Cryptanalyse de schémas de cryptographie à clé publique

Cryptanalysis of public-key cryptosystems is based on algorithmic and algebraic techniques in number theory. In the first part of this thesis, we present improvements to the LLL algorithm, originally designed by Lenstra, Lenstra, and Lovasz in the eighties to reduce a Euclidean lattice, i.e. to reduce the norm and orthogonalize the basis vectors as possible. We also show how to use this algorithm to reduce rank-2 module lattices in a cyclotomic number field with subfields using symplectic matrices. Indeed, some schemes such as NTRU or Falcon, whose security relies on this difficult problem, have been proposed in post-quantum cryptography and for homomorphic encryption. We also improve linear algebra techniques and propose better algorithms when the matrix is diagonally dominant. These advanced methods allow us to set new records in the computation of number fields: class number, generators of the unit group, generator of a principal ideal. In the second part, we study various classical problems in number theory: we improve algorithms for testing the primality of an integer, and in particular, the cyclotomic test initially proposed by Adleman and developed by Mihailescu. Then, we study different algorithms in a so-called black-box ring model, i.e. we study the number of additions and multiplications in the ring, without looking at how the elements are represented and how the computations are performed in the ring. This allows us in the final chapter, to instantiate these algorithms in different rings in order to propose efficient algorithms for cryptanalysis. In doing so, we are able to more easily distribute the computations of the entire algorithm, while so-called index-calculus algorithms use a linear algebra step which is difficult to parallelize.