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Author s Product display : Aiden A. Bruen ; David L. Abstract: This interdisciplinary volume contains papers from both a conference and special session on Error-Control Codes, Information Theory and Applied Cryptography. Book Series Name: Contemporary Mathematics. Volume: Publication Month and Year: Copyright Year: Page Count: Cover Type: Softcover. Print ISBN Online ISBN Print ISSN: Online ISSN: Primary MSC: 05 ; 94 ; 51 ; 81 ; Applied Math? MAA Book?
Apparel or Gift: false. Online Price 1 Label: List. Online Price 1: We give computationally efficient versions of the Jordan-Holder and Krull-Schmidt theorems in this context to describe all possible factorization.
Geometrically, we show how to compute a representation of the Frobenius operator on the space of roots, and how its Jordan form can be used to count the number of decompositions. We also describe an inverse theory, from which we can construct and count the number of additive polynomials with specified factorization patterns. The paper concerns the computation of something similar to Gaussian elimination but for matrices with entries which are polynomials univariate.go
Algebraic Geometry and its applications to error correcting codes and cryptography
These kind of computations were recently improved by Vincent for square matrices, but it was still open how to generalise these results to cover rectangular matrices. In the end, we found a very elegant randomized algorithm, which is fast in many cases, and a more sophisticated algorithm which works well in almost all cases. The issue is not completely settled, though, and there are still a spectrum of input for which we believe a slightly faster algorithm should exist. In this new paper we greatly expand the family of codes by mutilating Reed-Solomon codes multiple times.
The new codes are not much more difficult to analyse and we examine several interesting properties of them. These properties together lead us to conclude that some of the codes might be useful for public-key cryptography! The classical McEliece Cryptosystem is a methodology for turning any family of codes, for which one knows a good decoding algorithm, into a public-key cryptographic cipher. Moreover, if the cipher is secure against attackers using normal computers, then it is also secure against quantum computers!
Algebraic Geometry and Algebraic Coding Theory for Cryptography | ANR
That very clever but it has two draw-backs:. The public key of the cryptographic cipher can be large, especially if the family of codes does not allow a high decoding capability. Since many Twisted Reed-Solomon codes are MDS, they have excellent decoding capability and the resulting keys are therefore much smaller than competing suggestions for McEliece, e. Arguing that the codes are unbreakable is a much more dicey business, however: here we have made some headway by showing that all the properties which makes Reed-Solomon vulnerable do not apply to Twisted Reed-Solomon codes. Hermitian codes are the prime example of Algebraic Geometry codes, and they have the potential to supersede the widely used Reed-Solomon codes in many applications due to their longer length at comparable decoding capability.
Further, we demonstrate that this makes Interleaved Hermitian codes the best known codes in terms of decoding capability for a wide range of parameters, where one wishes relatively short codes over large alphabets. The paper also details how to implement the algorithm efficiently, achieving the rare sub-quadratic complexity in the code length. WordPress theme developed by Webriti.
Error-Correcting Codes, Finite Geometries and Cryptography
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