abelian varieties of GL(4) type. Novenas Jornadas de Teoría de Números.
• May 2022. Let's twist again: the problem with fake abelian surfaces. SIMBa Seminar.
  Abstract.

  The paramodularity conjecture of Brumer and Kramer proposes a connection between abelian surfaces and Siegel paramodular forms. In this correspondence there are some restrictions, for example, we ask that all the eigenvalues of a paramodular form are in Q. In theory, these forms could also correspond to fake surfaces, which are abelian fourfolds with quaternionic multiplication. This motivates the study of abelian varieties whose l-adic representations have dimension 4, which is the case of abelian surfaces.
  In this talk, I will explain the objects in the paramodularity conjecture: my goal is to put emphasis in the classification of abelian varieties of GL(4)-type, and in particular the ones that have similar representations to those of abelian surfaces. I will also present the examples we currently have at our disposal. Finally I will explain the difficulty in working with fake surfaces.

• 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. 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. 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).

• November 2019. Postquantum Cryptography: what, why, and how? 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.

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

  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).

Other writing

• October 2021. Random walks on supersingular isogeny graphs. Reports@SCM.
  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, Extended abstract with Gerard Finol.
  Abstract.

  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.
  Abstract.

  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 GL(2)-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 at some point 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. 3rd BYMAT. Poster with Gerard Finol.
  Abstract.

  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.

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

  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.
  Update: the field has advanced a lot since I did this project, in particular, SIDH has been completely broken. If you are interested in this, look for the papers by Castryck, Decru, Maino, Martindale and Robert.

  • 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.

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

  I had the opportunity to attend 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.

Seminars

• Serre's open image theorem, Spring 2025.
• Rational points on curves of high genus, Spring 2024.
• Learning seminar on Quadratic Chabauty, Spring 2023.
• Learning seminar on Automorphic Forms, Spring 2022.