Hachage vers les courbes elliptiques et cryptanalyse de schémas RSA

This thesis consists of two independent parts, devoted to both aspects of cryptology: construction and analysis.

Contributions to cryptography proper, on the one hand, address open questions in algebraic curve-based cryptography, particularly the problem of encoding and hashing to elliptic curves. We derive some quantitative results on curve-valued encoding functions, and give a satisfactory construction of hash functions based on those encodings, using a range of mathematical techniques from function field arithmetic, the algebraic geometry of curves and surfaces, and character sums. We also worked on a more implementation-related problem in elliptic curve cryptography, namely the construction of fast addition and doubling formulas.

Our cryptanalytic work, on the other hand, focuses on RSA-based cryptosystems—mostly encryption and signature schemes. We have obtained and carried out new attacks on standardized padding schemes that remain in widespread use, including ISO/IEC 9796-2 for signatures and PKCS#1 v1.5 for encryption. We also propose new physical fault attacks on RSA signature schemes using the Chinese Remainder Theorem, and a stronger attack on RSA schemes relying on small hidden-order subgroups. The tools involved include index calculus, lattice reduction techniques and efficient arithmetic of large degree polynomials.