## Section: Overall Objectives

### Scientific foundations

GRACE has two broad application domains—cryptography and coding
theory—linked by a common foundation in
algorithmic number theory and the geometry of algebraic curves.
In our research, which combines theoretical work
with practical software development,
we use algebraic curves
to *create better cryptosystems*,
to *provide better security assessments*
for cryptographic key sizes,
and to *build the best error-correcting codes*.

Coding and cryptography deal (in different ways) with securing communication systems for high-level applications. In our research, the two domains are linked by the computational issues related to algebraic curves (over various fields) and arithmetic rings. These fundamental number-theoretic algorithms, at the crossroads of a rich area of mathematics and computer science, have already proven their relevance in public key cryptography, with industrial successes including the RSA cryptosystem and elliptic curve cryptography. It is less well-known that the same branches of mathematics can be used to build very good codes for error correction. While coding theory has traditionally had an electrical engineering flavour, recent developments in computer science have shed new light on coding theory, leading to new applications more central to computer science.