# Project : tanc

## Section: Overall Objectives

### Overall Objectives

TANC is located in the Laboratoire d'Informatique de l'Ă‰cole polytechnique (LIX).

The aim of the TANC project is to promote the study, implementation and use of robust and verifyable asymmetric cryptosystems based on algorithmic number theory.

It is clear from this sentence that we combine high-level mathematics and efficient programming. Our main area of competence and interest is that of algebraic curves over finite fields, most notably the computational aspects of these objects, that appear as a substitute of good old fashioned cryptography based on modular arithmetic. One of the reasons for this change appears to be the key-size that is smaller for an equivalent security. We participate in the recent bio-diversity mood that tries to find substitutes for RSA, in case some attack would appear and destroy the products that employ it.

Whenever possible, we produce certificates (proofs) of validity for the objects and systems we build. For instance, an elliptic curve has many invariants, and their values need to be proved, since they may be difficult to compute.

Our research area comprises:

Fundamental algorithmic arithmetic: we are interested in primality proving algorithms based on elliptic curves (F. Morain being the world leader in this topic), integer factorization, and the computation of discrete logarithms over finite fields. These problems lie at the heart of the security of arithmetic based cryptosystems.

Complex multiplication: the theory of complex multiplication is a meeting point of algebra, complex analysis and algebraic geometry. Its applications range from primality proving to the efficient construction of elliptic cryptosystems.

Algebraic curves over finite fields: the algorithmic problems that we tackle deal with the efficient computation of group laws on Jacobians of curves, evaluation of the cardinality of these objects, and the study of the security of the discrete logarithm problem in such groups. These topics are the crucial points to be solved for potential use in real crypto-products.