DIRECT EXPONENT AND SCALAR MULTIPLICATION TRANSFORMATIONS OF MDS MATRICES: SOME GOOD CRYPTOGRAPHIC RESULTS FOR DYNAMIC DIFFUSION LAYERS OF BLOCK CIPHERS

Tran Thi Luong, Nguyen Ngoc Cuong

Abstract


Abstract: MDS (Maximum Distance Separable) matrices have an important role in the design of block ciphers and hash functions. The methods for transforming an MDS matrix into other ones have been proposed by many authors in the literature. In this paper, some new results about direct exponent and scalar multiplication transformations are given including the preservation of good cryptographic properties (the coefficient of fixed points and involutory property) of MDS matrices and other important cryptographic properties obtained from studying equivalence relations based on these transformations. An estimation of the number of  MDS matrices over  is also presented. In addition, these results are shown to be an important theoretical basis for building efficient dynamic diffusion layer algorithms for block ciphers.


Keywords


MDS matrix, direct exponent transformation, scalar multiplication transformation, dynamic algorithm

Full Text:

PDF

References


S. Vaudenay, “On the need for multipermutations: cryptanalysis of md4 and safer,” in Fast Software Encryption. Springer, 1995, pp. 286-297.

C. P. Schnorr and S. Vaudenay “Black box cryptanalysis of hash networks based on multipermutations,” in Advances in CryptologyEUROCRYPT’94. Springer, 1995, pp.47-57.

D. Kwon, S. H. Sung, J. H. Song and S. Park, “Design of block ciphers and coding theory”, Information Center for Mathematical Sciences, vol. 8, no. 1, pp. 13-20, 2005.

L. Keliher, “Linear cryptanalysis of substitution-permutation networks”, Queen's University, Kingston, Ontario, Canada, 2003.

J. Daemen and V. Rijmen, “Aes proposal: rijndael (version 2). nist aes website," 1999.

F. P. NIST, “197," advanced encryption standard (aes)," november 2001."

V. Rijmen, J. Daemen, B. Preneel, A. Bosselaers, E. De Win, “The cipher shark”, in Fast Software Encryption. Springer, 1996, pp. 99-111.

B. Schneier, J. Kelsey, D. Whiting, D. Wagner, C. Hall, and N. Ferguson, “Twofish: a 128-bit block cipher”, NIST AES Proposal, vol. 15, 1998.

G. Murtaza, A. A. Khan, S. W. Alam, A. Farooqi, “Fortification of aes with dynamic mix-column transformation,” IACR Cryptology ePrint Archive, vol. 2011, p. 184, 2011.

W. Mohamed, Ridza, M. Abdulrashid, “A method for linear transformation in substitution permutation network symmetric-key block cipher,” international application published under the patent cooperation treaty, 10 may 2012, pp. 3-14.

F. Ahmed and D. Elkamchouchi, “Strongest aes with s-boxes bank and dynamic key mds matrix (sdk-aes),” International Journal of Computer and Communication Engineering, vol. 2, no. 4, p. 530, 2013.

G. Murtaza and N. Ikram,“Direct exponent and scalar multiplication classes of an mds matrix,”IACR Cryptology ePrint Archive, vol. 2011, p. 151, 2011.

K. C. Gupta and I. G. Ray, “On constructions of mds matrices from companion matrices for lightweight cryptography,” in Security Engineering and Intlligence Informatics. Springer, 2013, pp. 29-43.

K. C. Gupta and I. G. Ray, “On constructions of mds matrices from circulant-like matrices for lightweight cryptography,” institution, Tech. Rep. ASU/2014/1, 2014.

F.J. MacWilliams, N.J.A. Sloane, The theory of error-correcting codes. Elsevier, 1977.

M. R. Z’aba, “Analysis of linear relationships in block ciphers”. Ph.D. Thesis, Queensland University of Technology, Brisbane, Australia, 2010.

T. T. Luong, N. N. Cuong, L. T. Dung, “A new statement about direct exponent of an MDS matrix in block ciphers”, in 2015 IEEE the Seventh International Conference on Knowledge and Systems Engineering (KSE), IEEE, 2015, pp. 340-343.

T. T. Luong, N. N. Cuong, L. T. Dung, “The preservation of good cryptographic properties of MDS matrix under direct exponent transformation”, Journal of Computer Science and Cybernetics, v.31, n.4 (2015), DOI: 10.15625/1813-9663/31/4/7059. (to appear).




Journal of Computer Science and Cybernetics ISSN: 1813-9663

Published by Vietnam Academy of Science and Technology