Applications of design theory for the constructions of MDS matrices for lightweight cryptography
In this paper, we observe simple yet subtle interconnections among design theory, coding theory and cryptography. Maximum distance separable (MDS) matrices have applications not only in coding theory but are also of great importance in the design of block ciphers and hash functions. It is nontrivial...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2017-06-01
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| Series: | Journal of Mathematical Cryptology |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/jmc-2016-0013 |
| _version_ | 1811279445114224640 |
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| author | Gupta Kishan Chand Pandey Sumit Kumar Ray Indranil Ghosh |
| author_facet | Gupta Kishan Chand Pandey Sumit Kumar Ray Indranil Ghosh |
| author_sort | Gupta Kishan Chand |
| collection | DOAJ |
| description | In this paper, we observe simple yet subtle interconnections among design theory, coding theory and cryptography.
Maximum distance separable (MDS) matrices have applications not only in coding theory but are also
of great importance in the design of block ciphers and hash functions. It is nontrivial
to find MDS matrices which could be used in lightweight cryptography. In the SAC 2004 paper [12], Junod and Vaudenay considered bi-regular matrices which are useful objects to build MDS matrices. Bi-regular matrices are those matrices all of whose entries are nonzero and all of whose 2×2{2\times 2} submatrices are nonsingular. Therefore MDS matrices are bi-regular matrices, but the converse is not true. They proposed the constructions of efficient MDS matrices by studying
the two major aspects of a d×d{d\times d} bi-regular matrix M, namely v1(M){v_{1}(M)}, i.e. the number of occurrences of 1 in M, and c1(M){c_{1}(M)}, i.e. the number of distinct elements in M other than 1. They calculated the maximum number of ones that can occur in a d×d{d\times d} bi-regular matrices, i.e. v1d,d{v_{1}^{d,d}} for d up to 8, but with their approach, finding v1d,d{v_{1}^{d,d}} for d≥9{d\geq 9} seems difficult. |
| first_indexed | 2024-04-13T00:54:45Z |
| format | Article |
| id | doaj.art-c75052ec444b4df2ad0c5b41e5652bed |
| institution | Directory Open Access Journal |
| issn | 1862-2976 1862-2984 |
| language | English |
| last_indexed | 2024-04-13T00:54:45Z |
| publishDate | 2017-06-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Journal of Mathematical Cryptology |
| spelling | doaj.art-c75052ec444b4df2ad0c5b41e5652bed2022-12-22T03:09:44ZengDe GruyterJournal of Mathematical Cryptology1862-29761862-29842017-06-011128511610.1515/jmc-2016-0013Applications of design theory for the constructions of MDS matrices for lightweight cryptographyGupta Kishan Chand0Pandey Sumit Kumar1Ray Indranil Ghosh2Applied Statistics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata700108, IndiaSchool of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, SingaporeApplied Statistics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata700108, IndiaIn this paper, we observe simple yet subtle interconnections among design theory, coding theory and cryptography. Maximum distance separable (MDS) matrices have applications not only in coding theory but are also of great importance in the design of block ciphers and hash functions. It is nontrivial to find MDS matrices which could be used in lightweight cryptography. In the SAC 2004 paper [12], Junod and Vaudenay considered bi-regular matrices which are useful objects to build MDS matrices. Bi-regular matrices are those matrices all of whose entries are nonzero and all of whose 2×2{2\times 2} submatrices are nonsingular. Therefore MDS matrices are bi-regular matrices, but the converse is not true. They proposed the constructions of efficient MDS matrices by studying the two major aspects of a d×d{d\times d} bi-regular matrix M, namely v1(M){v_{1}(M)}, i.e. the number of occurrences of 1 in M, and c1(M){c_{1}(M)}, i.e. the number of distinct elements in M other than 1. They calculated the maximum number of ones that can occur in a d×d{d\times d} bi-regular matrices, i.e. v1d,d{v_{1}^{d,d}} for d up to 8, but with their approach, finding v1d,d{v_{1}^{d,d}} for d≥9{d\geq 9} seems difficult.https://doi.org/10.1515/jmc-2016-0013bibdbi-regular matrixdesigndiffusionlatin squaremds matrixmixcolumn operation68r05 94b99 |
| spellingShingle | Gupta Kishan Chand Pandey Sumit Kumar Ray Indranil Ghosh Applications of design theory for the constructions of MDS matrices for lightweight cryptography Journal of Mathematical Cryptology bibd bi-regular matrix design diffusion latin square mds matrix mixcolumn operation 68r05 94b99 |
| title | Applications of design theory for the constructions of MDS matrices for lightweight cryptography |
| title_full | Applications of design theory for the constructions of MDS matrices for lightweight cryptography |
| title_fullStr | Applications of design theory for the constructions of MDS matrices for lightweight cryptography |
| title_full_unstemmed | Applications of design theory for the constructions of MDS matrices for lightweight cryptography |
| title_short | Applications of design theory for the constructions of MDS matrices for lightweight cryptography |
| title_sort | applications of design theory for the constructions of mds matrices for lightweight cryptography |
| topic | bibd bi-regular matrix design diffusion latin square mds matrix mixcolumn operation 68r05 94b99 |
| url | https://doi.org/10.1515/jmc-2016-0013 |
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