Computing discrete logarithms in subfields of residue class rings

Recent breakthrough methods \cite{gggz,joux,bgjt} on computing discrete logarithms in small characteristic finite fields share an interesting feature in common with the earlier medium prime function field sieve method \cite{jl}. To solve discrete logarithms in a finite extension of a finite field \F, a polynomial h(x) \in \F[x] of a special form is constructed with an irreducible factor g(x) \in \F[x] of the desired degree. The special form of $h(x)$ is then exploited in generating multiplicative relations that hold in the residue class ring \F[x]/h(x)\F[x] hence also in the target residue class field \F[x]/g(x)\F[x]. An interesting question in this context and addressed in this paper is: when and how does a set of relations on the residue class ring determine the discrete logarithms in the finite fields contained in it? We give necessary and sufficient conditions for a set of relations on the residue class ring to determine discrete logarithms in the finite fields contained in it. We also present efficient algorithms to derive discrete logarithms from the relations when the conditions are met. The derived necessary conditions allow us to clearly identify structural obstructions intrinsic to the special polynomial $h(x)$ in each of the aforementioned methods, and propose modifications to the selection of $h(x)$ so as to avoid obstructions.Comment: arXiv admin note: substantial text overlap with arXiv:1312.167

Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\widetilde{O}(n^{3/2}\log q + n \log^2 q)$ time to factor polynomials of degree $n$ over the finite field $\mathbb{F}_q$ with $q$ elements. A significant open problem is if the $3/2$ exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than $3/2$ would yield an algorithm for polynomial factorization with exponent better than $3/2$

Subquadratic time encodable codes beating the Gilbert-Varshamov bound

We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and 1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent. If \omega = 2, as widely believed, the encoding and decoding runtimes are respectively nearly linear and nearly quadratic. Prior to this work, encoding (resp. decoding) time of code families beating the Gilbert-Varshamov bound were quadratic (resp. cubic) or worse

Cosmic Origins Spectrograph Observations of Warm Intervening Gas Towards 3C263

We present HST/COS high S/N observations of the z = 0.32566 multi-phase absorber towards 3C263. The COS data shows absorption from H I, O VI, C III, N III, Si III and C II. The Ne VIII in this absorber is detected in the FUSE spectrum. The low and intermediate ions are kinematically aligned with each other and H I and display narrow line widths of 6 km/s. The O VI lines are kinematically offset by 12 km/s from the low ions and are a factor of four broader. All metal ions except O VI and Ne VIII are consistent with an origin in gas photoionized by the extragalactic background radiation. The bulk of the observed H I is also traced by this photoionized medium. The carbon abundance in this gas phase is near-solar. The O VI and Ne VIII favor an origin in collisionally ionized gas at T = 5.2 x 10^5 K. The H I absorption associated with this warm absorber is a BLA marginally detected in the COS spectrum. This warm gas phase has total hydrogen column density of N(H) ~ 3 x 10^19 which is 2 dex higher than what is traced by the photoionized gas. Simultaneous detection of O VI, Ne VIII and BLAs in an absorber can be a strong diagnostic of gas with temperature in the range of 10^5 - 10^6 K corresponding to the warm phase of the WHIM or shock-heated gas in the extended halos of galaxies.Comment: Accepted for publication in the Astrophysical Journa

A Survey of Weak MgII Absorbers at 0.4 < z < 2.4

We present results from a survey of weak MgII absorbers in the VLT/UVES spectra of 81 QSOs obtained from the ESO archive. In this survey, we identified 112 weak MgII systems within the redshift interval 0.4 < z < 2.4 with 86% completeness down to a rest-frame equivalent width of W_r(2796) = 0.02A, covering a cumulative redshift path length of deltaZ=77.3. From this sample, we estimate that the number of weak absorbers per unit redshift dN/dz increases from 1.06 +/- 0.04 at =1.9 to 1.76 +/- 0.08 at =1.2 and thereafter decreases to 1.51 +/- 0.09 at =0.9 and 1.06 +/- 0.10 at =0.6. Thus we find evidence for an evolution in the population of weak MgII absorbers, with their number density peaking at z=1.2. We also determine the equivalent width distribution of weak systems at =0.9 and =1.9. At 0.4 < z < 1.4, there is evidence for a turnover from a powerlaw of the form n(W_r) \propto W_r^{-1.04} at W_r(2796) < 0.1A. This turnover is more extreme at 1.4 < z < 2.4, where the equivalent width distribution is close to an extrapolation of the exponential distribution function found for strong MgII absorbers. Based on these results, we discuss the possibility that some fraction of weak MgII absorbers, particularly single cloud systems, are related to satellite clouds surrounding strong MgII systems. These structures could also be analogs to Milky Way high velocity clouds. In this context, the paucity of high redshift weak MgII absorbers is caused by a lack of isolated accreting clouds on to galaxies during that epoch.Comment: 14 pages, 11 figures, ApJ accepte