Abelian Surfaces and Isogeny-based Cryptography

Isogeny-based cryptography, built on the arithmetic of elliptic curves and higher-dimensional abelian varieties like abelian surfaces, is a cornerstone of post-quantum security. Abelian Surfaces and Isogeny-based Cryptography compiles cutting-edge research papers presented at a landmark conference held at the Einstein Institute of Mathematics, The Hebrew University of Jerusalem, from July 29-31, 2024, supported by the NATO Science for Peace and Security Programme. This event united mathematicians, cryptographers, and computer scientists to explore the interplay of algebraic geometry and cryptography, focusing on isogeny-based cryptographic systems. The volume features five rigorously peer-reviewed papers that advance the arithmetic of abelian surfaces, computational techniques for isogenies, and secure protocol design. Contributions include novel algorithms for (l, l)-isogenies using Kummer surfaces and theta functions, analyses of supersingular isogeny graphs for cryptographic applications, mappings of Prym varieties to Shimura varieties, and efficient computation of division polynomials in genus two. These works connect to broader mathematical structures, such as geometric symmetries and modular curves, enriching cryptographic innovation. This collection is an invaluable resource for researchers in cryptography and algebraic geometry. It seeks to inspire continued exploration of isogeny-based cryptography and abelian varieties, driving theoretical breakthroughs and practical solutions for secure communication in the quantum age.