Jens Zumbrägel

On the Function Field Sieve and the Impact of Higher Splitting Probabilities

In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as \(L_{q^n}(1/3,(4/9)^{1/3})\) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one and two elements when the field has a subfield of an appropriate size. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite fields with 21971 and 23164 elements, setting a record for binary fields.

Breaking `128-bit Secure' Supersingular Binary Curves

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 Thome. Using these developments, Adj, Menezes, Oliveira and Rodriguez-Henriquez 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.

ON THE DISCRETE LOGARITHM PROBLEM IN FINITE FIELDS OF FIXED CHARACTERISTIC

For

Characteristics of invariant weights related to code equivalence over rings

q

MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

a prime power, the discrete logarithm problem (DLP) in

Efficient recovering of operation tables of black box groups and rings

\mathbb{F}_{q}

Indiscreet logarithms in finite fields of small characteristic

consists in finding, for any

Profinite algebras and affine boundedness

g \in \mathbb{F}_{q}^{\times}

Every simple compact semiring is finite

and

Notes on the pseudoredundancy

h \in \langle g \rangle