## Cryptology ePrint Archive: Report 2013/446

Weakness of $\mbox{${\mathbb F}$}_{3^{6 \cdot 509}}$ for Discrete Logarithm Cryptography

Gora Adj and Alfred Menezes and Thomaz Oliveira and Francisco Rodr\'iguez-Henr\'iquez

Abstract: In 2013, Joux, and then Barbulescu, Gaudry, Joux and Thom\'{e}, 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 $\Fmain = \FF_{3^{3054}}$ 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 $\FF_{3^{509}}$ 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.

Category / Keywords: public-key cryptography /