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

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 quasipolynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thomé. Using these developments, Adj, Menezes, Oliveira and Rodŕıguez-Henŕıquez 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. At the 128-bit security level, they suggested that the new algorithms have no impact on the security of a genus one curve over F21223 , and reduce the security of a genus two curve over F2367 to 94.6 bits. In this paper we propose a new field representation and efficient descent principles, which together demonstrate that the new techniques can be made practical at the 128-bit security level. In particular, we show that the aforementioned genus one curve offers only 59 bits of security, and we report a total break of the genus two curve. Since these techniques are widely applicable, we conclude that small characteristic pairings should henceforth be considered completely insecure.