Special Session 40

New Trends in Cryptography

Organizers: Carina Alves (São Paulo State University, Brazil), Marialaura Noce (University of Salerno, Italy)

MSC codes: 94A60

Description:

Most well-known cryptographic protocols (such as RSA, Diffie–Hellman, and elliptic curve–based methods) rely on the structure of commutative groups and are closely related to the computational difficulty of integer factorization and discrete logarithms. In 1994, Shor introduced a quantum algorithm capable of solving these problems in polynomial time. As a result, the cryptographic community has been actively seeking alternative approaches that remain secure in a post-quantum setting. Some candidate solutions have been known for many years, while others are still emerging, driven by new mathematical insights and computational challenges. Lattice-based cryptography is a leading post-quantum approach, grounded on the presumed hardness of problems such as SVP, CVP, and LWE (and variants). These problems, while not always formulated explicitly in group-theoretic terms, still rely on rich algebraic structures and symmetries that naturally interact with group actions, module structures, and noncommutative frameworks. Group theory, and in particular the study of non-abelian groups, provides a rich source of complex and diverse problems for cryptography. Group-based cryptography introduces a new dimension to the cryptographic landscape, offering challenging problems for both group theorists and cryptographers. At the same time, there are strong conceptual and technical connections between group-based approaches and other areas of post-quantum cryptography, such as lattices, codes, and algebraic structures arising from number fields and non-commutative algebras. The aim of this session is to bring together specialists both in cryptography and in group-based cryptography to present an overview of the current state of the art in the field. More broadly, we also aim to include researchers working on lattice-based cryptography and other post-quantum paradigms grounded in hard computational problems from algebra, geometry, and number theory. On the one hand, the speakers will describe cryptographic protocols based on algorithmic problems in group theory; on the other hand, they will present families of groups that have been recently proposed as suitable platforms for cryptosystems. In addition, the program will highlight cryptosystems based on lattices, as well as related tools from coding theory and arithmetic geometry that support modern post-quantum design. This session is intended to consolidate and strengthen research collaborations between Brazil and Italy, fostering long-term scientific exchange and cooperation. The discussions and interactions among our lecturers and attendees are expected to pave the way for collaborative endeavors that will drive innovation and advancement in the realm of group-based cryptography. More generally, we expect these interactions to stimulate new joint projects on post-quantum cryptography, combining group-theoretic and lattice-theoretic perspectives. The meeting is part of discussions and collaborations among the UMI group “Crittografia e Codici”.