Skip to main content
Top

27-04-2024 | Original Paper

On circulant involutory and orthogonal MDS matrices over finite commutative rings

Authors: Shakir Ali, Atif Ahmad Khan, Bhupendra Singh

Published in: Applicable Algebra in Engineering, Communication and Computing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Let \(k>1\) be a fixed integer. In Gupta and Ray (Cryptography and Communications 7: 257–287, 2015), proved the non existence of \(2^k \times 2^k\) orthogonal circulant MDS matrices and involutory circulant MDS matrices over finite fields of characteristic 2. The main aim of this paper is to prove the non-existence of orthogonal circulant MDS matrices of order \(2^k\times 2^k\) and involutory circulant MDS matrices of order k over finite commutative rings of characteristic 2. Precisely, we prove that any circulant orthogonal matrix of order \(2^k\) over finite commutative rings of characteristic 2 with identity is not a MDS matrix. Moreover, some related results are also discussed. Finally, we provide some examples to prove that the assumed restrictions on our main results are not superfluous.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Adhiguna, I., Arifin, I.S.N., Yuliawan, F., Muchtadi-Alamsyah, I.: On orthogonal circulant MDS matrices. Int. J. Math. Comput. Sci. 17(4), 1619–1637 (2022)MathSciNet Adhiguna, I., Arifin, I.S.N., Yuliawan, F., Muchtadi-Alamsyah, I.: On orthogonal circulant MDS matrices. Int. J. Math. Comput. Sci. 17(4), 1619–1637 (2022)MathSciNet
2.
go back to reference Atiya, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. CRC Press (1989) Atiya, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. CRC Press (1989)
3.
go back to reference Barreto, P., Rijmen, V.: The Khazad legacy-level block cipher. In: Primitive submitted to NESSIE, 97(106) (2000) Barreto, P., Rijmen, V.: The Khazad legacy-level block cipher. In: Primitive submitted to NESSIE, 97(106) (2000)
4.
5.
go back to reference Tan, C.H., Prabowo, T.F.: Orthogonal MDS diffusion matrices over Galois rings. In: IMA International Conference on Cryptography and Coding, pp. 307–330. Springer (2017) Tan, C.H., Prabowo, T.F.: Orthogonal MDS diffusion matrices over Galois rings. In: IMA International Conference on Cryptography and Coding, pp. 307–330. Springer (2017)
6.
go back to reference Daemen, J., Knudsen, L., Rijmen, V.: The block cipher Square. In Fast Software Encryption: 4th International Workshop, FSE’ 97, 4, pp. 149–165. Springer Berlin Heidelberg (1997) Daemen, J., Knudsen, L., Rijmen, V.: The block cipher Square. In Fast Software Encryption: 4th International Workshop, FSE’ 97, 4, pp. 149–165. Springer Berlin Heidelberg (1997)
7.
go back to reference Daemen, J., Rijmen, V.: The design of Rijndael: AES - The Advanced Encryption Standard. Springer (2002) Daemen, J., Rijmen, V.: The design of Rijndael: AES - The Advanced Encryption Standard. Springer (2002)
8.
go back to reference Dong, X.-D., Son, C.B., Gunawan, E.: Matrix characterization of MDS linear codes over modules. Linear Algebra Appl. 277(1–3), 57–61 (1998)MathSciNetCrossRef Dong, X.-D., Son, C.B., Gunawan, E.: Matrix characterization of MDS linear codes over modules. Linear Algebra Appl. 277(1–3), 57–61 (1998)MathSciNetCrossRef
9.
go back to reference Filho, G.D., Barreto, P., Rijmen, V.: The MAELSTROM-0 hash function. In: Proceedings of the 6th Brazilian Symposium on Information and Computer Systems Security (2006) Filho, G.D., Barreto, P., Rijmen, V.: The MAELSTROM-0 hash function. In: Proceedings of the 6th Brazilian Symposium on Information and Computer Systems Security (2006)
10.
go back to reference Gauravaram, P., Knudsen, L.R., Matusiewicz, K., Mendel, F., Rechberger, C., Schlaffer, M., Thomsen, S.: Gr\(\phi\)stl a SHA-3 Candidate. Submission to NIST (2009) Gauravaram, P., Knudsen, L.R., Matusiewicz, K., Mendel, F., Rechberger, C., Schlaffer, M., Thomsen, S.: Gr\(\phi\)stl a SHA-3 Candidate. Submission to NIST (2009)
11.
go back to reference Guo, J., Peyrin, T., Poschmann, A.: The PHOTON family of lightweight hash functions. In: Advances in Cryptology- CRYPTO,: pp. 222–239. Springer, Berlin Heidelberg (2011) Guo, J., Peyrin, T., Poschmann, A.: The PHOTON family of lightweight hash functions. In: Advances in Cryptology- CRYPTO,: pp. 222–239. Springer, Berlin Heidelberg (2011)
12.
go back to reference Gupta, K.C., Ray, I.G.: Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications. Cryptogr. Commun. 7(2), 257–287 (2015)MathSciNetCrossRef Gupta, K.C., Ray, I.G.: Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications. Cryptogr. Commun. 7(2), 257–287 (2015)MathSciNetCrossRef
13.
go back to reference Kesarwani, A., Pandey, S., Sarkar, S., Venkateswarlu, A.: Recursive MDS matrices over finite commutative rings. Discret. Appl. Math. 304, 384–396 (2021)MathSciNetCrossRef Kesarwani, A., Pandey, S., Sarkar, S., Venkateswarlu, A.: Recursive MDS matrices over finite commutative rings. Discret. Appl. Math. 304, 384–396 (2021)MathSciNetCrossRef
14.
go back to reference Liu, M., Sim, S.M.: Lightweight MDS generalized circulant matrices. In: International Conference on Fast Software Encryption FSE-2016, LNCS 9783, pp. 101–120 Liu, M., Sim, S.M.: Lightweight MDS generalized circulant matrices. In: International Conference on Fast Software Encryption FSE-2016, LNCS 9783, pp. 101–120
15.
go back to reference Nakahara, J., Jr., Abrahao, E.: A new involutory MDS matrix for the AES. Int. J. Netw. Secur. 9(2), 109–116 (2009) Nakahara, J., Jr., Abrahao, E.: A new involutory MDS matrix for the AES. Int. J. Netw. Secur. 9(2), 109–116 (2009)
16.
go back to reference Rijmen, V., Daemen, J., Preneel, B., Bosselaers, A., Win, E.D.: The cipher SHARK. In: International workshop on fast software encryption FSE-1996, LNCS 1039, pp. 99–112. Springer Rijmen, V., Daemen, J., Preneel, B., Bosselaers, A., Win, E.D.: The cipher SHARK. In: International workshop on fast software encryption FSE-1996, LNCS 1039, pp. 99–112. Springer
17.
go back to reference Schneier, B., Kelsey, J., Whiting, D., Wagner, D., Hall, C., Ferguson, N.: Twofish: A 128-bit block cipher. In: The First AES Candidate Conference. National Institute for Standards and Technology (1998) Schneier, B., Kelsey, J., Whiting, D., Wagner, D., Hall, C., Ferguson, N.: Twofish: A 128-bit block cipher. In: The First AES Candidate Conference. National Institute for Standards and Technology (1998)
18.
go back to reference Schnorr, C., Vaudenay, S.: Black box cryptanalysis of hash networks based on multipermutations. In: Advances in cryptology-EUROCRYPT’94, LNCS 950, pp. 47–57, Springer (1995) Schnorr, C., Vaudenay, S.: Black box cryptanalysis of hash networks based on multipermutations. In: Advances in cryptology-EUROCRYPT’94, LNCS 950, pp. 47–57, Springer (1995)
20.
go back to reference Sony Corporation: The 128-bit block cipher CLEFIA algorithm specification. In: International workshop on fast software encryption, FSE-2007, pp. 181–195 Sony Corporation: The 128-bit block cipher CLEFIA algorithm specification. In: International workshop on fast software encryption, FSE-2007, pp. 181–195
21.
go back to reference Steven, T.: Dougherty, Algebraic Coding Theory over Finite Commutative Rings. Springer (2017) Steven, T.: Dougherty, Algebraic Coding Theory over Finite Commutative Rings. Springer (2017)
23.
go back to reference Vaudenay, S.: On the need for multipermutations: cryptanalysis of MD4 and SAFER. In: Preneel, B. (ed.) Proceedings of LNCS Fast Software Encryption, 1008, pp. 286–297, Springer (1995) Vaudenay, S.: On the need for multipermutations: cryptanalysis of MD4 and SAFER. In: Preneel, B. (ed.) Proceedings of LNCS Fast Software Encryption, 1008, pp. 286–297, Springer (1995)
Metadata
Title
On circulant involutory and orthogonal MDS matrices over finite commutative rings
Authors
Shakir Ali
Atif Ahmad Khan
Bhupendra Singh
Publication date
27-04-2024
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-024-00656-4

Premium Partner