Public key Cryptography
We are involved in the design of new public key cryptosystems.Some of them are based on modular arithmetic and have homomorphic properties which are useful in some applications (databases, electronic voting). Others are based on quadratic fields or error correcting codes.
Linear Feedback Shift Registers (LFSR) are basic tools widely used in the design of stream ciphers. We study the boolean functions used to extract enciphered data from such registers. Feedback with Carry Shift Registers (FCSR) are similar tools but their non-linear structure can be described using 2-adic integers. We have designed several stream ciphers based on FCSR and we continue to improve them. We study the mathematical properties of FCSR and use them to evaluate the security of these stream ciphers.
Error Correcting Codes
Error correcting are used in all communications to correct the errors introduced by the ambiant noise. We study mathematical properties of error correcting codes in order to find better ones. We are also interrested in the use of error correcting codes in public key ciphers and signature.
Number Theory provides functions which are difficult to compute andare useful in public key cryptography. We study different structures which can provide such functions : binary quadratic forms, elliptic curves, non-abelian groups, combinatoric and word geometry, p-adic numbers, linear recurring sequences.