A Reduction Theory for Codes and Lattices in Cryptography

Codes and lattices are two major mathematical platforms to build quantum-resistant cryptosystems.

Theose cryptosystems will soon be standardized and deployed, replacing historical solutions threatened by Shor's quantum algorithm.These new cryptosystems rely on the hardness of finding short vectors in a code or in a lattice, a computational problem called reduction.

Our confidence in their security relies on the relentless effort of cryptanalysis: quantifying, elucidating and trying to invalidate such hardness assumptions. *My ERC project aims at discovering faster cryptanalytic algorithms and at building better cryptography via the development of a unified reduction theory enabling a systematic transfer of techniques between codes and lattices.* As an underpinning example for this project, I have successfully transferred the seminal LLL reduction algorithm to binary codes, an algorithm still playing a central role in the cryptanalysis of lattices.

This unified theory will also lead to better mathematical and algorithmic abstractions, improve the clarity, generality, and composability of known techniques. Such qualitative enhancements will also help to obtain quantitative ones.

In turn, I will implement these enhancements into open-source libraries, designed for either high-performance or ease-of-use, in order to stimulate further algorithmic exploration by the community.Beyond algorithms, my approach will also create new connections between existing mathematical theories and the hardness of code and lattice problems.

For instance, it leads to consider moduli spaces as a new powerful tool for proving average-case hardness.By its contributions to both the theoretical and practical aspects of the hardness of lattice and code problems, this project will play a key role in the ongoing transition to quantum-resistant cryptography.