9 research outputs found
Twisted Reed-Solomon Codes
We present a new general construction of MDS codes over a finite field
. We describe two explicit subclasses which contain new MDS codes
of length at least for all values of . Moreover, we show that
most of the new codes are not equivalent to a Reed-Solomon code.Comment: 5 pages, accepted at IEEE International Symposium on Information
Theory 201
Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography
We present a generalisation of Twisted Reed-Solomon codes containing a new
large class of MDS codes. We prove that the code class contains a large
subfamily that is closed under duality. Furthermore, we study the Schur squares
of the new codes and show that their dimension is often large. Using these
structural properties, we single out a subfamily of the new codes which could
be considered for code-based cryptography: These codes resist some existing
structural attacks for Reed-Solomon-like codes, i.e. methods for retrieving the
code parameters from an obfuscated generator matrix.Comment: 5 pages, accepted at: IEEE International Symposium on Information
Theory 201
Novel Polynomial Basis and Its Application to Reed-Solomon Erasure Codes
In this paper, we present a new basis of polynomial over finite fields of
characteristic two and then apply it to the encoding/decoding of Reed-Solomon
erasure codes. The proposed polynomial basis allows that -point polynomial
evaluation can be computed in finite field operations with
small leading constant. As compared with the canonical polynomial basis, the
proposed basis improves the arithmetic complexity of addition, multiplication,
and the determination of polynomial degree from
to . Based on this basis, we then develop the encoding and
erasure decoding algorithms for the Reed-Solomon codes. Thanks to
the efficiency of transform based on the polynomial basis, the encoding can be
completed in finite field operations, and the erasure decoding
in finite field operations. To the best of our knowledge, this
is the first approach supporting Reed-Solomon erasure codes over
characteristic-2 finite fields while achieving a complexity of ,
in both additive and multiplicative complexities. As the complexity leading
factor is small, the algorithms are advantageous in practical applications
A compressive sensing algorithm for attitude determination
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 29-30).We propose a framework for compressive sensing of images with local distinguishable objects, such as stars, and apply it to solve a problem in celestial navigation. Specifically, let x [epsilon] RN be an N-pixel image, consisting of a small number of local distinguishable objects plus noise. Our goal is to design an m x N measurement matrix A with m << N, such that we can recover an approximation to x from the measurements Ax. We construct a matrix A and recovery algorithm with the following properties: (i) if there are k objects, the number of measurements m is O((klog N)/(log k)), undercutting the best known bound of O(klog(N/k)) (ii) the matrix A is ultra-sparse, which is important when the signal is weak relative to the noise, and (iii) the recovery algorithm is empirically fast and runs in time sub-linear in N. We also present a comprehensive study of the application of our algorithm to attitude determination, or finding one's orientation in space. Spacecraft typically use cameras to acquire an image of the sky, and then identify stars in the image to compute their orientation. Taking pictures is very expensive for small spacecraft, since camera sensors use a lot of power. Our algorithm optically compresses the image before it reaches the camera's array of pixels, reducing the number of sensors that are required.by Rishi Vijay Gupta.M.Eng
Compressive Sensing with Local Geometric Features
We propose a framework for compressive sensing of images with local
distinguishable objects, such as stars, and apply it to solve a problem in
celestial navigation. Specifically, let x be an N-pixel real-valued image,
consisting of a small number of local distinguishable objects plus noise. Our
goal is to design an m-by-N measurement matrix A with m << N, such that we can
recover an approximation to x from the measurements Ax.
We construct a matrix A and recovery algorithm with the following properties:
(i) if there are k objects, the number of measurements m is O((k log N)/(log
k)), undercutting the best known bound of O(k log(N/k)) (ii) the matrix A is
very sparse, which is important for hardware implementations of compressive
sensing algorithms, and (iii) the recovery algorithm is empirically fast and
runs in time polynomial in k and log(N).
We also present a comprehensive study of the application of our algorithm to
attitude determination, or finding one's orientation in space. Spacecraft
typically use cameras to acquire an image of the sky, and then identify stars
in the image to compute their orientation. Taking pictures is very expensive
for small spacecraft, since camera sensors use a lot of power. Our algorithm
optically compresses the image before it reaches the camera's array of pixels,
reducing the number of sensors that are required
Fast Fourier transform via automorphism groups of rational function fields
The Fast Fourier Transform (FFT) over a finite field computes
evaluations of a given polynomial of degree less than at a specifically
chosen set of distinct evaluation points in . If or
is a smooth number, then the divide-and-conquer approach leads to the fastest
known FFT algorithms. Depending on the type of group that the set of evaluation
points forms, these algorithms are classified as multiplicative (Math of Comp.
1965) and additive (FOCS 2014) FFT algorithms. In this work, we provide a
unified framework for FFT algorithms that include both multiplicative and
additive FFT algorithms as special cases, and beyond: our framework also works
when is smooth, while all known results require or to be
smooth. For the new case where is smooth (this new case was not
considered before in literature as far as we know), we show that if is a
divisor of that is -smooth for a real , then our FFT needs
arithmetic operations in . Our unified framework is
a natural consequence of introducing the algebraic function fields into the
study of FFT
Sur l'algorithme de décodage en liste de Guruswami-Sudan sur les anneaux finis
This thesis studies the algorithmic techniques of list decoding, first proposed by Guruswami and Sudan in 1998, in the context of Reed-Solomon codes over finite rings. Two approaches are considered. First we adapt the Guruswami-Sudan (GS) list decoding algorithm to generalized Reed-Solomon (GRS) codes over finite rings with identity. We study in details the complexities of the algorithms for GRS codes over Galois rings and truncated power series rings. Then we explore more deeply a lifting technique for list decoding. We show that the latter technique is able to correct more error patterns than the original GS list decoding algorithm. We apply the technique to GRS code over Galois rings and truncated power series rings and show that the algorithms coming from this technique have a lower complexity than the original GS algorithm. We show that it can be easily adapted for interleaved Reed-Solomon codes. Finally we present the complete implementation in C and C++ of the list decoding algorithms studied in this thesis. All the needed subroutines, such as univariate polynomial root finding algorithms, finite fields and rings arithmetic, are also presented. Independently, this manuscript contains other work produced during the thesis. We study quasi cyclic codes in details and show that they are in one-to-one correspondence with left principal ideal of a certain matrix ring. Then we adapt the GS framework for ideal based codes to number fields codes and provide a list decoding algorithm for the latter.Cette thèse porte sur l'algorithmique des techniques de décodage en liste, initiée par Guruswami et Sudan en 1998, dans le contexte des codes de Reed-Solomon sur les anneaux finis. Deux approches sont considérées. Dans un premier temps, nous adaptons l'algorithme de décodage en liste de Guruswami-Sudan aux codes de Reed-Solomon généralisés sur les anneaux finis. Nous étudions en détails les complexités de l'algorithme pour les anneaux de Galois et les anneaux de séries tronquées. Dans un deuxième temps nous approfondissons l'étude d'une technique de remontée pour le décodage en liste. Nous montrons que cette derni're permet de corriger davantage de motifs d'erreurs que la technique de Guruswami-Sudan originale. Nous appliquons ensuite cette même technique aux codes de Reed-Solomon généralisés sur les anneaux de Galois et les anneaux de séries tronquées et obtenons de meilleures bornes de complexités. Enfin nous présentons l'implantation des algorithmes en C et C++ des algorithmes de décodage en liste étudiés au cours de cette thèse. Tous les sous-algorithmes nécessaires au décodage en liste, comme la recherche de racines pour les polynômes univariés, l'arithmétique des corps et anneaux finis sont aussi présentés. Indépendamment, ce manuscrit contient d'autres travaux sur les codes quasi-cycliques. Nous prouvons qu'ils sont en correspondance biunivoque avec les idéaux à gauche d'un certain anneaux de matrices. Enfin nous adaptons le cadre proposé par Guruswami et Sudan pour les codes à base d'ideaux aux codes construits à l'aide des corps de nombres. Nous fournissons un algorithme de décodage en liste dans ce contexte