Elliptic Curve Discrete Logarithm Problem over Small Degree Extension Fields. Application to the Static Diffie-Hellman Problem on E(F q 5 )

Antoine Joux (1) , Vanessa Vitse (2)

1 PRISM - Parallélisme, Réseaux, Systèmes, Modélisation
2 IF - Institut Fourier

In 2008 and 2009, Gaudry and Diem proposed an index calculus method for the resolution of the discrete logarithm on the group of points of an elliptic curve defined over a small degree extension field Fqn. In this paper, we study a variation of this index calculus method, improving the overall asymptotic complexity when n = Ω(3 log 2 q). In particular, we are able to successfully obtain relations on E(F q 5), whereas the more expensive computational complexity of Gaudry and Diem's initial algorithm makes it impractical in this case. An important ingredient of this result is a variation of Faugère's Gröbner basis algorithm F4, which significantly speeds up the relation computation. We show how this index calculus also applies to oracle-assisted resolutions of the static Diffie-Hellman problem on these elliptic curves.

Mots clés: elliptic curve; discrete logarithm problem (DLP); index calculus; Gröbner basis computation; summation polynomials; static Diffie-Hellman problem (SDHP)

Domaines: Cryptographie et sécurité [cs.CR]; Théorie des nombres [math.NT]

Antoine Joux, Vanessa Vitse. Elliptic Curve Discrete Logarithm Problem over Small Degree Extension Fields. Application to the Static Diffie-Hellman Problem on E(F q 5 ). Journal of Cryptology, 2013, 26 (1), pp.119-143. ⟨hal-01981543⟩