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Chandrasekharappa T.G.S et al. / International Journal on Computer Science and Engineering (IJCSE)
S-boxes generated using Affine Transformation giving Maximum Avalanche Effect
Chandrasekharappa T.G.S., Prema K.V. and Kumara Shama
Department of Electronics and Communication Engineering Manipal Institute of Technology Manipal - 576104 INDIA tgscmpl@gmail.com
Abstract: The Advanced Encryption Standard (AES) was published by National Institute of Standards and Technology (NIST) in November 2001, to replace DES (Data Encryption Standard) and Triple DES. The S-box (Substitution box) used in AES is designed to be resistant to known cryptanalytic attacks [1][2]. The property of the S-box is that the output cannot be described as a simple mathematical function of the input. The S-box is designed to provide good avalanche effect. The Avalanche Criteria of S-box depends on the generator matrix A used in affine transformation to construct the S-box. This paper investigates the construction of Sboxes by affine transformation which satisfy maximum Avalanche Criteria.
Keywords: Affine transformation, AES algorithm, Irreducible polynomial, Avalanche Criteria, S-box. 1. Introduction: The S-box, constructed in AES algorithm uses the Affine transformation
y Ax C mod m( x)
(1).
where A is an 8 x 8 matrix with entries in GF(2) and C is a column matrix in GF(2), m(x) is an irreducible polynomial in GF(29). The entries used in A matrix are [f8h; 7ch, 3eh, 1fh, 8fh, c7h, e1h, f1h]T and C = [63h]T (2) To be useful as S-box generator, matrix A should be non-singular. We can generate approximately 263 such non-singular matrices with each irreducible polynomials. The polynomials that result in non-singular matrices are bound by [01; 02; 04; 08; 10; 20; 40; 80]T on lower end and [feh, 7fh, bfh, dfh, efh, f7h, fbh, fdh]T on higher end. As per Avalanche criteria a one bit change in input should result in at least 50% changes in the output bits. A cryptographic
S-boxes generated using Affine Transformation giving Maximum Avalanche Effect
Chandrasekharappa T.G.S., Prema K.V. and Kumara Shama
Department of Electronics and Communication Engineering Manipal Institute of Technology Manipal - 576104 INDIA tgscmpl@gmail.com
Abstract: The Advanced Encryption Standard (AES) was published by National Institute of Standards and Technology (NIST) in November 2001, to replace DES (Data Encryption Standard) and Triple DES. The S-box (Substitution box) used in AES is designed to be resistant to known cryptanalytic attacks [1][2]. The property of the S-box is that the output cannot be described as a simple mathematical function of the input. The S-box is designed to provide good avalanche effect. The Avalanche Criteria of S-box depends on the generator matrix A used in affine transformation to construct the S-box. This paper investigates the construction of Sboxes by affine transformation which satisfy maximum Avalanche Criteria.
Keywords: Affine transformation, AES algorithm, Irreducible polynomial, Avalanche Criteria, S-box. 1. Introduction: The S-box, constructed in AES algorithm uses the Affine transformation
y Ax C mod m( x)
(1).
where A is an 8 x 8 matrix with entries in GF(2) and C is a column matrix in GF(2), m(x) is an irreducible polynomial in GF(29). The entries used in A matrix are [f8h; 7ch, 3eh, 1fh, 8fh, c7h, e1h, f1h]T and C = [63h]T (2) To be useful as S-box generator, matrix A should be non-singular. We can generate approximately 263 such non-singular matrices with each irreducible polynomials. The polynomials that result in non-singular matrices are bound by [01; 02; 04; 08; 10; 20; 40; 80]T on lower end and [feh, 7fh, bfh, dfh, efh, f7h, fbh, fdh]T on higher end. As per Avalanche criteria a one bit change in input should result in at least 50% changes in the output bits. A cryptographic
References: [1] [2] [3] [4] [5] [6] [7] [8] William Stallings, “Cryptography and Network Security Principles and Practices”, Fourth Edition, Pearson, Prentice Hall, 2006. Daemen, J. and V. Rijmen, “The Design of RIJNDAEL- AES The Advanced Encryption Standard”, Springer-Verlag, Berlin, 2002. Chandrasekharappa T G S, Prema K V, Kumara Shama, “Possible S-boxes generated from Affine transformation those satisfy Maximum Strict Avalanche Criteria, Proceedings of World Academy of Science, Engineering and Technology 60, pp.880-883 Dec. 2009. Elif Saygi, Zulfukar Saygi, Meltem SOnmezturan, Ali doganaksoy, Statistical approach on the number of SAC satisfying functions, IEEE transactions on computers, vol. 44. No. 9, September 1995. Rejane Forre, The strict avalanche criterion: spectral properties of Boolean Functions and an Extended De_nitio, Advances in cryptologycrypto 88, Lecture notes in computer science, volume 403, pp. 450-468 Springer-verlag. Isil VERGILI, Melek D. YUCEL, Avalanche and Bit Independence Properties for the Ensembles of Randomly Chosen n _ n S-boxes, Turk J Elec. Engin, VOL.9, NO.2 2001. ISSN : 0975-3397 Vol. 3 No. 9 september 2011 3193