Section: Application Domains
Cryptography and Cryptanalysis
In the twentyfirst century, cryptography plays two essential roles: it is used to ensure security and integrity of communications and communicating entities. Contemporary cryptographic techniques can be used to hide private data, and to prove that public data has not been modified; to provide anonymity, and to assert and prove public identities. The creation and testing of practical cryptosystems involves

The design and analysis of compact and efficient algorithms to implement those protocols, and to attack their underlying mathematical and computational problems;

The robust implementation of those algorithms in lowlevel software and hardware, and their deployment in the wild.
While these layers are interdependent, GRACE's cryptographic research is focused heavily on the middle layer: we design, implement, and analyze the most efficient algorithms for fundamental tasks in contemporary cryptography. Our “clients”, in a sense, are protocol designers on the one hand, and software and hardware engineers on the other.
F. Morain and B. Smith work primarily on the numbertheoretic algorithms that underpin the current stateoftheart in publickey cryptography (which is used to establish secure connections, and create and verify digital signatures, among other applications). For example, their participation in the ANR CATREL project aims to give a realistic assessment of the security of systems based on the Discrete Logarithm Problem, by creating a free, open, algorithmic package implementing the fastest known algorithms for attacking DLP instances. This will have an extremely important impact on contemporary pairingbased cryptosystems, as well as legacy finite fieldbased cryptosystems. On a more constructive note, F. Morain' elliptic curve point counting and primality proving algorithms are essential tools in the everyday construction of strong publickey cryptosystems, while B. Smith's recent work on elliptic and genus 2 curves aims to improve the speed of curvebased cryptosystems (such as Elliptic Curve Diffie–Hellman key exchange, a crucial step in establishing secure internet connections) without compromising their security.
D. Augot, F. LevyditVehel, and A. Couvreur's research on codes has farreaching applications in codebased cryptography. This is a field which is growing rapidly in importance—partly due to the supposed resistance of codebased cryptosystems to attacks from quantum computing, partly due to the range of new techniques on offer, and partly because the fundamental problem of parameter selection is relatively poorly understood. For example, A. Couvreur's work on filtration attacks on codes has an important impact on the design of codebased systems using wild Goppa codes or algebraic geometry codes, and on the choice of parameter sizes for secure implementations.
Coding theory also has important practical applications in the improvement of conventional symmetric cryptosystems. For example, D. Augot's recent work on MDS matrices via BCH codes gives a more efficient construction of optimal diffusion layers in block ciphers. Here we use combinatorial, nonalgorithmic properties of codes, in the internals of designs of block ciphers.
While coding theory brings tools as above for the classical problems of encryption, authentication, and so on, it can also provide solutions to new cryptographic problems. This is classically illustrated by the use of ReedSolomon codes in secret sharing schemes. Grace is involved in the study, construction and implementation of locally decodable codes, which have applications in quite a few cryptographic protocols : Private Information Retrieval, Proofs of Retrievability, Proofs of Ownership, etc.