## CryptoDB

### Vanessa Vitse

#### Publications

Year
Venue
Title
2014
EUROCRYPT
2012
EUROCRYPT
2010
EPRINT
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 $\F_{q^n}$. In this paper, we study a variation of this index calculus method, improving the overall asymptotic complexity when $\log q \leq c n^3$. In particular, we are able to successfully obtain relations on $E(\F_{p^5})$, 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 new variation of Faugère's Gröbner basis algorithm F4, which significantly speeds up the relation computation and might be of independent interest. As an application, we show how this index calculus leads to a practical example of an oracle-assisted resolution of the elliptic curve static Diffie-Hellman problem over a finite field on $130$ bits, which is faster than birthday-based discrete logarithm computations on the same curve.
2010
EPRINT
Algebraic cryptanalysis usually requires to find solutions of several similar polynomial systems. A standard tool to solve this problem consists of computing the Gröbner bases of the corresponding ideals, and Faugère's F4 and F5 are two well-known algorithms for this task. In this paper, we present a new variant of the F4 algorithm which is well suited to algebraic attacks of cryptosystems since it is designed to compute Gröbner bases of a set of polynomial systems having the same shape. It is faster than F4 as it avoids all reductions to zero, but preserves its simplicity and its computation efficiency, thus competing with F5.

#### Coauthors

Jean-Charles Faugère (1)
Louise Huot (1)
Antoine Joux (4)
Guénaël Renault (1)