# Using Symmetries in the Index Calculus for Elliptic Curves Discrete Logarithm

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## Abstract

In 2004, an algorithm is introduced to solve the DLP for elliptic curves defined over a non-prime finite field \(\mathbb{F}_{q^{n}}\). One of the main steps of this algorithm requires decomposing points of the curve \(E(\mathbb{F}_{q^{n}})\) 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 well-known 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^{ω(n−1)} to solve the corresponding PDP where *ω* 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 \(\mathbb{F}_{q^{5}}\), with *q*≈2^{64}, is supposed to be ∼ 2^{160} operations in \(E(\mathbb{F}_{q^{5}})\) using generic algorithms compared to 2^{130} operations (multiplications 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 quasi-homogeneous 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 a Gröbner basis which provides a better heuristic complexity of the total solving process.

### Key words

ECDLP Edwards curves Elliptic curves Decomposition attack Gröbner basis with symmetries Index calculus Jacobi intersections curves## Notes

### Acknowledgements

The authors would like to thank the reviewers for their fruitful remarks which helped us to improve the complexity analysis and practical results of this work. This work was partly supported by the HPAC grant of the French National Research Agency (HPAC ANR-11-BS02-013).

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