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A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets
Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices
1. | Department of Applied Mathematics, Malek Ashtar University of Technology, Isfahan, Iran |
2. | Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran |
3. | Department of Electrical & Computer Engineering, University of Victoria, Victoria, BC, Canada |
One of the best methods for constructing maximum distance separable ($ \operatorname{MDS} $) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of $ \operatorname{MDS} $ matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce $ 2n \times 2n $ involutory $ \operatorname{MDS} $ matrices in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct multiple $ \operatorname{MDS} $ matrices which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as Cauchy-light matrices. The introduced Cauchy-light matrices are applied to construct $ n \times n $ $ \operatorname{MDS} $ matrices having at least $ 3n-3 $ entries equal to the unit element $ 1 $; such a matrix is called a lightweight $ \operatorname{MDS} $ matrix and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.
References:
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D. Augot and M. Finiasz, Direct construction of recursive MDS diffusion layers using shortened BCH codes, in Fast Software Encryption. FSE 2014, Vol. 8540, Springer, Berlin, Heidelberg, 2014, 3-17.
doi: 10.1007/978-3-662-46706-0_1. |
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C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in $GF(2^n)$ with applications to MDS matrices, in Advances in Cryptology. CRYPTO 2016. Part 1, Lecture Notes in Comput. Sci., Vol. 9814, Springer, Berlin, 2016,625-653.
doi: 10.1007/978-3-662-53018-4_23. |
[5] |
T. P. Berger, G. Paul and S. Vaudenay, eds., Construction of recursive MDS diffusion layers from gabidulin codes, in Progress in Cryptology. INDOCRYPT 2013, Vol. 8250, Springer, Cham, 2013,274-285.
doi: 10.1007/978-3-319-03515-4. |
[6] |
J. Daemen and V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04722-4. |
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G. Filho, P. Barreto and V. Rijmen, The Maelstrom-$0$ hash function, in Proceedings of the Sixth Brazilian Symposium on Information and Computer Systems Security, (2006). Google Scholar |
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J. Guo, T. Peyrin and A. Poschmann, The PHOTON family of lightweight hash functions, in Advances in Cryptology. CRYPTO 2011, Vol. 6841, Springer, Heidelberg, 2011,222-239.
doi: 10.1007/978-3-642-22792-9. |
[9] |
K. C. Gupta and I. G. Ray, On constructions of MDS matrices from companion matrices for lightweight cryptography, in Security Engineering and Intelligence Informatics. CD-ARES 2013, Vol. 8128, Springer, Berlin, Heidelberg, 2013, 29-43.
doi: 10.1007/978-3-642-40588-4_3. |
[10] |
K. C. Gupta and I. G. Ray, On constructions of involutory MDS matrices, in Progress in Cryptology. AFRICACRYPT 2013, Vol. 7918, Springer, Heidelberg, 2013, 43-60.
doi: 10.1007/978-3-642-38553-7_3. |
[11] |
K. C. Gupta and I. G. Ray,
Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications, Cryptogr. Commun., 7 (2015), 257-287.
doi: 10.1007/s12095-014-0116-3. |
[12] |
K. C. Gupta, S. K. Pandey, I. G. Ray and S. Samanta,
Cryptographically significant MDS matrices over finite fields: A brief survey and some generalized results, Adv. Math. Commun., 13 (2019), 779-843.
doi: 10.3934/amc.2019045. |
[13] |
H. Hou and S. Y. Han,
A new construction and an efficient decoding method for Rabin-like codes, IEEE Transactions on Communications, 66 (2018), 521-533.
doi: 10.1109/TCOMM.2017.2766140. |
[14] |
P. Junod and S. Vaudenay, Perfect diffusion primitives for block ciphers building efficient MDS matrices, in Selected Areas in Cryptography. SAC 2004, Vol. 3357, Springer, Berlin, 2005, 84-99.
doi: 10.1007/978-3-540-30564-4_6. |
[15] |
K. Khoo, T. Peyrin, A. Y. Poschmann and H. Yap, FOAM: Searching for hardware-optimal SPN structures and components with a fair comparison, in Cryptographic Hardware and Embedded Systems. CHES 2014, Vol. 8731, Springer, Berlin, Heidelberg, 2014,433-450.
doi: 10.1007/978-3-662-44709-3_24. |
[16] |
L. Kölsch, XOR-counts and lightweight multiplication with fixed elements in binary finite fields, in Advances in Cryptology. EUROCRYPT 2019, Vol. 11476, Springer, Cham, 2019,285-312.
doi: 10.1007/978-3-030-17653-2_10. |
[17] |
H. Kranz, G. Leander, K. Stoffelen and F. Wiemer, Shorter linear straight-line programs for MDS matrices, IACR Transactions on Symmetric Cryptology, 2017 (2017), 188-211. Google Scholar |
[18] |
J. Lacan and J. Fimes,
Systematic MDS erasure codes based on Vandermonde matrices, IEEE Communications Letters, 8 (2004), 570-572.
doi: 10.1109/LCOMM.2004.833807. |
[19] |
S. Li, S. Sun, C. Li, Z. Wei and L. Hu, Constructing low-latency involutory MDS matrices with lightweight circuits, IACR Transactions on Symmetric Cryptology, 2019 (2019), 84-117. Google Scholar |
[20] |
M. Liu and S. M. Sim, Lightweight MDS generalized circulant matrices, in Fast Software Encryption. FSE 2016, Vol. 9783, Springer, Berlin, Heidelberg, 2016,101-120.
doi: 10.1007/978-3-662-52993-5_6. |
[21] |
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, The Clarendon Press, Oxford University Press, New York, 1995. |
[22] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes II, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[23] |
C. Paar, Optimized arithmetic for Reed-Solomon encoders, in Proceedings of IEEE International Symposium on Information Theory, Ulm, Germany, (1997), 250-250.
doi: 10.1109/ISIT.1997.613165. |
[24] |
R. M. Roth and G. Seroussi,
On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830.
doi: 10.1109/TIT.1985.1057113. |
[25] |
R. M. Roth and A. Lempel,
On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory, 35 (1989), 1314-1319.
doi: 10.1109/18.45291. |
[26] |
M. Sajadieh, M. Dakhilalian, H. Mala and B. Omoomi,
On construction of involutory MDS matrices from Vandermonde Matrices in GF($2^q$), Des. Codes Cryptogr., 64 (2012), 287-308.
doi: 10.1007/s10623-011-9578-x. |
[27] |
C. Schindelhauer and C. Ortolf, Maximum distance separable codes based on circulant Cauchy matrices, in Structural Information and Communication Complexity. SIROCCO 2013, Vol. 8179, Springer, Cham, 2013,334-345.
doi: 10.1007/978-3-319-03578-9_28. |
[28] |
C. E. Shannon,
Communication theory of secrecy systems, Bell System Tech. J., 28 (1949), 656-715.
doi: 10.1002/j.1538-7305.1949.tb00928.x. |
[29] |
T. Shirai and K. Shibutani, Improving immunity of feistel ciphers against differential cryptanalysis by using multiple MDS matrices, in Fast Software Encryption. FSE 2004, Vol. 3017, Springer, Berlin, Heidelberg, 2004,260-278.
doi: 10.1007/978-3-540-25937-4_17. |
[30] |
S. M. Sim, K. Khoo, F. Oggier and T. Peyrin, Lightweight MDS involution matrices, in Fast Software Encryption. FSE 2015, Vol. 9054, Springer, Berlin, Heidelberg, 2015,471-493.
doi: 10.1007/978-3-662-48116-5_23. |
[31] |
J. R. Stembridge,
A concise proof of the Littlewood-Richardson rule, Electron. J. Combin., 9 (2002), 1-4.
doi: 10.37236/1666. |
[32] |
S. Wu, M. Wang and W. Wu, Recursive diffusion layers for (lightweight) block ciphers and hash functions, in Selected Areas in Cryptography. SAC 2012, Vol. 7707, Springer, Heidelberg, 2012,355-371.
doi: 10.1007/978-3-642-35999-6. |
show all references
References:
[1] |
D. Augot and M. Finiasz, Direct construction of recursive MDS diffusion layers using shortened BCH codes, in Fast Software Encryption. FSE 2014, Vol. 8540, Springer, Berlin, Heidelberg, 2014, 3-17.
doi: 10.1007/978-3-662-46706-0_1. |
[2] |
P. Barreto and V. Rijmen, The Khazad legacy-level block cipher, in Proceedings of the First Open NESSIE Workshop, Belgium, (2000). Google Scholar |
[3] |
P. Barreto and V. Rijmen, The Anubis block cipher, in Proceedings of the First Open NESSIE Workshop, Belgium, (2000). Google Scholar |
[4] |
C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in $GF(2^n)$ with applications to MDS matrices, in Advances in Cryptology. CRYPTO 2016. Part 1, Lecture Notes in Comput. Sci., Vol. 9814, Springer, Berlin, 2016,625-653.
doi: 10.1007/978-3-662-53018-4_23. |
[5] |
T. P. Berger, G. Paul and S. Vaudenay, eds., Construction of recursive MDS diffusion layers from gabidulin codes, in Progress in Cryptology. INDOCRYPT 2013, Vol. 8250, Springer, Cham, 2013,274-285.
doi: 10.1007/978-3-319-03515-4. |
[6] |
J. Daemen and V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04722-4. |
[7] |
G. Filho, P. Barreto and V. Rijmen, The Maelstrom-$0$ hash function, in Proceedings of the Sixth Brazilian Symposium on Information and Computer Systems Security, (2006). Google Scholar |
[8] |
J. Guo, T. Peyrin and A. Poschmann, The PHOTON family of lightweight hash functions, in Advances in Cryptology. CRYPTO 2011, Vol. 6841, Springer, Heidelberg, 2011,222-239.
doi: 10.1007/978-3-642-22792-9. |
[9] |
K. C. Gupta and I. G. Ray, On constructions of MDS matrices from companion matrices for lightweight cryptography, in Security Engineering and Intelligence Informatics. CD-ARES 2013, Vol. 8128, Springer, Berlin, Heidelberg, 2013, 29-43.
doi: 10.1007/978-3-642-40588-4_3. |
[10] |
K. C. Gupta and I. G. Ray, On constructions of involutory MDS matrices, in Progress in Cryptology. AFRICACRYPT 2013, Vol. 7918, Springer, Heidelberg, 2013, 43-60.
doi: 10.1007/978-3-642-38553-7_3. |
[11] |
K. C. Gupta and I. G. Ray,
Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications, Cryptogr. Commun., 7 (2015), 257-287.
doi: 10.1007/s12095-014-0116-3. |
[12] |
K. C. Gupta, S. K. Pandey, I. G. Ray and S. Samanta,
Cryptographically significant MDS matrices over finite fields: A brief survey and some generalized results, Adv. Math. Commun., 13 (2019), 779-843.
doi: 10.3934/amc.2019045. |
[13] |
H. Hou and S. Y. Han,
A new construction and an efficient decoding method for Rabin-like codes, IEEE Transactions on Communications, 66 (2018), 521-533.
doi: 10.1109/TCOMM.2017.2766140. |
[14] |
P. Junod and S. Vaudenay, Perfect diffusion primitives for block ciphers building efficient MDS matrices, in Selected Areas in Cryptography. SAC 2004, Vol. 3357, Springer, Berlin, 2005, 84-99.
doi: 10.1007/978-3-540-30564-4_6. |
[15] |
K. Khoo, T. Peyrin, A. Y. Poschmann and H. Yap, FOAM: Searching for hardware-optimal SPN structures and components with a fair comparison, in Cryptographic Hardware and Embedded Systems. CHES 2014, Vol. 8731, Springer, Berlin, Heidelberg, 2014,433-450.
doi: 10.1007/978-3-662-44709-3_24. |
[16] |
L. Kölsch, XOR-counts and lightweight multiplication with fixed elements in binary finite fields, in Advances in Cryptology. EUROCRYPT 2019, Vol. 11476, Springer, Cham, 2019,285-312.
doi: 10.1007/978-3-030-17653-2_10. |
[17] |
H. Kranz, G. Leander, K. Stoffelen and F. Wiemer, Shorter linear straight-line programs for MDS matrices, IACR Transactions on Symmetric Cryptology, 2017 (2017), 188-211. Google Scholar |
[18] |
J. Lacan and J. Fimes,
Systematic MDS erasure codes based on Vandermonde matrices, IEEE Communications Letters, 8 (2004), 570-572.
doi: 10.1109/LCOMM.2004.833807. |
[19] |
S. Li, S. Sun, C. Li, Z. Wei and L. Hu, Constructing low-latency involutory MDS matrices with lightweight circuits, IACR Transactions on Symmetric Cryptology, 2019 (2019), 84-117. Google Scholar |
[20] |
M. Liu and S. M. Sim, Lightweight MDS generalized circulant matrices, in Fast Software Encryption. FSE 2016, Vol. 9783, Springer, Berlin, Heidelberg, 2016,101-120.
doi: 10.1007/978-3-662-52993-5_6. |
[21] |
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, The Clarendon Press, Oxford University Press, New York, 1995. |
[22] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes II, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[23] |
C. Paar, Optimized arithmetic for Reed-Solomon encoders, in Proceedings of IEEE International Symposium on Information Theory, Ulm, Germany, (1997), 250-250.
doi: 10.1109/ISIT.1997.613165. |
[24] |
R. M. Roth and G. Seroussi,
On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830.
doi: 10.1109/TIT.1985.1057113. |
[25] |
R. M. Roth and A. Lempel,
On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory, 35 (1989), 1314-1319.
doi: 10.1109/18.45291. |
[26] |
M. Sajadieh, M. Dakhilalian, H. Mala and B. Omoomi,
On construction of involutory MDS matrices from Vandermonde Matrices in GF($2^q$), Des. Codes Cryptogr., 64 (2012), 287-308.
doi: 10.1007/s10623-011-9578-x. |
[27] |
C. Schindelhauer and C. Ortolf, Maximum distance separable codes based on circulant Cauchy matrices, in Structural Information and Communication Complexity. SIROCCO 2013, Vol. 8179, Springer, Cham, 2013,334-345.
doi: 10.1007/978-3-319-03578-9_28. |
[28] |
C. E. Shannon,
Communication theory of secrecy systems, Bell System Tech. J., 28 (1949), 656-715.
doi: 10.1002/j.1538-7305.1949.tb00928.x. |
[29] |
T. Shirai and K. Shibutani, Improving immunity of feistel ciphers against differential cryptanalysis by using multiple MDS matrices, in Fast Software Encryption. FSE 2004, Vol. 3017, Springer, Berlin, Heidelberg, 2004,260-278.
doi: 10.1007/978-3-540-25937-4_17. |
[30] |
S. M. Sim, K. Khoo, F. Oggier and T. Peyrin, Lightweight MDS involution matrices, in Fast Software Encryption. FSE 2015, Vol. 9054, Springer, Berlin, Heidelberg, 2015,471-493.
doi: 10.1007/978-3-662-48116-5_23. |
[31] |
J. R. Stembridge,
A concise proof of the Littlewood-Richardson rule, Electron. J. Combin., 9 (2002), 1-4.
doi: 10.37236/1666. |
[32] |
S. Wu, M. Wang and W. Wu, Recursive diffusion layers for (lightweight) block ciphers and hash functions, in Selected Areas in Cryptography. SAC 2012, Vol. 7707, Springer, Heidelberg, 2012,355-371.
doi: 10.1007/978-3-642-35999-6. |
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