Thorsten Kleinjung, Benjamin Wesolowski
We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. More generally, we prove that it can be solved in the field of cardinality $p^n$ in expected time $(pn)^{2\log_2(n) + O(1)}$.
Robert Granger, Thorsten Kleinjung, Jens Zumbrägel
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 quasi-polynomial time algorithm, due to Barbulescu, Gaudry, Joux and Thomé. Using these developments, Adj, Menezes, Oliveira and Rodríguez-Henríquez 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, which were originally thought to be $128$-bit secure. In particular, they suggested that the new algorithms have no impact on the security of a genus one curve over ${\mathbb F}_{2^{1223}}$, and reduce the security of a genus two curve over ${\mathbb F}_{2^{367}}$ to $94.6$ bits. In this paper we propose a new field representation and efficient general descent principles which together make the new techniques far more practical. Indeed, at the `128-bit security level' our analysis shows that the aforementioned genus one curve has approximately $59$ bits of security, and we report a total break of the genus two curve.
Razvan Barbulescu, Joppe W. Bos, Cyril Bouvier et al.
In this paper we prove some divisibility properties of the cardinality of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas of the proofs help us to find new families of elliptic curves with good division properties which increase the success probability of ECM.