Abstract
The aim of the research was to conduct a cryptographic analysis of an encryption scheme developed on the basis of nonpositional polynomial notations to estimate the algorithm strength. Nonpositional polynomial notations (NPNs) are residue number systems (RNSs) based on irreducible polynomials over GF(2). To evaluate if the algorithms developed on the basis of NPNs are secure, mathematical models of cryptanalysis involving algebraic, linear and differential methods have been designed. The cryptanalysis is as follows. A system of nonlinear equations is obtained from a function transforming plaintext into ciphertext with a key. Next, a possibility of transition of the nonlinear system to a linear one is considered. The cryptanalysis was conducted for the cases with known: 1) ciphertext; 2) plaintext and the related ciphertext; 3) plaintext file format; and 4) ASCII-encoded plaintext.
References
[1] Bijashev R.G., “Development and investigation of methods of the overall increase in reliability in data exchange systems of distributed ACSs,” Doctoral Dissertation in Technical Sciences, Moscow, 1985 Search in Google Scholar
[2] Akushskii I.Ya. and Juditskii D. I.,Machine Arithmetic in Residue Classes [in Russian], Sov. Radio, Moscow, 1968,439 Search in Google Scholar
[3] Biyashev R. G. and Nyssanbayeva S .E., “Algorithm for Creation a Digital Signature with Error Detection and Correction”, Cybernetics and Systems Analysis,2012, 48, 4, 489–497 10.1007/s10559-012-9428-5Search in Google Scholar
[4] Moisil Gr. C., Algebraic Theory of Discrete Automatic Devices [Russian translation], Inostr. Lit., Moscow,1963, 680 Search in Google Scholar
[5] Nyssanbayev R. K., “Cryptographical method on the basis of polynomial bases”, Herald of the Ministry of Science and Higher Education and National Academy of Science of the Republic of Kazakhstan, 1999, 5, 63–65 Search in Google Scholar
[6] Amerbayev V.M., Biyashev R.G., Nyssanbayeva S.E. Implementation of nonpositional notations for cryptographic security // Proceedings of the National Academy of Science of the Republic of Kazakhstan. Ser. Physics and Mathematics, Almaty: Gylym, 2005, 3, 84-89 Search in Google Scholar
[7] Biyashev R., Nyssanbayeva S., Kapalova N. The Key Exchange Algorithm on Basis of Modular Arithmetic // Proceedings of International Conference on Electrical, Control and Automation Engineering (1-2 December,2013, Hong Kong- Lancaster), U.S.A.:DEStech Publications, 2013, 501–505 Search in Google Scholar
[8] Biyashev R., Kalimoldayev M., NyssanbayevaS., Kapalova N., Khakimov R. Program Modeling of the Cryptography Algorithms on Basis of Polynomial Modular Arithmetic / The 5th International Conference on Society and Information Technologies (4-7 March 2014) Orlando, Florida, USE – IIIS, 49–54 Search in Google Scholar
[9] Biyashev R.G., Nyssanbayeva S.E., Kapalova N.A. Secret keys for nonpositional cryptosystems. Development, investigation and implementation, LAB LAMBERT Academic Publishing, 2014 Search in Google Scholar
[10] Ivanov M.A. Cryptographic methods of information security in computer systems and networks, M.:KUDIT-OBRAZ, 2001, 308 Search in Google Scholar
[11] Rostovtsev A.G., Mahovenko E.B. Theoretical cryptography. NPO «Professional», Saint Petersburg, 2004 Search in Google Scholar
[12] https://ru.wikipedia.org/wiki/Cryptoanalysis Search in Google Scholar
[13] Babenko L.K., Ischukova E.A. Modern block encryption algorithms and methods of analysis thereof,Moscow, Helios, ARV, 2006 Search in Google Scholar
[14] Matsui M., «Linear Cryptanalysis Method for DES Cipher» (Advances in Cryptology | EUROCRYPT ’93 Proceedings, Springer- Verlag, 1994 10.1007/3-540-48285-7_33Search in Google Scholar
[15] Voronin R. I., Algebraic cryptoanalysis of one-round S-AES, PDM, 2011, 4, 29–31 Search in Google Scholar
[16] Kleiman E. The XL and XSL attacks on Baby Rijndael. lowa State University, Ames, lowa, 2005 Search in Google Scholar
[17] Murphy S., «The Cryptanalysis of FEAL-4 with 20 Chosen Plaintexts » (Journal of Cryptology, V. 2, N. 3, 1990, 145 10.1007/BF00190801Search in Google Scholar
[18] Matsui M., «The First Experimental Cryptanalysis of the Data Encryption Standard» (Advances in Cryptology | CRYPTO ’94 Proceedings, Springer-Verlag, 1994, 125–129 10.1007/3-540-48658-5_1Search in Google Scholar
[19] Brown L., Kwan M., Pieprzyk J., and Seberry J., «Improving Resistance to Dierential Cryptanalysis and the Redesign of LOKI»,Advances in Cryptology | ASIACRYPT ’91 Proceedings, Springer-Verlag, 1993, 36–50 10.1007/3-540-57332-1_3Search in Google Scholar
[20] KnudsenL.R., «Cryptanalysis of LOKI-91», Advances in Cryptology | AUSCRYPT ’92, Springer-Verlag, 1993, 196–208 10.1007/3-540-57220-1_62Search in Google Scholar
[21] Tokita T. , T. Sorimachi, and M. Matsui, «Linear Cryptanalysis of LOKI and s2DES», Advances in Cryptology | ASIACRYPT ’94, Springer-Verlag, 1995, 293–303 10.1007/BFb0000442Search in Google Scholar
[22] J. Borst, L.R. Knudsen, and V. Rijmen, «Two Attacks on Reduced IDEA», Advances in Cryptology | EUROCRYPT ’97, Springer- Verlag, 1997, 1–13 10.1007/3-540-69053-0_1Search in Google Scholar
[23] Hawkes P., «Dierential-Linear Weak Key Classes of IDEA», Advances in Cryptology EUROCRYPT ’98 Proceedings, Springer- Verlag, 1998, 112–126 10.1007/BFb0054121Search in Google Scholar
[24] Nyberg K., «Linear Approximation of Block Ciphers», Advances in Cryptology | EUROCRYPT ’94 Proceedings, Springer- Verlag, 1995, 439–444 10.1007/BFb0053460Search in Google Scholar
[25] Nyberg K. and KnudsenL., «Provable Security Against a Dierential Attack», Journal of Cryptology, V. 8, N. 1, 1995, 27–37 10.1007/BF00204800Search in Google Scholar
[26] Nyberg K. and Knudsen L., «Provable Security Against a Dierential Cryptanalysis», Advances in Cryptology | CRYPTO ’92 Proceedings, Springer- Verlag, 1993, 566–574 10.1007/3-540-48071-4_41Search in Google Scholar
[27] Rivest R.L., «The RC5 Encryption Algorithm», Fast Software Encryption, 2nd International Workshop Proceedings, Springer- Verlag, 1995, 86–96 10.1007/3-540-60590-8_7Search in Google Scholar
[28] Kaliski B.S. and Yin Y.L., «On Differential and Linear Cryptanalysis of the RC5 Encryption Algorithm», Advances in Cryptology | CRYPTO ’95 Proceedings, Springer-Verlag, 1995, 445–454 10.1007/3-540-44750-4_14Search in Google Scholar
[29] Knudsen L.R. and Meier W., «Improved Differential Attacks on RC5», Advances in Cryptology | CRYPTO ’96 Proceedings, Springer-Verlag, 1996, 216–228 10.1007/3-540-68697-5_17Search in Google Scholar
[30] Selcuk A.A., «New Results in Linear Cryptanalysis of RC5», Fast Software Encryption, 5th International Workshop Proceedings, Springer-Verlag, 1998, 1–16 10.1007/3-540-69710-1_1Search in Google Scholar
© 2016 N. Kapalova and D. Dyusenbayev
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.