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Information set decoding of Lee-metric codes over finite rings [article]

Violetta Weger, Massimo Battaglioni, Paolo Santini, Franco Chiaraluce, Marco Baldi, Edoardo Persichetti
2021 arXiv   pre-print
In this paper we switch from the Hamming metric to the Lee metric, and study ISD algorithms and their complexity for codes measured with the Lee metric over finite rings.  ...  Information set decoding (ISD) algorithms are the best known procedures to solve the decoding problem for general linear codes.  ...  Conclusion We generalized Prange's and Stern's ISD algorithms considering codes over finite rings endowed with the Lee metric. We analyzed the complexity of these ISD algorithms.  ... 
arXiv:2001.08425v2 fatcat:fevcgoge6vaujhafzjnqswtklm

Page 729 of Mathematical Reviews Vol. , Issue 99a [page]

1991 Mathematical Reviews  
The aim of the paper is to find properties of the Lee metric, which are important for constructing nonlinear binary codes from linear codes over rings.  ...  Some properties of the Lee metric are not necessary for the representation of non- linear codes by linear codes over rings. For example, the triangle inequality is not required.  ... 

Page 5701 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
Tonchev (1-MITU; Houghton, MI) 2003g:94059 94B35 Byrne, Eimear (IRL-MNTH; Maynooth) Decoding a class of Lee metric codes over a Galois ring. (English summary) IEEE Trans. Inform.  ...  The author proves an extension theorem for linear codes over a finite quasi-Frobenius module, namely: Let rgQ be a quasi- Frobenius module over a finite commutative ring R with | and K Cr Q be a linear  ... 

Page 5064 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
The author introduces the notion of a relatively self-dual code over a finite commutative ring. He develops a gluing technique for the construction of such codes.  ...  (IL-TECH-C; Haifa); Siegel, Paul H. (1-IBM2; San Jose, CA) Lee-metric BCH codes and their application to constrained and channels. (English summary) IEEE Trans. Inform.  ... 

Page 3313 of Mathematical Reviews Vol. , Issue 80H [page]

1980 Mathematical Reviews  
Considering Ax, the set of all n-tuples over A, as a module over A, the paper first exhibits Lee metric codes generated by (i) a single vector u=(1,2,--- , N—1) for N an odd prime or N =2', i an integer  ...  Using finite ring theory, a method for constructing BCH-type cyclic codes over the ring Z,. of integers modulo p*, p a prime, is given in terms of their generator polynomial.  ... 

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.  ...  Information set decoding (ISD) algorithms are the main method for decoding random linear codes in the Hamming metric, whenever the problem has only a few solutions.  ... 
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.  ...  Information set decoding (ISD) algorithms are the main method for decoding random linear codes in the Hamming metric, whenever the problem has only a few solutions.  ... 
doi:10.3934/amc.2020089 fatcat:nlnyopxblffcnn2ngpdlqdfrjq

On the Properties of Error Patterns in the Constant Lee Weight Channel [article]

Jessica Bariffi, Hannes Bartz, Gianluigi Liva, Joachim Rosenthal
2022 arXiv   pre-print
This problem is of particular interest for cryptographic applications, like for example Lee metric code-based cryptosystems, since an attacker may use scalar multiplication to reduce the Lee weight of  ...  The problem of scalar multiplication applied to vectors is considered in the Lee metric.  ...  New families of error correcting codes endowed with the Lee metric together with an iterative decoding algorithm were proposed [12] while information set decoding (ISD) in the Lee metric has been analyzed  ... 
arXiv:2110.01878v2 fatcat:pezyaqdxwrgxra5vvqziorvo7i

On the Rank Decoding Problem Over Finite Principal Ideal Rings [article]

Hervé Tale Kalachi, Hermann Tchatchiem Kamche
2022 arXiv   pre-print
But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings  ...  This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields.  ...  metric codes over finite rings.  ... 
arXiv:2106.11569v3 fatcat:n6hoaffotrfgbid4lsrmdlz6pm

List Decoding of Quaternary Codes in the Lee Metric [article]

Marcus Greferath, Jens Zumbrägel
2022 arXiv   pre-print
We present a list decoding algorithm for quaternary negacyclic codes over the Lee metric.  ...  To achieve this result, we use a Sudan-Guruswami type list decoding algorithm for Reed-Solomon codes over certain ring alphabets.  ...  Codes over integer residue rings equipped with the Lee metric have currently received increasing attention in the community.  ... 
arXiv:2202.03977v1 fatcat:vigflvxyxnbtxagryedrpawabm

Page 2469 of Mathematical Reviews Vol. , Issue 81F [page]

1981 Mathematical Reviews  
The reviewer and the author earlier introduced a general way of defining a metric between n-vectors over the ring Z,, the ring of integers mod q.  ...  This general method of defining distance will give, in particular, the Hamming distance and the Lee distance, the two widely used metrics in coding, for partitions: P= P,, = { Bo, B,}, B, = {1,2,---,q—  ... 

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.  ...  Then, we study the complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite fields to the Lee metric over finite rings, as well as proposing some novel  ...  Coding Theory in the Lee Metric. In this section we recall the definitions and main properties of linear codes over finite rings endowed with the Lee metric, see also [8] .  ... 
arXiv:2002.12785v5 fatcat:xrtv6beys5h4hl5k7rmc5smfuu

Page 2800 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews  
These equations are lin- ear Diophantine equations over the ring of polynomials F(D) with coefficients in a finite field. A Viterbi-like algorithm is developed.  ...  The authors define a general class of metrics which includes the Hamming and Lee metrics.  ... 

Page 2244 of Mathematical Reviews Vol. , Issue 87d [page]

1987 Mathematical Reviews  
From the summary: “We present several methods of constructing codes for the Lee metric.  ...  Some of the methods are analogues of those used to construct codes for the Hamming metric while others take the special properties of the Lee metric into account.” 87d:94038 Knuth, Donald E. (1-STF-C)  ... 

Page 684 of Mathematical Reviews Vol. 56, Issue 2 [page]

1978 Mathematical Reviews  
Information and Control 35 (1977), no. 1, 48-51. A code is defined as an ideal in RG, the group ring of the finite Abelian group G over R, a commutative ring with identity.  ...  B (N.S.) 12 (1976), no. 4, 365-390. 94 INFORMATION AND COMMUNICATION, CIRCUITS, AUTOMATA 56 #4/5044-5051 Author’s summary: “Given a subcalculable probability measure over the set of finite binary objects  ... 
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