Discrete logarithm computations over finite fields using Reed-Solomon codes

Cheng and Wan have related the decoding of Reed-Solomon codes to the computation of discrete logarithms over finite fields, with the aim of proving the hardness of their decoding. In this work, we experiment with solving the discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2) operations over GF(q), operating on a q x q matrix with (h+2) q non-zero coefficients. We give faster variants including an incremental version and another one that uses auxiliary finite fields that need not be subfields of GF(q^h); this variant is very practical for moderate values of q and h. We include some numerical results of our first implementations

On points at infinity of real spectra of polynomial rings

Let R be a real closed field and A=R[x_1,...,x_n]. Let sper A denote the real spectrum of A. There are two kinds of points in sper A : finite points (those for which all of |x_1|,...,|x_n| are bounded above by some constant in R) and points at infinity. In this paper we study the structure of the set of points at infinity of sper A and their associated valuations. Let T be a subset of {1,...,n}. For j in {1,...,n}, let y_j=x_j if j is not in T and y_j=1/x_j if j is in T. Let B_T=R[y_1,...,y_n]. We express sper A as a disjoint union of sets of the form U_T and construct a homeomorphism of each of the sets U_T with a subspace of the space of finite points of sper B_T. For each point d at infinity in U_T, we describe the associated valuation v_{d*} of its image d* in sper B_T in terms of the valuation v_d associated to d. Among other things we show that the valuation v_{d*} is composed with v_d (in other words, the valuation ring R_d is a localization of R_{d*} at a suitable prime ideal)

On the Pierce-Birkhoff Conjecture

This paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points$\alpha,\beta\in\sper\ A$and their separating ideal$$; we refer to this statement as the Local Pierce-Birkhoff conjecture at$\alpha,\beta$. In this paper, for each pair$(\alpha,\beta)$with$ht()=\dim A$, we define a natural number, called complexity of$(\alpha,\beta)$. Complexity 0 corresponds to the case when one of the points$\alpha,\beta$is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n-1 implies the connectedness conjecture in dimension n in the case when$ht()$is less than n-1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce--Birkhoff conjectures in any dimension in the case when$ht()$less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to$ht()=3$, the pair$(\alpha,\beta)$is of complexity 1 and$A$ is excellent with residue field the field of real numbers

Impedance active control of flight control devices

The work presented in this paper concerns the active control of flight control devices (sleeves, yokes, side-sticks, rudder pedals,...). The objective is to replace conventional technologies by active technology to save weight and to feedback kinesthetic sensations to the pilot. Some architectures are proposed to control the device mechanical impedance felt by pilot and to couple pilot and co-pilot control devices. A first experimental test-bed was developed to validate and illustrate control laws and theirs limitations due to dynamic couplings with the pilot own-impedance

Updated estimate of the duration of the meningo-encephalitic stage in gambiense human African trypanosomiasis

Background: The duration of the stages of HAT is an important factor in epidemiological studies and intervention planning. Previously, we published estimates of the duration of the haemo-lymphatic stage 1 and meningo-encephalitic stage 2 of the gambiense form of human African trypanosomiasis (HAT), in the absence of treatment. Here we revise the estimate of stage 2 duration, computed based on data from Uganda and South Sudan, by adjusting observed infection prevalence for incomplete case detection coverage and diagnostic inaccuracy. Findings: The revised best estimate for the mean duration of stage 2 is 252 days (95% CI 171–399), about half of our initial best estimate, giving a total mean duration of untreated gambiense HAT infection of approximately 2 years and 2 months. Conclusions: Our new estimate provides improved information on the transmission dynamics of this neglected tropical disease in Uganda and South Sudan. We stress that there remains considerable variability around the estimated mean values, and that one must be cautious in applying these results to other foci

Growth of quasiconvex subgroups

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel, and Osin on rotating families.Comment: 28 pages, 1 figure. v3 is the final version, to appear in Math Proc. Cambridge Philos. So