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Information Set Decoding for Lee-Metric Codes using Restricted Balls [article]

Jessica Bariffi, Karan Khathuria, Violetta Weger
2022 arXiv   pre-print
in restricted balls.  ...  The observation that such decoders come with a larger cost than their Hamming metric counterparts make the Lee metric a promising alternative for classical code-based cryptography.  ...  Restricted-Balls Algorithm The idea of the new information set decoding algorithms is to use the information on the uniformly chosen e ∈ S(0, t, n, p s ).  ... 
arXiv:2205.12903v1 fatcat:cs7zwbm3xjexplhqxoiuy4tv2m

Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

Anxiao Jiang, Moshe Schwartz, Jehoshua Bruck
2010 IEEE Transactions on Information Theory  
We show bounds on the size of such codes, and use metric-embedding techniques to give constructions which translate a wealth of knowledge of codes in the Lee metric to codes over permutations in Kendall's  ...  In this scheme, a set of n cells stores information in the permutation induced by the different charge levels of the individual cells.  ...  This establishes a general method for designing CCRM codes using an abundance of well-known codes over the Lee metric.  ... 
doi:10.1109/tit.2010.2043764 fatcat:cjzmzz76sbggtjespfgtahitby

Page 5923 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
(K WT-KUWA-CS; Kuwait); Bose, Bella (1-ORS-EE:; Corvallis, OR) Quasi-perfect Lee distance codes. (English summary) IEEE Trans. Inform. Theory 49 (2003), no. 6, 1535-1539.  ...  In particular, for n >no(r), the authors give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r +2 in {0,1}”.  ... 

Information Set Decoding in the Lee Metric with Applications to Cryptography [article]

Anna-Lena Horlemann-Trautmann, Violetta Weger
2020 arXiv   pre-print
We convert Stern's information set decoding (ISD) algorithm to the ring Z/4 Z equipped with the Lee metric.  ...  We show that using Lee metric codes can drastically decrease the key size, compared to Hamming metric codes. In the end we explain how our results can be generalized to other Galois rings Z/p^sZ.  ...  Acknowledgments We would like to thank Karan Khathuria for fruitful discussions and technical support.  ... 
arXiv:1903.07692v2 fatcat:g5bvb6qaanayfdalll7jc5zone

Information set decoding in the lee metric with applications to cryptography

Anna-Lena Horlemann-Trautmann, ,Faculty of Mathematics and Statistics, University of St. Gallen, Bodanstr. 6, St. Gallen, Switzerland, Violetta Weger, ,Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
2019 Advances in Mathematics of Communications  
We convert Stern's information set decoding (ISD) algorithm to the ring Z/4Z equipped with the Lee metric.  ...  We show that using Lee metric codes can substantially decrease the key size, compared to Hamming metric codes. In the end we explain how our results can be generalized to other Galois rings Z/p s Z.  ...  Acknowledgments We would like to thank Karan Khathuria for fruitful discussions and technical support.  ... 
doi:10.3934/amc.2020089 fatcat:nlnyopxblffcnn2ngpdlqdfrjq

On the Hardness of the Lee Syndrome Decoding Problem [article]

Violetta Weger, Karan Khathuria, Anna-Lena Horlemann, Massimo Battaglioni, Paolo Santini, Edoardo Persichetti
2022 arXiv   pre-print
In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric.  ...  For the analyzed algorithms, we assess the computational complexity in the asymptotic regime and compare it to the corresponding algorithms in the Hamming metric.  ...  In order to define information set decoders for the codes we consider, we need a suitable notion of information set, which is given next. Definition 2.4.  ... 
arXiv:2002.12785v5 fatcat:xrtv6beys5h4hl5k7rmc5smfuu

A New Path to Code-based Signatures via Identification Schemes with Restricted Errors [article]

Marco Baldi, Massimo Battaglioni, Franco Chiaraluce, Anna-Lena Horlemann-Trautmann, Edoardo Persichetti, Paolo Santini, Violetta Weger
2021 arXiv   pre-print
We study the properties of random codes under this new decoding perspective, in the fashion of traditional coding theory results, and assess the complexity of solving a random R-SDP instance.  ...  In this paper we introduce a variant of the Syndrome Decoding Problem (SDP), that we call Restricted SDP (R-SDP), in which the entries of the searched vector are defined over a subset of the underlying  ...  In particular, we prove the NP-completeness of the restricted syndrome decoding problem and provide adaptions of known Information Set Decoding (ISD) algorithms to this case.  ... 
arXiv:2008.06403v3 fatcat:cbrrlayonfdjnltu6cav6slb74

Smaller Decoding Exponents: Ball-Collision Decoding [chapter]

Daniel J. Bernstein, Tanja Lange, Christiane Peters
2011 Lecture Notes in Computer Science  
This paper introduces "ball-collision decoding" and shows that it has a smaller exponent for each (R, W ): the speedup from Stern's algorithm to ball-collision decoding is exponential in n.  ...  The best attacks known against this system are generic decoding attacks that treat McEliece's hidden binary Goppa codes as random linear codes.  ...  Ball-collision decoding has its roots in information-set decoding, which was used against the McEliece system in, e.g., [50] , [17] , [18] , and [8] .  ... 
doi:10.1007/978-3-642-22792-9_42 fatcat:lfw3mfi5yrgkzn3u4k373dw3ru

Generic Decoding in the Cover Metric [article]

Sebastian Bitzer, Julian Renner, Antonia Wachter-Zeh, Violetta Weger
2022 arXiv   pre-print
We then provide a generic decoder, following the information set decoding idea from Prange's algorithm in the Hamming metric.  ...  As the cover metric lies in between the Hamming and rank metric, it presents itself as a promising candidate for code-based cryptography.  ...  As an information set decoding equivalent in the new metric, we provide a Prange-like generic decoding algorithm in the cover metric.  ... 
arXiv:2205.12738v1 fatcat:siid6xcu7rd6tdjkbudz3fzjuy

Systematic Error-Correcting Codes for Rank Modulation [article]

Hongchao Zhou, Moshe Schwartz, Anxiao Jiang, Jehoshua Bruck
2013 arXiv   pre-print
Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures.  ...  We use non-constructive arguments to show the existence of [n,k,n-k]-systematic codes for general parameters.  ...  For a decoding procedure, map the received permutation to its factoradic representation, and use the decoding for the Leemetric code (essentially, a BCH decoding procedure) given in [19] .  ... 
arXiv:1310.6817v1 fatcat:z4p2kvlhvfc45nteoeprfd3haa

Systematic error-correcting codes for rank modulation

Hongchao Zhou, Anxiao Jiang, J. Bruck
2012 2012 IEEE International Symposium on Information Theory Proceedings  
We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes.  ...  We study systematic codes for rank modulation under Kendall's τ -metric as well as under the ∞ -metric.  ...  For a decoding procedure, map the received permutation to its factoradic representation, and use the decoding for the Lee-metric code (essentially, a BCH decoding procedure) given in [24] .  ... 
doi:10.1109/isit.2012.6284106 dblp:conf/isit/ZhouJB12 fatcat:vgqltkjhdnazjklmfdfrkltwbq

Systematic Error-Correcting Codes for Rank Modulation

Hongchao Zhou, Moshe Schwartz, Anxiao Andrew Jiang, Jehoshua Bruck
2015 IEEE Transactions on Information Theory  
We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes.  ...  We study systematic codes for rank modulation under Kendall's τ -metric as well as under the ∞ -metric.  ...  For a decoding procedure, map the received permutation to its factoradic representation, and use the decoding for the Lee-metric code (essentially, a BCH decoding procedure) given in [24] .  ... 
doi:10.1109/tit.2014.2365499 fatcat:gfdetsgmfrdstkwwbgyxzsdc7a

The Art of Signaling: Fifty Years of Coding Theory [chapter]

2009 Information Theory  
The proper allocation of these resources to the different obstacles is fertile ground for information theory and coding, but for the past 50 years the focus of coding theory in particular has been reliable  ...  Now there were codes before there was a theory of coding, and the mathematical framework for decoding certain algebraic codes (Bose-Chaudhuri-Hocquengham (BCH) codes) was written down in the late 18th  ...  ACKNOWLEDGMENT The author is grateful to Peter Cameron for information on Fisher and the Hamming codes, to Jim Reeds for historical detail on telegraph codebooks, to Alexander Vardy for the statistics  ... 
doi:10.1109/9780470544907.ch20 fatcat:eeuyqk35orhmdleeiktamxgv6u

Graphs, Tessellations, and Perfect Codes on Flat Tori

S.I.R. Costa, M. Muniz, E. Agustini, R. Palazzo
2004 IEEE Transactions on Information Theory  
Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney [13].  ...  We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori.  ...  Battail and the referees for pertinent comments and fruitful suggestions.  ... 
doi:10.1109/tit.2004.834754 fatcat:cyv7n2ospbaclebe2kdzjvvdmq

The art of signaling: fifty years of coding theory

A.R. Calderbank
1998 IEEE Transactions on Information Theory  
The proper allocation of these resources to the different obstacles is fertile ground for information theory and coding, but for the past 50 years the focus of coding theory in particular has been reliable  ...  Now there were codes before there was a theory of coding, and the mathematical framework for decoding certain algebraic codes (Bose-Chaudhuri-Hocquengham (BCH) codes) was written down in the late 18th  ...  ACKNOWLEDGMENT The author is grateful to Peter Cameron for information on Fisher and the Hamming codes, to Jim Reeds for historical detail on telegraph codebooks, to Alexander Vardy for the statistics  ... 
doi:10.1109/18.720549 fatcat:utsr3jz3frexjezv3bznqkaxie
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