The integer factorization and discrete logarithm problems are cornerstones of several public-key cryptography algorithms. In the realm of algorithms targeted at solving these highly difficult challenges, the number field sieve algorithm and its siblings are of utmost importance. Part 1 of this work presents the family of algorithms around the number field sieve, together with several personal contributions in this research area. Other works are detailed in part 2, notably in relationship with the discrete logarithm problem on Jacobians of curves, and my contributions to this problem in some special cases. Some aspects of my contributions on sparse linear algebra in finite fields, motivated by the aforementioned algorithms, are discussed in part 3 of this work. Part 4 covers my research on computer arithmetic, and in particular efficient arithmetic for binary polynomials. Parts 3 and 4 of this work emphasize a strong connection with the goal of efficient implementation.
