Paper 2012/199
Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm
JeanCharles Faugère, Pierrick Gaudry, Louise Huot, and Guénaël Renault
Abstract
In 2004, an algorithm is introduced to solve the DLP for elliptic curves defined over a non prime finite field $\F_{q^n}$. One of the main steps of this algorithm requires decomposing points of the curve $E(\F_{q^n})$ with respect to a factor base, this problem is denoted PDP. In this paper, we will apply this algorithm to the case of Edwards curves, the wellknown family of elliptic curves that allow faster arithmetic as shown by Bernstein and Lange. More precisely, we show how to take advantage of some symmetries of twisted Edwards and twisted Jacobi intersections curves to gain an exponential factor \(2^{\omega (n1)}\) to solve the corresponding PDP where $\omega$ is the exponent in the complexity of multiplying two dense matrices. Practical experiments supporting the theoretical result are also given. For instance, the complexity of solving the ECDLP for twisted Edwards curves defined over $\F_{q^5}$, with \(q\approx2^{64}\), is supposed to be $\sim$ $2^{160}$ operations in $E(\F_{q^5})$ using generic algorithms compared to \(2^{130}\) operations (multiplication of two $32$bits words) with our method. For these parameters the PDP is intractable with the original algorithm. The main tool to achieve these results relies on the use of the symmetries and the quasihomogeneous structure induced by these symmetries during the polynomial system solving step. Also, we use a recent work on a new algorithm for the change of ordering of Gröbner basis which provides a better heuristic complexity of the total solving process.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 Published elsewhere. Unknown where it was published
 Keywords
 ECDLPIndex CalculusGröbner Basis with symmetries
 Contact author(s)
 louise huot @ lip6 fr
 History
 20130618: last of 2 revisions
 20120413: received
 See all versions
 Short URL
 https://ia.cr/2012/199
 License

CC BY
BibTeX
@misc{cryptoeprint:2012/199, author = {JeanCharles Faugère and Pierrick Gaudry and Louise Huot and Guénaël Renault}, title = {Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm}, howpublished = {Cryptology ePrint Archive, Paper 2012/199}, year = {2012}, note = {\url{https://eprint.iacr.org/2012/199}}, url = {https://eprint.iacr.org/2012/199} }