DIVERSITY AES IN MIXCOLUMNS STEP WITH 8X8 CIRCULANT MATRIX

  • Yan-Wen Chen student https://orcid.org/0000-0002-0961-6157
  • Jeng-Jung Wang Department of Information Engineering I-Shou University, Kaohsiung, Taiwan 84008, Republic of China. https://orcid.org/0000-0001-9424-4177
  • Yan-Haw Chen Department of Information Engineering I-Shou University, Kaohsiung, Taiwan 84008, Republic of China.
  • Chong-Dao Lee Department of Information Engineering I-Shou University, Kaohsiung, Taiwan 84008, Republic of China.
Keywords: Involutory Matrix, Dynamic Matrix, Finite Field, Horner Rule, Mixcolumns, Multiplication.

Abstract

In AES MixColumns operation, the branch number of circulant matrix is raised from 5 to 9 with 8´8 circulant matrices that can be enhancing the diffusion power. An efficient method to compute the circulant matrices in AES MixColumns transformation for speeding encryption is presented. Utilizing 8´8 involutory matrix multiplication is required 64 multiplications and 56 additions in in AES Mix-Columns transformation. We proposed the method with diversity 8´8 circulant matrices is only needed 19 multiplications and 57 additions. It is not only to encryption operations but also to decryption operations. Therefore, 8´8 circlant matrix operation with AES key sizes of 128bits, 192bits, and 256 bits are above 29.1%, 29.3%, and 29.8% faster than using 4´4 involutory matrix operation (16 multiplications, 12 additions), respectively. 8´8 circulant matrix encryption/decryption speed is above 78% faster than 8´8 involutory matrix operation. Ultimately, the proposed method for evaluating matrix multiplication can be made regular, simple and suitable for software implementations on embedded systems.

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References

A. Biryukov, D. Khovratovich (2009), “Related-Key cryptanalysis of the full AES-192 and AES-256,” In: Matsui, M. (ed.) ASIACRYPT 2009 LNCS, 5912, pp. 1-18 https://eprint.iacr.org/2009/317.pdf. Retrieved from https://doi.org/10.1007/978-3-642-10366-7_1 DOI: https://doi.org/10.1007/978-3-642-10366-7_1

A. Mahboob, N. Ikram (2006), “Lookup table based multiplication technique for GF(2m) with cryptographic significance,” IEE Proc. Commun, vol. 52, no. 6, pp. 965-974. Retrieved from https://doi.org/10.1049/ip-com:20050022 DOI: https://doi.org/10.1049/ip-com:20050022

A. Maximov (2019), “AES MixColumn with 92 XOR gates,” Cryptology ePrint Archive, Report 2019/833, Retrieved from https://eprint.iacr.org/2019/833 , Jul.

A. Stepanov, D. Rose (2015), From mathematics to generic programming. Pearson Education, New York, 3st edn, pp. 9.

B. Langenberg, H. Pham, and R. Steinwandt (2020), "Reducing the Cost of Implementing the Advanced Encryption Standard as a Quantum Circuit," in IEEE Trans. on Quantum Engineering, vol. 1, no. 2500112, pp. 1-12. Retrieved from https://doi.org/10.1109/TQE.2020.2965697 DOI: https://doi.org/10.1109/TQE.2020.2965697

B. Schneier, J. Kelsey, D. Whiting, D. Wagner, and C. Hall (1998), “Twofish: a 128-Bit block cipher,” Available NIST's AES homepage, Retrieved from https://www.schneier.com/academic/paperfiles/paper-twofish-paper.pdf.

C. C. Wang, T. K. Truong, H. M. Shao, L. J. Deutsch, J. K. Omura, and I. S. Reed (1983), “VLSI architectures for computing multiplications and inverses in GF(2m),” TDA Progress Report, pp. 42-75. Retrieved from https://doi.org/10.1109/tc.1985.1676616 DOI: https://doi.org/10.1109/TC.1985.1676616

C. H. Yang and Y. S. Chien (2020), “FPGA Implementation and Design of a Hybrid Chaos-AES Color Image Encryption Algorithm,” Symmetry, vol. 12, no. 2, 187, pp. 1-17. Retrieved from https://doi.org/10.3390/sym12020189 DOI: https://doi.org/10.3390/sym12020189

D. Augot, M. Finiasz (2013), “Exhaustive search for small dimension recursive MDS diffusion layers for block ciphers and hash functions,” IEEE Int. Conf. on Information Theory, Turkey, pp 1551-1555, Jul. Retrieved from https://doi.org/10.1109/ISIT.2013.6620487 DOI: https://doi.org/10.1109/ISIT.2013.6620487

D. Yin, Y. Gao (2017), “A new construction of lightweight MDS matrices,” IEEE Int. Conf. on Computer and Communication, pp. 2560-2563. Retrieved from https://doi.org/10.1109/CompComm.2017.8322997 DOI: https://doi.org/10.1109/CompComm.2017.8322997

F. J. MacWilliams, N. J. Sloane (1978), The theory of error-correcting codes: North-Holland, 1nd edn.

G. N. Selimis, A. P. Fournaris, and O. Koufopavlou (2006), “Applying low power techniques in AES MixColumn/InvMixColumn transformations,” IEEE Int. Conf, Electronics, Circuits and Systems ICECS’06, France, pp. 10-13, Dec. Retrieved from https://doi.org/10.1109/ICECS.2006.379628 DOI: https://doi.org/10.1109/ICECS.2006.379628

I. S. Reed, T. K. Truong (1978), “A fast computation of complex convolution using a hybrid transform,” DNS Progress Report, pp. 42-46. Retrieved from https://doi.org/10.1109/TASSP.1978.1163150 DOI: https://doi.org/10.1109/TASSP.1978.1163150

I. S. Reed, X. Chen (1999), Error-control coding for data networks, Kluwer Academic Publishers, Boston. Retrieved from https://doi.org/10.1007/978-1-4615-5005-1 DOI: https://doi.org/10.1007/978-1-4615-5005-1

J. Daemen, V. Rijmen (1999), AES proposal: Rijndael, document version 2. Retrieved from https://doi.org/10.1109/LCOMM.2004.833807

J. Lacan and J. Fimes (2004), “Systematic MDS erasure codes based on vandermonde matrices,” IEEE Trans. Commun. Lett., vol. 8, no. 9, pp. 570-572. Retrieved from https://doi.org/10.1109/LCOMM.2004.833807 DOI: https://doi.org/10.1109/LCOMM.2004.833807

J. Nakahara Jr, E. Abrahao (2009), “A New involutory MDS matrix for the AES,” International Journal of Network Security, vol.9, no.2, pp.109–116. Retrieved from https://d1wqtxts1xzle7.cloudfront.net/30902835/ijns-2009-v9-n2-p109-116.pdf?1362934357=&response-content-disposition=inline%3B+filename%3DA_New_Involutory_MDS_Matrix_for_the_AES.pdf&Expires=1632550400&Signature=fMBdhnUJNMZwPR2Vty-P-3dLJ9EKIaeLeFVGoFXz4oo1fFu1Y71GuCtdiYnzUBL4Byh63sc~Y0LUYFXShECE5c6~s3m8zYWmZVwepIX1czUfQbIK~2Ei5crxbZqRxxISHNMAeCcLEh0Y0yQvA5iXVEb0D9-wphLT46rurVt3MDtgxtx-YKWzVAiP1bSzpBtaFa84OZJc8dRsE60uontP90CwrfMmeqmLaqrvkB1GSie45RPP5x398x6RVy73Y~B4TSlu2mCUmXq1fOdwIue~ykBbjjopEa1iH9PdFgV6TCRYdFSaeIZaHF1-o-9J817X4LJERCSUTUY8MGALlWTYKw__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA

Jeng-Jung Wang, Yan-Haw Chen, Guan-Hsiung Liaw, Jack Chang, Cheng-Chih Lee (2020), "Efficient schemes with diverse of a pair of circulant matrices for AES MixColumns-InvMixcolumns transformation," Communications_of_the_CCISA, vol. 26, no. 2, pp. 1-20. Retrieved from https://cccisa.ccisa.org.tw/article/view/2314

M. H. Jing, Z. H. Chen, J. H. Chen, and Y. H. Chen (2007), “System for high-speed and diversified AES using FPGA,” Microprocessors and Microsystems, vol. 31, pp. 94–102, Mar. Retrieved from https://doi.org/10.1016/j.micpro.2006.02.018 DOI: https://doi.org/10.1016/j.micpro.2006.02.018

National Institute of Standards and Technology (NIST) (2001) “Advanced Encryption Standard (AES),” PUBS FIPS 197, Nov.

P. Junod, S. Vaudenay (2004), Perfect diffusion primitives for block ciphers. building efficient MDS Matrices. Federalede Lausanne, Switzerland. Retrieved from https://doi.org/10.1007/978-3-540-30564-4_6 DOI: https://doi.org/10.1007/978-3-540-30564-4_6

S. Winograd (1978), “On computing the discrete Fourier transform,” Mathematics of computation, vol. 32, no.141, pp. 175-199. Retrieved from https://doi.org/10.1090/S0025-5718-1978-0468306-4 DOI: https://doi.org/10.1090/S0025-5718-1978-0468306-4

T. Luong (2016), “Constructing effectively MDS and recursive MDS matrices by Reed-Solomon codes,” Journal of Science and Technology on Information security, pp. 10-15. Retrieved from http://tailieu.antoanthongtin.vn/Files/files/site-2/files/MDS%20matric.pdf

Y. H. Chen, C. H. Huang (2020), "Efficient operations in large finite field for elliptic curve cryptographic,” International Journal of Engineering Technologies and Management Research, vol. 7, no. 6, pp. 141-151. Retrieved from https://doi.org/10.29121/ijetmr.v7.i6.2020.712 DOI: https://doi.org/10.29121/ijetmr.v7.i6.2020.712

Y. Wang, L. Ni, C. H. Chang, and H. Yu (2016), “DW-AES: A Domain-Wall Nanowire-Based AES for high throughput and energy-efficient data encryption in Non-Volatile memory,” IEEE T INF FOREN SEC, vol. 11, no. 11, pp. 2426-2440. Retrieved from https://doi.org/10.1109/TIFS.2016.2576903 DOI: https://doi.org/10.1109/TIFS.2016.2576903

Published
2021-09-25
How to Cite
Chen, Y.-W., Wang, J.-J., Chen, Y.-H., & Lee, C.-D. (2021). DIVERSITY AES IN MIXCOLUMNS STEP WITH 8X8 CIRCULANT MATRIX. International Journal of Engineering Technologies and Management Research, 8(9), 19-35. https://doi.org/10.29121/ijetmr.v8.i9.2021.1037