**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

**Available format(s): **PDF | BibTeX Citation

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

**Short URL: **ia.cr/2016/914

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