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Identity-based key aggregate cryptosystem from multilinear maps
Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results
1. | Applied Statistics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata-700108, India |
2. | Ashoka University, Sonepat, Haryana, India |
3. | School of Engineering and Mathematical Sciences, City University London, London EC1V 0HB, United Kingdom |
A matrix is MDS or super-regular if and only if every square submatrices of it are nonsingular. MDS matrices provide perfect diffusion in block ciphers and hash functions. In this paper we provide a brief survey on cryptographically significant MDS matrices - a first to the best of our knowledge. In addition to providing a summary of existing results, we make several contributions. We exhibit some deep and nontrivial interconnections between different constructions of MDS matrices. For example, we prove that all known Vandermonde constructions are basically equivalent to Cauchy constructions. We prove some folklore results which are used in MDS matrix literature. Wherever possible, we provide some simpler alternative proofs. We do not discuss efficiency issues or hardware implementations; however, the theory accumulated and discussed here should provide an easy guide towards efficient implementations.
References:
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C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in $GF(2^n)$ with applications to MDS matrices, Advances in cryptology–CRYPTO 2016. Part I, 625–653, Lecture Notes in Comput. Sci., 9814, Springer, Berlin, 2016.
doi: 10.1007/978-3-662-53018-4_23. |
[7] |
T. P. Berger, Construction of recursive MDS diffusion layers from gabidulin codes, In INDOCRYPT 2013, LNCS, Springer, 8250 (2013), 274–285.
doi: 10.1007/978-3-319-03515-4_18. |
[8] |
T. P. Berger and A. Ourivski, Construction of new MDS codes from Gabidulin codes, In Proceedings of ACCT 2009, Kranevo, Bulgaria, 40–47, June, 2004. Google Scholar |
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doi: 10.1006/jsco.1996.0125. |
[10] |
G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seeman, On repeated-root cyclic codes, In IEEE Transactions on Inform. Theory, 37 (1991), 337–342.
doi: 10.1109/18.75249. |
[11] |
V. Cauchois and P. Loidreau,
About circulant involutory MDS matrices, Des. Codes Cryptogr., 87 (2019), 249-260.
doi: 10.1007/s10623-018-0520-3. |
[12] |
J. Choy, H. Yap, K. Khoo, J. Guo, T. Peyrin, A. Poschmann and C. H. Tan, SPN-hash: Improving the provable resistance against differential collision attacks, Progress in cryptology–AFRICACRYPT 2012, 270–286, Lecture Notes in Comput. Sci., 7374, Springer, Heidelberg, 2012.
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Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications, Cryptography and Communications, 7 (2015), 257-287.
doi: 10.1007/s12095-014-0116-3. |
[25] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu, Towards a general construction of recursive MDS diffusion layers, In WCC2015, https://hal.inria.fr/hal-01276436. Google Scholar |
[26] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu, On the direct construction of recursive
MDS matrices, In Des. Codes Crypto., 82 (2017), 77–94.
doi: 10.1007/s10623-016-0233-4. |
[27] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu, Towards a general construction of recursive
MDS diffusion layers, In Des. Codes Crypto., 82 (2017), 179–195.
doi: 10.1007/s10623-016-0261-0. |
[28] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu,
Almost involutory recursive MDS diffusion layers, Design, Codes and Cryptography, 87 (2019), 609-626.
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show all references
References:
[1] |
D. Augot and M. Finiasz, Exhaustive search for small dimension recursive MDS diffusion layers or block ciphers and hash functions, In Proc. of the 2013 IEEE International Symposium on Information Theory, IEEE, 2013, 1551–1555.
doi: 10.1109/ISIT.2013.6620487. |
[2] |
D. Augot and M. Finiasz, Direct construction of recursive mds diffusion layers using shortened BCH codes, In FSE 2014, LNCS, Springer, 8540 (2015), 3–17, Also available http://eprint.iacr.org/2014/566.pdf.
doi: 10.1007/978-3-662-46706-0_1. |
[3] |
P. S. L. M. Barreto and V. Rijmen, The Khazad Legacy-Level Block Cipher, Submission to the NESSIE Project, 2000. Available at https://www.cosic.esat.kuleuven.be/nessie/workshop/submissions.html. Google Scholar |
[4] |
P. S. L. M. Barreto and V. Rijmen, The Anubis block cipher, Submission to the NESSIE Project, 2000. Available at https://www.cosic.esat.kuleuven.be/nessie/workshop/submissions.html. Google Scholar |
[5] |
P. S. L. M. Barreto and V. Rijmen, The Whirlpool hashing function, In Proceedings of the 1st NESSIE Workshop, 15 pages, 2000. Available at https://www.cosic.esat.kuleuven.be/nessie/workshop/submissions.html. Google Scholar |
[6] |
C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in $GF(2^n)$ with applications to MDS matrices, Advances in cryptology–CRYPTO 2016. Part I, 625–653, Lecture Notes in Comput. Sci., 9814, Springer, Berlin, 2016.
doi: 10.1007/978-3-662-53018-4_23. |
[7] |
T. P. Berger, Construction of recursive MDS diffusion layers from gabidulin codes, In INDOCRYPT 2013, LNCS, Springer, 8250 (2013), 274–285.
doi: 10.1007/978-3-319-03515-4_18. |
[8] |
T. P. Berger and A. Ourivski, Construction of new MDS codes from Gabidulin codes, In Proceedings of ACCT 2009, Kranevo, Bulgaria, 40–47, June, 2004. Google Scholar |
[9] |
W. Bosma, J. Cannon and C. Playoust, The magma algebra system I: The user language, J. Symbolic Comput, 24 (1997), 235–265, Computational algebra and number theory (London, 1993).
doi: 10.1006/jsco.1996.0125. |
[10] |
G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seeman, On repeated-root cyclic codes, In IEEE Transactions on Inform. Theory, 37 (1991), 337–342.
doi: 10.1109/18.75249. |
[11] |
V. Cauchois and P. Loidreau,
About circulant involutory MDS matrices, Des. Codes Cryptogr., 87 (2019), 249-260.
doi: 10.1007/s10623-018-0520-3. |
[12] |
J. Choy, H. Yap, K. Khoo, J. Guo, T. Peyrin, A. Poschmann and C. H. Tan, SPN-hash: Improving the provable resistance against differential collision attacks, Progress in cryptology–AFRICACRYPT 2012, 270–286, Lecture Notes in Comput. Sci., 7374, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-31410-0_17. |
[13] |
T. Cui, C. Jin and Z. Kong, On compact cauchy matrices for substitution-permutation networks, In IEEE Transactions on Computers, 64 (2015), 2098–2102.
doi: 10.1109/TC.2014.2346180. |
[14] |
J. Daemen, L. R. Knudsen and V. Rijmen, The block cipher SQUARE, In 4th Fast Software Encryption Workshop, LNCS, 1267 (1997), 149–165, Springer-Verlag.
doi: 10.1007/BFb0052343. |
[15] |
J. Daemen and V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard, Springer-Verlag, 2002.
doi: 10.1007/978-3-662-04722-4. |
[16] |
G. D. Filho, P. Barreto and V. Rijmen, The Maelstrom-0 Hash Function, In Proceedings of the 6th Brazilian Symposium on Information and Computer Systems Security, 2006. Google Scholar |
[17] |
P. Gauravaram, L. R. Knudsen, K. Matusiewicz, F. Mendel, C. Rechberger, M. Schlaffer and S. Thomsen, Gr$\phi$stl a SHA-3 Candidate, Submission to NIST, 2008, Available at http://www.groestl.info/. Google Scholar |
[18] |
R. M. Gray, Toeplitz and Circulant Matrices: A Review Foundations and Trends in Communications and Information Theory, NOW, 2005.
doi: 10.1561/0100000006. |
[19] |
J. Guo, T. Peyrin and A. Poschmann, The PHOTON family of lightweight hash functions, In CRYPTO, Springer, 2011 (2011), 222–239.
doi: 10.1007/978-3-642-22792-9_13. |
[20] |
J. Guo, T. Peyrin, A. Poshmann and M. J. B. Robshaw, The LED block cipher, In CHES 2011, LNCS, 6917 (2011), 326–341, Springer.
doi: 10.1007/978-3-642-23951-9_22. |
[21] |
K. C. Gupta and I. G. Ray, On constructions of involutory MDS matrices, Progress in cryptology–AFRICACRYPT 2013, 7918 (2013), 43–60.
doi: 10.1007/978-3-642-38553-7_3. |
[22] |
K. C. Gupta and I. G. Ray, On constructions of MDS matrices from companion matrices for lightweight cryptography, In CD-ARES 2013 Workshops: MoCrySEn, Springer, 8128 (2013), 29–43.
doi: 10.1007/978-3-642-40588-4_3. |
[23] |
K. C. Gupta and I. G. Ray, On constructions of circulant MDS matrices for lightweight cryptography, In ISPEC 2014, Springer, 2014, 564–576. Google Scholar |
[24] |
K. C. Gupta and I. G. Ray,
Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications, Cryptography and Communications, 7 (2015), 257-287.
doi: 10.1007/s12095-014-0116-3. |
[25] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu, Towards a general construction of recursive MDS diffusion layers, In WCC2015, https://hal.inria.fr/hal-01276436. Google Scholar |
[26] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu, On the direct construction of recursive
MDS matrices, In Des. Codes Crypto., 82 (2017), 77–94.
doi: 10.1007/s10623-016-0233-4. |
[27] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu, Towards a general construction of recursive
MDS diffusion layers, In Des. Codes Crypto., 82 (2017), 179–195.
doi: 10.1007/s10623-016-0261-0. |
[28] |
K. C. Gupta, S. K. Pandey and A. Venkateswarlu,
Almost involutory recursive MDS diffusion layers, Design, Codes and Cryptography, 87 (2019), 609-626.
doi: 10.1007/s10623-018-0582-2. |
[29] |
H. Han and H. Zhang, The research on the maximum branch number of P-permutations, 2010 2nd International Workshop on Intelligent Systems and Applications, Wuhan, 2010, 1–4.
doi: 10.1109/IWISA.2010.5473354. |
[30] |
H. M. Heys and S. E. Tavares, The design of substitution-permutation networks resistant to differential and linear cryptanalysis, Proceedings of 2nd ACM Conference on Computer and Communications Security, Fairfax, Virginia, 1994, 148–155.
doi: 10.1145/191177.191206. |
[31] |
H. M. Heys and S. E. Tavares,
Avalanche characteristics of substitution-permutation encryption networks, IEEE Trans. Comp., 44 (1995), 1131-1139.
doi: 10.1109/12.464391. |
[32] |
H. M. Heys and S. E. Tavares,
The design of product ciphers resistant to differential and linear cryptanalysis, Journal of Cryptology, 9 (1996), 1-19.
doi: 10.1007/BF02254789. |
[33] |
J. W. P. Hirschfeld, The main conjecture for MDS codes, Cryptography and Coding (Cirencester, 1995),, Lecture Notes in Comput. Sci., 1025 (1995), 44–52.
doi: 10.1007/3-540-60693-9_7. |
[34] |
P. Junod and S. Vaudenay, Perfect diffusion primitives for block ciphers building efficient MDS matrices, Selected Areas in Cryptography, 84–99, Lecture Notes in Comput. Sci., 3357, Springer, Berlin, 2005.
doi: 10.1007/978-3-540-30564-4_6. |
[35] |
P. Junod and S. Vaudenay, FOX: A new family of block ciphers, Selected Areas in Cryptography, 114–129, Lecture Notes in Comput. Sci., 3357, Springer, Berlin, 2005.
doi: 10.1007/978-3-540-30564-4_8. |
[36] |
P. Junod and M. Macchetti, Revisiting the IDEA philosophy, International Workshop on Fast Software Encryption, FSE 2009: Fast Software Encryption, Lecture Notes in Computer Science, 5665 (2009), 277–295.
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Construction Type | Vandermonde based Construction |
Cauchy based Construction |
Type 1: No extra condition | 1. Need not be involutory 2. Need not be Hadamard 3. Need not be compact |
1. Need not be involutory 2. Need not be Hadamard 3. Need not be compact |
Type 2: |
1. Involutory and equal 2. Need not be Hadamard 3. Need not be compact |
1. Need not be involutory, whereas 2. Need not be Hadamard 3. Need not be compact |
Type 3: |
1. Involutory and equal 2. Need not be Hadamard 3. Compact |
1. Need not be involutory, whereas 2. Need not be Hadamard 3. Compact |
Type 4: |
1. Involutory and equal 2. Hadamard 3. Compact |
1. Need not be involutory, whereas 2. Hadamard 3. Compact |
Construction Type | Vandermonde based Construction |
Cauchy based Construction |
Type 1: No extra condition | 1. Need not be involutory 2. Need not be Hadamard 3. Need not be compact |
1. Need not be involutory 2. Need not be Hadamard 3. Need not be compact |
Type 2: |
1. Involutory and equal 2. Need not be Hadamard 3. Need not be compact |
1. Need not be involutory, whereas 2. Need not be Hadamard 3. Need not be compact |
Type 3: |
1. Involutory and equal 2. Need not be Hadamard 3. Compact |
1. Need not be involutory, whereas 2. Need not be Hadamard 3. Compact |
Type 4: |
1. Involutory and equal 2. Hadamard 3. Compact |
1. Need not be involutory, whereas 2. Hadamard 3. Compact |
Type | Dimension | Involutory MDS | Orthogonal MDS |
Circulant | do not exist | do not exist | |
do not exist | may exist (Remark 24) | ||
do not exist | may exist (Remark 24) | ||
Type-Ⅰ | do not exist | do not exist | |
do not exist | do not exist | ||
Type-Ⅱ | do not exist | do not exist | |
may exist (Example 9) | may exist | ||
left-Circulant | do not exist | do not exist | |
may exist (Remark 31) | may exist (Remark 31) | ||
may exist (Remark 31) | may exist (Remark 31) | ||
Toeplitz | do not exist | do not exist | |
do not exist | may exist (Remark 34) | ||
do not exist | may exist (Remark 34) | ||
Hankel | do not exist | do not exist | |
may exist (Remark 36) | may exist (Remark 36) | ||
may exist (Remark 36) | may exist (Remark 36) |
Type | Dimension | Involutory MDS | Orthogonal MDS |
Circulant | do not exist | do not exist | |
do not exist | may exist (Remark 24) | ||
do not exist | may exist (Remark 24) | ||
Type-Ⅰ | do not exist | do not exist | |
do not exist | do not exist | ||
Type-Ⅱ | do not exist | do not exist | |
may exist (Example 9) | may exist | ||
left-Circulant | do not exist | do not exist | |
may exist (Remark 31) | may exist (Remark 31) | ||
may exist (Remark 31) | may exist (Remark 31) | ||
Toeplitz | do not exist | do not exist | |
do not exist | may exist (Remark 34) | ||
do not exist | may exist (Remark 34) | ||
Hankel | do not exist | do not exist | |
may exist (Remark 36) | may exist (Remark 36) | ||
may exist (Remark 36) | may exist (Remark 36) |
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