# American Institute of Mathematical Sciences

May  2021, 15(2): 311-327. doi: 10.3934/amc.2020068

## Cryptographic properties of cyclic binary matrices

 1 Department of Sciences, Shahid Rajaee Teacher Training University, Tehran, Iran 2 Department of Mathematicsl and Computer Sciences, Kharazmi University, Tehran, Iran 3 Department of Mathematics and Computer Sciences, Shahid Beheshti University, Tehran, Iran 4 Electrical Engineering Department, Shahid Rajaee Teacher Training University, Tehran, Iran

* Corresponding author: Nasour Bagheri

Received  September 2019 Revised  December 2019 Published  January 2020

Fund Project: The fourth author is supported by Shahid Rajaee Teacher Training University

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.

Citation: Akbar Mahmoodi Rishakani, Seyed Mojtaba Dehnavi, Mohmmadreza Mirzaee Shamsabad, Nasour Bagheri. Cryptographic properties of cyclic binary matrices. Advances in Mathematics of Communications, 2021, 15 (2) : 311-327. doi: 10.3934/amc.2020068
##### References:
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Mala, Perfect involutory diffusion layers based on invertibility of some linear functions, IET Information Security, 5 (2011), 228-236. doi: 10.1049/iet-ifs.2010.0289.  Google Scholar [24] M. Sajadieh, M. Dakhilalian, H. Mala and P. Sepehrdad, Efficient recursive diffusion layers for block ciphers and hash functions, J. Cryptology, 28 (2015), 240-256.  doi: 10.1007/s00145-013-9163-8.  Google Scholar [25] M. T. Sakalli and B. Aslan, On the algebraic construction of cryptographically good 32${\times}$32 binary linear transformations, J. Computational Applied Mathematics, 259 (2014), 485-494.  doi: 10.1016/j.cam.2013.05.008.  Google Scholar [26] R. Schürer and W. C. Schmid, MinT: A Database for Optimal Net Parameters, In: Niederreiter H., Talay D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. Google Scholar [27] B. Schneier, J. Kelsey, D. Whiting, D. Wagner, C. Hall and N. Ferguson, Twofish: A 128-bit block cipher., Google Scholar

show all references

##### References:
 [1] Specification of the 3GPP confidentiality and integrity algorithms 128-EEA3 and 128-EIA3. document 2: ZUC specification, Online available at https://www.gsma.com/aboutus/wp-content/uploads/2014/12/eea3eia3zucv16.pdf. Google Scholar [2] K. Aoki, T. Ichikawa, M. Kanda, M. Matsui, S. Moriai, J. Nakajima and T. Tokita, Camellia: A 128-bit block cipher suitable for multiple platforms - design and analysis, in Selected Areas in Cryptography (Waterloo, ON, 2000), 39-56, Lecture Notes in Comput. Sci., 2012, Springer, Berlin, 2001. doi: 10.1007/3-540-44983-3_4.  Google Scholar [3] B. Aslan and M. T. Sakalli, Algebraic construction of cryptographically good binary linear transformations, Security and Communication Networks, 7 (2014), 53-63.  doi: 10.1002/sec.556.  Google Scholar [4] S. Banik, A. Bogdanov, T. Isobe, K. Shibutani, H. Hiwatari, T. Akishita and F. Regazzoni, Midori: A block cipher for low energy (extended version), Advances in Cryptology-ASIACRYPT 2015. Part Ⅱ, 9453 (2015), 411-436.  doi: 10.1007/978-3-662-48800-3_17.  Google Scholar [5] 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. doi: 10.1007/978-3-662-53018-4_23.  Google Scholar [6] B. Bilgin, A. Bogdanov, M. Knezevic, F. Mendel and Q. Wang, FIDES: Lightweight authenticated cipher with side-channel resistance for constrained hardware, Cryptographic Hardware and Embedded Systems - CHES 2013, 8086 (2013), 142-158.  doi: 10.1007/978-3-642-40349-1_9.  Google Scholar [7] J. Daemen and V. Rijmen, Rijndael for AES, in AES Candidate Conference, 2000,343-348. doi: 10.1007/0-387-23483-7_358.  Google Scholar [8] C. Dobraunig, M. Eichlseder, F. Mendel and M. Schl ffer, Ascon v1.2, Online available at http://competitions.cr.yp.to/round3/asconv12.pdf. Google Scholar [9] P. Ekdahl and T. Johansson, A new version of the stream cipher SNOW, in Selected Areas in Cryptography, 9th Annual International Workshop, SAC 2002, St. John's, Newfoundland, Canada, August 15-16, 2002. Revised Papers (eds. K. Nyberg and H. M. Heys), vol. 2595 of Lecture Notes in Computer Science, Springer, 2003, 47-61. doi: 10.1007/3-540-36492-7_5.  Google Scholar [10] D. Feng, X. Feng, W. Zhang, X. Fan and C. Wu, Loiss: A byte-oriented stream cipher, in Coding and Cryptology - Third International Workshop, IWCC 2011, Qingdao, China, May 30-June 3, 2011. Proceedings, 6639 (2011), 109-125. doi: 10.1007/978-3-642-20901-7_7.  Google Scholar [11] M. Grassl, Code Tables: Bounds on the parameters of various types of codes, Online available at http://www.codetables.de/main.html, last access Jan. 2020. Google Scholar [12] J. Guo, J. Jean, I. Nikolic, K. Qiao, Y. Sasaki and S. M. Sim, Invariant subspace attack against midori64 and the resistance criteria for s-box designs, IACR Trans. Symmetric Cryptol., 2016 (2016), 33-56. Google Scholar [13] Z. Guo, R. Liu, S. Gao, W. Wu and D. Lin, Direct construction of optimal rotational-xor diffusion primitives, IACR Trans. Symmetric Cryptol., 2017 (2017), 169-187. Google Scholar [14] Z. Guo, W. Wu and S. Gao, Constructing lightweight optimal diffusion primitives with feistel structure, in Selected Areas in Cryptography - SAC 2015 - 22nd International Conference, Sackville, NB, Canada, August 12-14, 2015, Revised Selected Papers (eds. O. Dunkelman and L. Keliher), vol. 9566 of Lecture Notes in Computer Science, Springer, 2016,352-372. doi: 10.1007/978-3-319-31301-6_21.  Google Scholar [15] D. Hong, J. Sung, S. Hong, J. Lim, S. Lee, B. Koo, C. Lee, D. Chang, J. Lee, K. Jeong, H. Kim, J. Kim and S. Chee, HIGHT: A new block cipher suitable for low-resource device, in Cryptographic Hardware and Embedded Systems - CHES 2006, 8th International Workshop, Yokohama, Japan, October 10-13, 2006, Proceedings, 2006, 46-59. doi: 10.1007/11894063_4.  Google Scholar [16] W. Ji and L. Hu, New description of SMS4 by an embedding overgf(28), in Progress in Cryptology - INDOCRYPT 2007, 8th International Conference on Cryptology in India, Chennai, India, December 9-13, 2007, Proceedings (eds. K. Srinathan, C. P. Rangan and M. Yung), vol. 4859 of Lecture Notes in Computer Science, Springer, 2007,238-251. Google Scholar [17] M. Kanda, S. Moriai, K. Aoki, H. Ueda, Y. Takashima, K. Ohta and T. Matsumoto, E2 - a new 128-bit block cipher, E83-A, (2000), 48-59. Google Scholar [18] B. Koo, H. S. Jang and J. H. 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. doi: 10.1007/11927587_7.  Google Scholar [19] 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. doi: 10.1007/978-3-540-24691-6_32.  Google Scholar [20] G. Leander, B. Minaud and S. Rønjom, A generic approach to invariant subspace attacks: Cryptanalysis of robin, iscream and zorro, in Advances in Cryptology - EUROCRYPT 2015 - 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Sofia, Bulgaria, April 26-30, 2015, Proceedings, Part I, 2015,254-283. doi: 10.1007/978-3-662-46800-5_11.  Google Scholar [21] S. Ling and C. Xing, Coding Theory: A First Course, Cambridge University Press, 2004, https://books.google.com/books?id=N1jiL8V3ISwC. doi: 10.1017/CBO9780511755279.  Google Scholar [22] A. M. Rishakani, S. M. Dehnavi, M. R. M. Shamsabad, H. Maimani and E. Pasha, New concepts in design of lightweight mds diffusion layers, 2014 11th International ISC Conference on Information Security and Cryptology, 2014. doi: 10.1109/ISCISC.2014.6994017.  Google Scholar [23] M. Sajadieh, M. Dakhilalian and H. Mala, Perfect involutory diffusion layers based on invertibility of some linear functions, IET Information Security, 5 (2011), 228-236. doi: 10.1049/iet-ifs.2010.0289.  Google Scholar [24] M. Sajadieh, M. Dakhilalian, H. Mala and P. Sepehrdad, Efficient recursive diffusion layers for block ciphers and hash functions, J. Cryptology, 28 (2015), 240-256.  doi: 10.1007/s00145-013-9163-8.  Google Scholar [25] M. T. Sakalli and B. Aslan, On the algebraic construction of cryptographically good 32${\times}$32 binary linear transformations, J. Computational Applied Mathematics, 259 (2014), 485-494.  doi: 10.1016/j.cam.2013.05.008.  Google Scholar [26] R. Schürer and W. C. Schmid, MinT: A Database for Optimal Net Parameters, In: Niederreiter H., Talay D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. Google Scholar [27] B. Schneier, J. Kelsey, D. Whiting, D. Wagner, C. Hall and N. Ferguson, Twofish: A 128-bit block cipher., Google Scholar
The corresponding diffusion layer of ASCON
An ASCON-like diffusion layer
Parallel use of diffusion layers
Our proposed integrated diffusion layer
Diffusion layers of HIGHT and ASCON
 n Source The mapping Number of fixed-points 8 HIGHT circ(1, 6, 7) 8 8 HIGHT circ(2, 4, 5) 2 64 ASCON $f_1$=circ(0, 19, 28) 2 64 ASCON $f_2$=circ(0, 39, 61) 4 64 ASCON $f_3$=circ(0, 1, 6) 2 64 ASCON $f_4$=circ(0, 10, 17) 2 64 ASCON $f_5$=circ(0, 7, 41) 4
 n Source The mapping Number of fixed-points 8 HIGHT circ(1, 6, 7) 8 8 HIGHT circ(2, 4, 5) 2 64 ASCON $f_1$=circ(0, 19, 28) 2 64 ASCON $f_2$=circ(0, 39, 61) 4 64 ASCON $f_3$=circ(0, 1, 6) 2 64 ASCON $f_4$=circ(0, 10, 17) 2 64 ASCON $f_5$=circ(0, 7, 41) 4
Number of $16 \times 16$ CBMs with fixed branch number
 $b_f=4$ $b_f=6$ $b_f=8$ $w_f=3$ 105 - - $w_f=5$ 145 1220 - $w_f=7$ 301 3332 1372 $w_f=9$ 495 4644 1296 $w_f=11$ 297 2134 572 $w_f=13$ 169 286 - $w_f=15$ 15 - -
 $b_f=4$ $b_f=6$ $b_f=8$ $w_f=3$ 105 - - $w_f=5$ 145 1220 - $w_f=7$ 301 3332 1372 $w_f=9$ 495 4644 1296 $w_f=11$ 297 2134 572 $w_f=13$ 169 286 - $w_f=15$ 15 - -
$n \times n$ CBMs with maximal branch number and efficient implementation
 n mapping $\#$XORs $\#$fixed-points 16 $circ(0,1,2)(0,2,9)$ 49 4 16 $circ(0,14,15)(0,7,14)$ 49 4 16 $circ(0,1,2)(0,7,14)$ 49 16 16 $circ(0,14,15)(0,2,9)$ 49 16 16 $circ(0,3,6)(0,5,10)$ 51 16 16 $circ(0,3,6)(0,6,11)$ 51 4 32 $circ(0,3,6)(0,4,8)(0,5,10)$ 146 64 32 $circ(0,3,6)(0,4,8)(0,22,27)$ 146 4 32 $circ(0,26,29)(0,24,28)(0,22,27)$ 146 64 32 $circ(0,4,8)(0,5,10)(0,13,26)$ 148 4 32 $circ(0,4,8)(0,22,27)(0,6,19)$ 148 4 32 $circ(0,1,2)(0,7,14)(0,12,24)$ 150 64 32 $circ(0,30,31)(0,18,25)(0,8,20)$ 150 64 32 $circ(0,1,2)(0,9,18)(0,12,24)$ 150 4 32 $circ(0,30,31)(0,14,23)(0,8,20)$ 150 4 32 $circ(0,3,6)(0,4,8)(0,11,22)$ 150 4 32 $circ(0,7,14)(0,12,24)(0,15,30)$ 151 4 32 $circ(0,9,18)(0,12,24)(0,15,30)$ 151 64 32 $circ(0,4,8 )(0,11,22)(0,13,26)$ 152 64
 n mapping $\#$XORs $\#$fixed-points 16 $circ(0,1,2)(0,2,9)$ 49 4 16 $circ(0,14,15)(0,7,14)$ 49 4 16 $circ(0,1,2)(0,7,14)$ 49 16 16 $circ(0,14,15)(0,2,9)$ 49 16 16 $circ(0,3,6)(0,5,10)$ 51 16 16 $circ(0,3,6)(0,6,11)$ 51 4 32 $circ(0,3,6)(0,4,8)(0,5,10)$ 146 64 32 $circ(0,3,6)(0,4,8)(0,22,27)$ 146 4 32 $circ(0,26,29)(0,24,28)(0,22,27)$ 146 64 32 $circ(0,4,8)(0,5,10)(0,13,26)$ 148 4 32 $circ(0,4,8)(0,22,27)(0,6,19)$ 148 4 32 $circ(0,1,2)(0,7,14)(0,12,24)$ 150 64 32 $circ(0,30,31)(0,18,25)(0,8,20)$ 150 64 32 $circ(0,1,2)(0,9,18)(0,12,24)$ 150 4 32 $circ(0,30,31)(0,14,23)(0,8,20)$ 150 4 32 $circ(0,3,6)(0,4,8)(0,11,22)$ 150 4 32 $circ(0,7,14)(0,12,24)(0,15,30)$ 151 4 32 $circ(0,9,18)(0,12,24)(0,15,30)$ 151 64 32 $circ(0,4,8 )(0,11,22)(0,13,26)$ 152 64
$2^m \times 2^m$, $m \geq 6$ CBMs with provable branch numbers and determined number of fixed-points
 $f$ $b_{f}$ $\#$XORs $\#$fixed-points $circ(0,1,2)(0,2,9)$ 8 $49 \times 2^{m-4}$ 4 $circ(0,14,15)(0,7,14)$ $\geq8$ $49 \times 2^{m-4}$ 4 $circ(0,3,6)(0,4,8)(0,22,27)$ $\geq12$ $146 \times 2^{m-5}$ 4 $circ(0,4,8)(0,5,10)(0,13,26)$ $\geq12$ $148 \times 2^{m-5}$ 4 $circ(0,1,2)(0,9,18)(0,12,24)$ $\geq12$ $150 \times 2^{m-5}$ 4
 $f$ $b_{f}$ $\#$XORs $\#$fixed-points $circ(0,1,2)(0,2,9)$ 8 $49 \times 2^{m-4}$ 4 $circ(0,14,15)(0,7,14)$ $\geq8$ $49 \times 2^{m-4}$ 4 $circ(0,3,6)(0,4,8)(0,22,27)$ $\geq12$ $146 \times 2^{m-5}$ 4 $circ(0,4,8)(0,5,10)(0,13,26)$ $\geq12$ $148 \times 2^{m-5}$ 4 $circ(0,1,2)(0,9,18)(0,12,24)$ $\geq12$ $150 \times 2^{m-5}$ 4
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