Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs

A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

An introduction to linear and cyclic codes

International audienceOur purpose is to recall some basic aspects about linear and cyclic codes. We first briefly describe the role of error-correcting codes in communication. To do this we introduce, with examples, the concept of linear codes and their parameters, in particular the Hamming distance. A fundamental subclass of linear codes is given by cyclic codes, that enjoy a very interesting algebraic structure. In fact, cyclic codes can be viewed as ideals in a residue classes ring of univariate polynomials. BCH codes are the most studied family of cyclic codes, for which some efficient decoding algorithms are known, as the method of Sugiyama

On the complexity of arithmetic secret sharing

Since the mid 2000s, asymptotically-good strongly-multiplicative linear (ramp) secret sharing schemes over a fixed finite field have turned out as a central theoretical primitive in numerous constant-communication-rate results in multi-party cryptographic scenarios, and, surprisingly, in two-party cryptography as well. Known constructions of this most powerful class of arithmetic secret sharing schemes all rely heavily on algebraic geometry (AG), i.e., on dedicated AG codes based on asymptotically good towers of algebraic function fields defined over finite fields. It is a well-known open question since the first (explicit) constructions of such schemes appeared in CRYPTO 2006 whether the use of “heavy machinery” can be avoided here. i.e., the question is whether the mere existence of such schemes can also be proved by “elementary” techniques only (say, from classical algebraic coding theory), even disregarding effective construction. So far, there is no progress. In this paper we show the theoretical result that, (1) no matter whether this open question has an affirmative answer or not, these schemes can be constructed explicitly by elementary algorithms defined in terms of basic algebraic coding theory. This pertains to all relevant operations associated to such schemes, including, notably, the generation of an instance for a given number of players n, as well as error correction in the presence of corrupt shares. We further show that (2) the algorithms are quasi-linear time (in n); this is (asymptotically) significantly more efficient than the known constructions. That said, the analysis of the mere termination of these algorithms does still rely on algebraic geometry, in the sense that it requires “blackbox application” of suitable existence results for these schemes. Our method employs a nontrivial, novel adaptation of a classical (and ubiquitous) paradigm from coding theory that enables transformation of existence results on asymptotically good codes into explicit construction of such codes via concatenation, at some constant loss in parameters achieved. In a nutshell, our generating idea is to combine a cascade of explicit but “asymptotically-bad-yet-good-enough schemes” with an asymptotically good one in such a judicious way that the latter can be selected with exponentially small number of players in that of the compound scheme. This opens the door t

A p-adic quasi-quadratic point counting algorithm

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level $2^\nu p$ where $\nu >0$ is an integer and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.Comment: 32 page

Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

Explicit constructions of asymptotically good towers of function fields

Thesis (MSc)--Stellenbosch University, 2003ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensions of algebraic function fields of one variable such that the constituent function fields have the same (finite) field of constants and the genus of these tend to infinity. A study can be made of the asymptotic behaviour of the ratio of the number of places of degree one over the genus of FJWq as i tends to infinity. A tower is called asymptotically good if this limit is a positive number. The well-known Drinfeld- Vladut bound provides a general upper bound for this limit. In practise, asymptotically good towers are rare. While the first examples were non-explicit, we focus on explicit towers of function fields, that is towers where equations recursively defining the extensions Fi+d F; are known. It is known that if the field of constants of the tower has square cardinality, it is possible to attain the Drinfeld- Vladut upper bound for this limit, even in the explicit case. If the field of constants does not have square cardinality, it is unknown how close the limit of the tower can come to this upper bound. In this thesis, we will develop the theory required to construct and analyse the asymptotic behaviour of explicit towers of function fields. Various towers will be exhibited, and general families of explicit formulae for which the splitting behaviour and growth of the genus can be computed in a tower will be discussed. When the necessary theory has been developed, we will focus on the case of towers over fields of non-square cardinality and the open problem of how good the asymptotic behaviour of the tower can be under these circumstances.AFRIKAANSE OPSOMMING: 'n Toring van globale funksieliggame F = (FI, F2' ... ) is 'n oneindige toring van skeibare uitbreidings van algebraïese funksieliggame van een veranderlike sodat die samestellende funksieliggame dieselfde (eindige) konstante liggaam het en die genus streef na oneindig. 'n Studie kan gemaak word van die asimptotiese gedrag van die verhouding van die aantal plekke van graad een gedeel deur die genus van Fi/F q soos i streef na oneindig. 'n Toring word asimptoties goed genoem as hierdie limiet 'n positiewe getal is. Die bekende Drinfeld- Vladut grens verskaf 'n algemene bogrens vir hierdie limiet. In praktyk is asimptoties goeie torings skaars. Terwyl die eerste voorbeelde nie eksplisiet was nie, fokus ons op eksplisiete torings, dit is torings waar die vergelykings wat rekursief die uitbreidings Fi+d F; bepaal bekend is. Dit is bekend dat as die kardinaliteit van die konstante liggaam van die toring 'n volkome vierkant is, dit moontlik is om die Drinfeld- Vladut bogrens vir die limiet te behaal, selfs in die eksplisiete geval. As die konstante liggaam nie 'n kwadratiese kardinaliteit het nie, is dit onbekend hoe naby die limiet van die toring aan hierdie bogrens kan kom. In hierdie tesis salons die teorie ontwikkel wat benodig word om eksplisiete torings van funksieliggame te konstrueer, en hulle asimptotiese gedrag te analiseer. Verskeie torings sal aangebied word en algemene families van eksplisiete formules waarvoor die splitsingsgedrag en groei van die genus in 'n toring bereken kan word, sal bespreek word. Wanneer die nodige teorie ontwikkel is, salons fokus op die geval van torings oor liggame waarvan die kardinaliteit nie 'n volkome vierkant is nie, en op die oop probleem aangaande hoe goed die asimptotiese gedrag van 'n toring onder hierdie omstandighede kan wees

Some Applications of Coding Theory in Cryptography

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