## Section: Overall Objectives

### Scientific foundations

Grace combines expertise and deep knowledge in algorithmic number theory and algebraic geometry, to build and analyse (public-key) cryptosystems, design new error correcting codes, with real-world concerns like cybersecurity or blockchains (software and hardware implemntations, secure implementations in constrained environments, countermeasures against side channel attacks, white box cryptography).

The foundations of Grace therefore lie in algorithmic number theory (fundamental algorithms primality, factorization), number fields, the arithmetic geometry of curves, algebraic gemoetry and the theory of algebraic codes.

Arithmetic Geometry is the meeting point of algebraic geometry and number theory: the study of geometric objects defined over arithmetic number systems. In our case, the most important objects are curves and their Jacobians over finite fields; these are fundamental to our applications in both coding theory and cryptology. Jacobians of curves are excellent candidates for cryptographic groups when constructing efficient instances of public-key cryptosystems, of which Diffieâ€“Hellman key exchange is an instructive example.

Coding Theory studies originated with the idea of using redundancy in messages to protect them against noise and errors. While the last decade of the 20th century has seen the success of so-called iterative decoding methods, we see now many new ideas in the realm of algebraic coding, with the foremost example being list decoding, (zero knowledge or not) proofs of computation.

Part of the activities of the team are oriented towards post-quantum cryptography, either based on elliptic curves (isogenies) or code-based. Also the team study relevant cryptography for the blockchain arena.

The group is strongly invested in cybersecurity: software security, secure hardware implementations, privacy, etc.