Ebook: Abelian Surfaces and Isogeny-based Cryptography

This proceedings volume, Abelian Surfaces and Isogeny-based Cryptography, compiles research papers presented at a conference held in Jerusalem in 2024, supported by a NATO grant. The event brought together mathematicians, cryptographers, and computer scientists to examine the intersection of algebraic geometry and cryptography, with a focus on isogeny-based cryptographic systems. These systems, leveraging the arithmetic of elliptic curves and higher-dimensional abelian varieties such as abelian surfaces, are increasingly vital for post-quantum cryptography. This volume, encompassing the papers [1], [2], [3], [4], and [5], addresses theoretical and practical aspects of abelian surfaces and their cryptographic applications.

Each paper underwent rigorous peer review to meet high academic standards. The contributions span topics such as the arithmetic of abelian surfaces, computational techniques for isogenies, and the design of secure cryptographic protocols. The conference also highlighted connections to broader mathematical areas, such as geometric structures and their symmetries, which inform the development of cryptographic methods. We express gratitude to NATO for their funding, the organizing committee for their efforts, and the authors and reviewers for their commitment to this volume’s quality.

This collection aims to serve as a resource for researchers and to encourage further investigation into isogeny-based cryptography and abelian varieties. The results presented here are intended to contribute to both theoretical advancements and practical solutions for secure communication.

In [1] the authors explore the geometry and computation of (ℓ, ℓ)-isogenies between abelian surfaces, focusing on Jacobians of genus 2 curves and their Kummer surfaces. They establish a comprehensive framework integrating abelian varieties, theta functions, and Kummer surface embeddings, leveraging the Torelli theorem to connect genus 2 curves with their principally polarized Jacobians. Theta functions of level 2 embed Kummer surfaces into P³, enabling both theoretical analysis and practical computations. We generalize Richelot’s (2,2)-isogenies to arbitrary odd ℓ, developing efficient algorithms for computing these isogenies.

In [2] Donagi and Livné study low-dimensional abelian varieties with an action of the quaternion group, realized as Prym varieties from quaternion-Galois covers of curves. They describe a map from a parameter space of such varieties to a Shimura variety, identifying the modular curve Y₀(2)/ω₂ as the Shimura variety in the unramified genus 2 case. Additionally, they establish a connection between these Prym varieties and cubic threefolds with nine nodes, showing that their moduli spaces are isomorphic to the same modular curve, with implications for isogeny-based cryptographic constructions.

In [3] the authors examine the arithmetic of the loci L_n, parameterizing genus 2 curves with (n, n)-split Jacobians over finite fields F_q. They compute rational points |L_n(F_q)| over F₃, F₉, F₂₇, F₈₁, F₅, F₂₅, and F₁₂₅, and derive zeta functions Z(L_n, t) for n = 2, 3. Utilizing these findings, they explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

In [4] Goren and Love present a survey of supersingular isogeny graphs associated to supersingular elliptic curves and their various applications to cryptography. Within limitation of space, they attempt to address a broad audience and make this part widely accessible. For those graphs they also present three recent results and sketch their proofs. They then discuss a generalization to superspecial isogeny graphs associated to superspecial abelian varieties with real multiplication. These graphs were introduced by Charles, Goren and Lauter and so our discussion is brief. Motivated by their cryptographic applications, they prove a general theorem concerning generation of lattices over totally real fields by elements of specified norm. Throughout the paper they have attempted to clarify certain considerations that are either vaguely stated in the literature, or are folklore. This paper will hopefully be useful both to a novice wishing to familiarize themselves with this very active area, and to an expert who may enjoy some vignettes and an overview of some new results.

In [5] the author determines an effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for 3- and 4-torsion divisors on a genus two curve are obtained explicitly in terms of the Mumford coordinates and the x- and y-coordinates of the support of torsion divisors. As a result, n-torsion divisors on a given curve can be computed directly from the division polynomials. Alternatively, these divisors are obtained by solving the Jacobi inversion problem at points of the Jacobian variety of order n.

Acknowledgments: We are deeply grateful to the NATO Science for Peace and Security Programme for their generous support through Grant G6218, which enabled the conference Isogenies of Abelian Varieties held at The Hebrew University of Jerusalem, July 29–31, 2024, organized in collaboration with Oakland University, Rochester, Michigan, and the Einstein Institute of Mathematics, Hebrew University, Jerusalem. This event facilitated critical discussions and collaborations that greatly enriched this volume. We are honored to contribute to the NATO Science for Peace and Security Series - D: Information and Communication Security, and we sincerely thank NATO for their dedication to advancing research in information and communication security.

Editors

Tony Shaska and Shaul Zemel

May 2025

References

[1] Clingher, A., Malmendier, A., & Shaska, T. (2025). Isogenies, Kummer surfaces, and theta functions. In T. Shaska & S. Zemel (Eds.), Abelian Surfaces and Isogeny-based Cryptography. IOS Press

[2] Donagi, R., & Livné, R. (2025). Abelian varieties with quaternion multiplication. In T. Shaska & S. Zemel (Eds.), Abelian Surfaces and Isogeny-based Cryptography. IOS Press

[3] Mello, J., Salami, S., Shaska, E., & Shaska, T. (2025). Rational points and zeta functions on Humbert surfaces with square discriminant. In T. Shaska & S. Zemel (Eds.), Abelian Surfaces and Isogeny-based Cryptography. IOS Press

[4] Goren, E., & Love, J. (2025). Supersingular elliptic curves, quaternion algebras, and applications to cryptography. In T. Shaska & S. Zemel (Eds.), Abelian Surfaces and Isogeny-based Cryptography. IOS Press

[5] Bernatska, J. (2025). Division polynomials in genus two. In T. Shaska & S. Zemel (Eds.), Abelian Surfaces and Isogeny-based Cryptography. IOS Press

The paper discusses geometric and computational aspects associated with (n, n)-isogenies for principally polarized Abelian surfaces and related Kummer surfaces. We start by reviewing the comprehensive Theta function framework for classifying genus-two curves, their principally polarized Jacobians, as well as for establishing explicit quartic normal forms for associated Kummer surfaces. This framework is then used for practical isogeny computations. A particular focus of the discussion is the (n, n)-Split isogeny case. We also explore possible extensions of Richelot’s (2, 2)-isogenies to higher order cases, with a view towards developing efficient isogeny computation algorithms.

In this article we use a Prym construction to study low dimensional abelian varieties with an action of the quaternion group. In special cases we describe the Shimura variety parameterizing such abelian varieties, as well as a map to this Shimura variety from a natural parameter space of quaternionic abelian varieties. Our description is based on the moduli of cubic threefolds with nine nodes, a subject going back to C. Segre, which we study in some detail.

This paper examines the arithmetic of the loci L_n, parameterizing genus 2 curves with (n, n)-split Jacobians over finite fields F_q. We compute rational points |L_n(F_q)| over F₃, F₉, F₂₇, F₈₁, and F₅, F₂₅, F₁₂₅, derive zeta functions Z(L_n, t) for n = 2, 3. Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

This paper contains a survey of supersingular isogeny graphs associated to supersingular elliptic curves and their various applications to cryptography. Within limitation of space, we attempt to address a broad audience and make this part widely accessible. For those graphs we also present three recent results and sketch their proofs. We then discuss a generalization to superspecial isogeny graphs associated to superspecial abelian varieties with real multiplication. These graphs were introduced by Charles, Goren and Lauter and so our discussion is brief. Motivated by their cryptographic applications, we prove a general theorem concerning generation of lattices over totally real fields by elements of specified norm. Throughout the paper we have attempted to clarify certain considerations that are either vaguely stated in the literature, or are folklore. We hope this paper will be useful both to a novice wishing to familiarize themselves with this very active area, and to the expert who may enjoy some vignettes and an overview of some new results.

An effective algebraic method of computing division polynomials in terms of Mumford coordinates is presented. The division polynomials in question define n-torsion divisors on a curve. The proposed method produces division polynomials explicitly, in terms of x, y-coordinates of a given curve. The method is illustrated by an example of a genus two canonical curve, and has a straightforward extension to a canonical hyperelliptic curve of an arbitrary genus. Explicit expressions of the division polynomials which define 3- and 4-torsion divisors in terms of Mumford coordinates, and x-, y-coordinates of the support are proposed. Such an algebraic representation of division polynomials has not been presented in the literature before.