Section: New Results

Discrete logarithm in finite fields of small extension degree

Participant : Pierrick Gaudry [contact] .

Together with Razvan Barbulescu (CNRS, IMJ-PRG), Aurore Guillevic and François Morain (Grace project-team), we investigated the discrete logarithm problem in the case of finite fields of the form ${𝔽}_{p^n}$, where $n>1$ is a small integer. We proposed in a preprint — a part of which was accepted to Eurocrypt 2015 — various theoretical and practical improvements [25] :

• new methods for selecting polynomials,

• better (heuristic) asymptotic complexity in the case where $n\approx logp$, and

• use of algebraic number theory to show that in some cases we can skip the Schirokauer maps.

We have adapted Cado-nfs in order to perform a record computation in a field of the form ${𝔽}_{p^2}$, where $p^2$ has 180 digits. To our knowledge, this is the first time that the number field sieve algorithm is used in practice for record-size computations in this type of fields.