Cryptology ePrint Archive: Report 2016/914

Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields

Gora Adj and Isaac Canales-Martínez and Nareli Cruz-Cortés and Alfred Menezes and Thomaz Oliveira and Luis Rivera-Zamarripa and Francisco Rodríguez-Henríquez

Abstract: Since 2013 there have been several developments in algorithms for computing discrete logarithms in small-characteristic finite fields, culminating in a quasi-polynomial algorithm. In this paper, we report on our successful computation of discrete logarithms in the cryptographically-interesting characteristic-three finite field ${\mathbb F}_{3^{6 \cdot 509}}$ using these new algorithms; prior to 2013, it was believed that this field enjoyed a security level of 128 bits. We also show that a recent idea of Guillevic can be used to compute discrete logarithms in the cryptographically-interesting finite field ${\mathbb F}_{3^{6 \cdot 709}}$ using essentially the same resources as we expended on the ${\mathbb F}_{3^{6 \cdot 509}}$ computation. Finally, we argue that discrete logarithms in the finite field ${\mathbb F}_{3^{6 \cdot 1429}}$ can feasibly be computed today; this is significant because this cryptographically-interesting field was previously believed to enjoy a security level of 192 bits.

Category / Keywords: discrete logarithm problem, bilinear pairings, cryptanalysis, implementation

Date: received 21 Sep 2016, last revised 21 Sep 2016

Contact author: francisco at cs cinvestav mx

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Version: 20160922:010425 (All versions of this report)

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