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In late 2012 and early 2013 the discrete logarithm problem (DLP) in finitefields of small characteristic underwent a dramatic series of breakthroughs,culminating in a heuristic quasi-polynomial time algorithm, due to Barbulescu,Gaudry, Joux and Thom\'e. Using these developments, Adj, Menezes, Oliveira andRodr\'iguez-Henr\'iquez analysed the concrete security of the DLP, as it arisesfrom pairings on (the Jacobians of) various genus one and two supersingularcurves in the literature, which were originally thought to be $128$-bit secure.In particular, they suggested that the new algorithms have no impact on thesecurity of a genus one curve over ${\mathbb F}_{2^{1223}}$, and reduce thesecurity of a genus two curve over ${\mathbb F}_{2^{367}}$ to $94.6$ bits. Inthis paper we propose a new field representation and efficient general descentprinciples which together make the new techniques far more practical. Indeed,at the `128-bit security level' our analysis shows that the aforementionedgenus one curve has approximately $59$ bits of security, and we report a totalbreak of the genus two curve.

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