Computing Individual Discrete Logarithms Faster in GF(pn)(p^n) with the NFS-DL Algorithm

International audienceThe Number Field Sieve (NFS) algorithm is the best known method to compute discrete logarithms (DL) in finite fields $\mathbb{F}_{p^n}$, with $p$ medium to large and $n \geq 1$ small. This algorithm comprises four steps: polynomial selection, relation collection, linear algebra and finally, individual logarithm computation. The first step outputs two polynomials defining two number fields, and a map from the polynomial ring over the integers modulo each of these polynomials to $\mathbb{F}_{p^n}$. After the relation collection and linear algebra phases, the (virtual) logarithm of a subset of elements in each number field is known. Given the target element in $\mathbb{F}_{p^n}$, the fourth step computes a preimage in one number field. If one can write the target preimage as a product of elements of known (virtual) logarithm, then one can deduce the discrete logarithm of the target. As recently shown by the Logjam attack, this final step can be critical when it can be computed very quickly. But we realized that computing an individual DL is much slower in medium-and large-characteristic non-prime fields $\mathbb{F}_{p^n}$ with $n \geq 3$, compared to prime fields and quadratic fields $\mathbb{F}_{p^2}$. We optimize the first part of individual DL: the \emph{booting step}, by reducing dramatically the size of the preimage norm. Its smoothness probability is higher, hence the running-time of the booting step is much improved. Our method is very efficient for small extension fields with $2 \leq n \leq 6$ and applies to any $n > 1$, in medium and large characteristic

Low-oxygen waters limited habitable space for early animals

The oceans at the start of the Neoproterozoic Era (1,000–541 million years ago, Ma) were dominantly anoxic, but may have become progressively oxygenated, coincident with the rise of animal life. However, the control that oxygen exerted on the development of early animal ecosystems remains unclear, as previous research has focussed on the identification of fully anoxic or oxic conditions, rather than intermediate redox levels. Here we report anomalous cerium enrichments preserved in carbonate rocks across bathymetric basin transects from nine localities of the Nama Group, Namibia (~550–541 Ma). In combination with Fe-based redox proxies, these data suggest that low-oxygen conditions occurred in a narrow zone between well-oxygenated surface waters and fully anoxic deep waters. Although abundant in well-oxygenated environments, early skeletal animals did not occupy oxygen impoverished regions of the shelf, demonstrating that oxygen availability (probably >10 μM) was a key requirement for the development of early animal-based ecosystems

Asymptotic complexities of discrete logarithm algorithms in pairing-relevant finite fields

International audienceWe study the discrete logarithm problem at the boundary case between small and medium characteristic finite fields, which is precisely the area where finite fields used in pairing-based cryptosystems live. In order to evaluate the security of pairing-based protocols, we thoroughly analyze the complexity of all the algorithms that coexist at this boundary case: the Quasi-Polynomial algorithms, the Number Field Sieve and its many variants, and the Function Field Sieve. We adapt the latter to the particular case where the extension degree is composite, and show how to lower the complexity by working in a shifted function field. All this study finally allows us to give precise values for the characteristic asymptotically achieving the highest security level for pairings. Surprisingly enough, there exist special characteristics that are as secure as general ones