## Cryptology ePrint Archive: Report 2014/119

Breaking 128-bit Secure' Supersingular Binary Curves (or how to solve discrete logarithms in ${\mathbb F}_{2^{4 \cdot 1223}}$ and ${\mathbb F}_{2^{12 \cdot 367}}$)

Robert Granger and Thorsten Kleinjung and Jens Zumbr\"agel

Abstract: In late 2012 and early 2013 the discrete logarithm problem (DLP) in finite fields of small characteristic underwent a dramatic series of breakthroughs, culminating in a heuristic quasi-polynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thom\'e. Using these developments, Adj, Menezes, Oliveira and Rodr\'iguez-Henr\'iquez analysed the concrete security of the DLP, as it arises from pairings on (the Jacobians of) various genus one and two supersingular curves in the literature, which were originally thought to be $128$-bit secure. In particular, they suggested that the new algorithms have no impact on the security of a genus one curve over ${\mathbb F}_{2^{1223}}$, and reduce the security of a genus two curve over ${\mathbb F}_{2^{367}}$ to $94.6$ bits. In this paper we propose a new field representation and efficient general descent principles which together make the new techniques far more practical. Indeed, at the 128-bit security level' our analysis shows that the aforementioned genus one curve has approximately $59$ bits of security, and we report a total break of the genus two curve.

Category / Keywords: Discrete logarithm problem, finite fields, supersingular binary curves, pairings

Date: received 14 Feb 2014, last revised 12 Jun 2014

Contact author: robbiegranger at gmail com

Available format(s): PDF | BibTeX Citation

Note: This is the full version of the CRYPTO 2014 paper.

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