Algebraic Methods for Stronger Crypto

2026-07-22

Our field is cryptology. Our overarching objective is to advance significantly the frontiers in design and analysis of high-security cryptography for the future generation. Particularly, we wish to enhance the efficiency, functionality, and, last-but-not-least, fundamental understanding of cryptographic security against very powerful adversaries. Our approach here is to develop completely novel methods by deepening, strengthening and broadening the algebraic foundations of the field. Concretely, our lens builds on the arithmetic codex. This is a general, abstract cryptographic primitive whose basic theory we recently developed and whose asymptotic part, which relies on algebraic geometry, enjoys crucial applications in surprising foundational results on constant communication-rate two-party cryptography. A codex is a linear (error correcting) code that, when endowing its ambient vector space just with coordinate-wise multiplication, can be viewed as simulating, up to some degree, richer arithmetical structures such as finite fields (or products thereof), or generally, finite-dimensional algebras over finite fields. Besides this degree, coordinate-localities for which simulation holds and for which it does not at all are also captured. Our method is based on novel perspectives on codices which significantly widen their scope and strengthen their utility. Particularly, we bring symmetries, computational- and complexity theoretic aspects, and connections with algebraic number theory, -geometry, and -combinatorics into play in novel ways. Our applications range from public-key cryptography to secure multi-party computation. Our proposal is subdivided into 3 interconnected modules: (1) Algebraic- and Number Theoretical Cryptanalysis (2) Construction of Algebraic Crypto Primitives (3) Advanced Theory of Arithmetic Codices

Netherlands

2026-01-15