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Improved algorithms of post-quantum cryptographic group actions

par Gilchrist, Valerie
Président du jury Dricot, Jean-Michel
Promoteur Petit, Christophe
Publication Non publié, 2025-11-25

Thèse de doctorat

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Résumé :

Our modern world is dependent on online communications and transactions. To ensure privacy in these digital actions, it is extremely common to use protocols such as the Diffie-Hellman key exchange scheme. Unfortunately, these systems have been shown to be vulnerable to attacks from a quantum adversary. This means that an overhaul of the field of cryptography is needed in order to anticipate and mitigate this threat.Over the past years, the community has begun to answer this call by proposing and analyzing new techniques that are conjectured to be quantum safe. One such example is the study of cryptographic group actions. These mappings mimic some of the structure from the Diffie-Hellman protocol,but abstract it enough so that new mathematical operations can be employed that avoid quantum attacks. Currently, the main examples of group actions come from isogenies, codes, and lattices.In order for a protocol to be used in the real-world we would hope for it to be secure, fast, and requiring little storage. This means that in these early stages of research and development of cryptographic group actions, it is essential to begin to answer some of the questions surrounding security and efficiency. This thesis contributes to this effort by focusing on these aspects across different group actions that have been proposed for use in the literature.It is certainly true that each individual group action needs its own study of security. When designers are trying to achieve some advanced functionalities in a protocol, they may sometimes alter the underlying security assumptions.Each time this is done, the new assumptions need their own careful study as well. In this thesis we consider four different variants of group action security assumptions spanning the isogeny, tensor, linear code, and matrix code group actions. We analyze their security by either offering concrete attacks or giving a polynomial time reduction to a more well-studied assumption. This helps to indicate whether a problem is appropriate for use in cryptography, and what parameters to use if so.In addition to security, this thesis is also inscribed in the effort to study the efficiency of group actions by including three works studying the isogeny group action and related subroutines. They span the core isogeny problem for oriented elliptic curves, evaluating isogenies from a kernel generator, and supersingularity testing. With an effort from the community to improve overall runtimes, we can hope to get closer to reasonable speeds for real-life use.By studying the security and efficiency of cryptographic group actions, this thesis shines light on group actions as a promising framework for post-quantum cryptographic solutions.