Weakness of 𝔽36·509 for Discrete Logarithm Cryptography

In 2013, Joux, and then Barbulescu, Gaudry, Joux and Thome, presented new algorithms for computing discrete logarithms in finite fields of small and medium characteristic. We show that these new algorithms render the finite field ${\mathbb{F}}_{3^{6 \cdot 509}} = {\mathbb{F}}_{3^{3054}}$ weak for discrete logarithm cryptography in the sense that discrete logarithms in this field can be computed significantly faster than with the previous fastest algorithms. Our concrete analysis shows that the supersingular elliptic curve over ${\mathbb{F}}_{3^{509}}$ with embedding degree 6 that had been considered for implementing pairing-based cryptosystems at the 128-bit security level in fact provides only a significantly lower level of security. Our work provides a convenient framework and tools for performing a concrete analysis of the new discrete logarithm algorithms and their variants.