Cryptographic properties of cyclic binary matrices
Many modern symmetric ciphers apply MDS or almost MDS matrices as diffusion layers. The performance of a diffusion layer depends on its diffusion property measured by branch number and implementation cost which is usually measured by the number of XORs required. As the implementation cost of MDS mat...
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| Published in | Advances in mathematics of communications Vol. 15; no. 2; pp. 311 - 327 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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01.05.2021
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| ISSN | 1930-5338 1930-5346 |
| DOI | 10.3934/amc.2020068 |
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| Abstract | Many modern symmetric ciphers apply MDS or almost MDS matrices as diffusion layers. The performance of a diffusion layer depends on its diffusion property measured by branch number and implementation cost which is usually measured by the number of XORs required. As the implementation cost of MDS matrices of large dimensions is high, some symmetric ciphers use binary matrices as diffusion layers to trade-off efficiency versus diffusion property. In the current paper, we investigate cyclic binary matrices (CBMs for short), mathematically. Based upon this theorical study, we provide efficient matrices with provable lower bound on branch number and minimal number of fixed-points. We consider the product of sparse CBMs to construct efficiently implementable matrices with the desired cryptographic properties. |
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| AbstractList | Many modern symmetric ciphers apply MDS or almost MDS matrices as diffusion layers. The performance of a diffusion layer depends on its diffusion property measured by branch number and implementation cost which is usually measured by the number of XORs required. As the implementation cost of MDS matrices of large dimensions is high, some symmetric ciphers use binary matrices as diffusion layers to trade-off efficiency versus diffusion property. In the current paper, we investigate cyclic binary matrices (CBMs for short), mathematically. Based upon this theorical study, we provide efficient matrices with provable lower bound on branch number and minimal number of fixed-points. We consider the product of sparse CBMs to construct efficiently implementable matrices with the desired cryptographic properties. |
| Author | Rishakani, Akbar Mahmoodi Bagheri, Nasour Shamsabad, Mohmmadreza Mirzaee Dehnavi, Seyed Mojtaba |
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| Cites_doi | 10.1007/978-3-642-40349-1_9 10.1007/s00145-013-9163-8 10.1007/978-3-662-48800-3_17 10.1016/j.cam.2013.05.008 10.1002/sec.556 |
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| Keywords | Cyclic binary matrix matrix decomposition branch number symmetric cipher diffusion layer |
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Song, On constructing of a 32 $\times$32 binary matrix as a diffusion layer for a 256-bit block cipher, in Information Security and Cryptology - ICISC 2006, 9th International Conference, Busan, Korea, November 30 - December 1, 2006, Proceedings (eds. M. S. Rhee and B. Lee), vol. 4296 of Lecture Notes in Computer Science, Springer, 2006, 51-64. – reference: C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in gf(2^n) with applications to MDS matrices, in Advances in Cryptology - CRYPTO 2016 - 36th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 14-18, 2016, Proceedings, Part I, 9814 (2016), 625-653. – volume: 8086 start-page: 142 year: 2013 end-page: 158 ident: 6 article-title: FIDES: Lightweight authenticated cipher with side-channel resistance for constrained hardware publication-title: Cryptographic Hardware and Embedded Systems - CHES 2013 doi: 10.1007/978-3-642-40349-1_9 – reference: Z. Guo, R. Liu, S. Gao, W. Wu and D. 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Heys), vol. 2595 of Lecture Notes in Computer Science, Springer, 2003, 47-61. – reference: D. Kwon, J. Kim, S. Park, S. H. Sung, Y. Sohn, J. H. Song, Y. Yeom, E. Yoon, S. Lee, J. Lee, S. Chee, D. Han and J. Hong, New block cipher: ARIA, in Information Security and Cryptology - ICISC 2003, 6th International Conference, Seoul, Korea, November 27-28, 2003, Revised Papers (eds. J. I. Lim and D. H. Lee), vol. 2971 of Lecture Notes in Computer Science, Springer, 2003,432-445. – reference: S. Ling and C. Xing, Coding Theory: A First Course, Cambridge University Press, 2004, https://books.google.com/books?id=N1jiL8V3ISwC. – volume: 259 start-page: 485 year: 2014 end-page: 494 ident: 25 article-title: On the algebraic construction of cryptographically good 32${\times}$32 binary linear transformations publication-title: J. Computational Applied Mathematics doi: 10.1016/j.cam.2013.05.008 – reference: M. 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