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On the Families of Stable Multivariate Transformations of Large Order and Their Cryptographical Applications


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[1] DING, J.-GOWER, V.-SCHMIDT, D. S.: Multivariate Public Key Cryptosystems. In: Adv. Inform. Sec. Vol. 25, Springer-Verlag, New York, 2006. Search in Google Scholar

[2] COSSIDENTE, A.-DE RESMINI, M. J.: Remarks on Singer cyclic groups and their normalizers, Des. Codes Cryptogr. 32 (2004), no. 1-3, 97-102, 10.1023/B:DESI.0000029214.50635.17Search in Google Scholar

[3] KANTOR, W.: Linear groups containing a Singer cycle, J. Algebra 62 (1980), no. 1, 232-234.10.1016/0021-8693(80)90214-8Search in Google Scholar

[4] BABAI, L.: Graph isomorphism in quasipolinomial time, arXive: 1512 03547v1 [cs. DS], December 11, 2015.Search in Google Scholar

[5] KIPNIS, A.-SHAMIR, A.: Cryptanalisis of the Oil and Vinegar Signature Scheme, Adv. in Cryptology-Crypto ’96, Lect. Notes in Comput. Sci. Vol. 1462, 1996, pp. 257-266.10.1007/BFb0055733Search in Google Scholar

[6] PATARIN, J.-GOUBIN, L.: Trapdoor one-way permutations and multivariate polynominals, ICICS 1997, pp. 356-368.10.1007/BFb0028491Search in Google Scholar

[7] Asymmetric cryptography with S-Boxes, ICICS 1997, pp. 369-380.Search in Google Scholar

[8] PATARIN, J.: Asymmetric cryptography with a hidden monomial, and a candidate algorithm for 64 bits asymmetric signatures. In: Advances in cryptologyCRYPTO’96, Santa Barbara, CA), Lect. Notes in Comput. Sci. Vol. 1109, Springer-Verlag, Berlin, 1996, pp. 45-60. 10.1007/3-540-68697-5_4Search in Google Scholar

[9] GOUBIN, L.-PATARIN, J.-YANG, B.-Y.: Multivariate Cryptography. Encyclopedia of Cryptography and Security (2nd ed.), Springer-Verlag Berlin 2011, pp. 824-828.10.1007/978-1-4419-5906-5_421Search in Google Scholar

[10] PORRAS, J.-BAENA, J.-DING, J.: New candidates for multivariate trapdoor functions, Rev. Colombiana Mat. 49 (2015), no. 1, 57-76.10.15446/recolma.v49n1.54163Search in Google Scholar

[11] USTIMENKO, V. A.: Explicit constructions of extremal graphs and new multivariate cryptosystems, In: Proceedings of The Central European Conference, 2014, Budapest”, Studia Sci. Math. Hungar. 52 (2015), 185-204.10.1556/012.2015.52.2.1312Search in Google Scholar

[12] On multivariate cryptosystems based on computable maps with invertible decompositions, Ann. Univ. Mariae Curie-Sk_lodowska Sect. AI-Inform. 14 (2014), 7-18.10.2478/umcsinfo-2014-0001Search in Google Scholar

[13] On Shubert cells in grassmanians and new algorithm of multivariatecryptography, Tr. Inst. Mat. 23 (2015), no. 2, 137-148Search in Google Scholar

[14] On Linguistic Dynamical Systems, Graphs of Large Girth and Cryptography, Zap. Nauchn. Semin. POMI 326, (2005), 214-234; translated in J. Math. Sci., 140 (2007), no. 3, 461-471.Search in Google Scholar

[15] USTIMENKO, V.-WROBLEWSKA, A.: On the key exchange with nonlinear polynomial maps of stable degree, Ann. Univ. Mariae Curie-Skodowska Sect. AI-Inform. 13 (2013), no. 1, 63-80.10.2478/v10065-012-0047-6Search in Google Scholar

[16] WROBLEWSKA, A.: On some properties of graph based public keys, Albanian J. Math. 2 (2008), no. 3, 229-234.Search in Google Scholar

[17] USTIMENKO, V.: Coordinatisation of trees and their quotients. In: the ”Voronoj’s Impact on Modern Science”, Vol 2, 1998, Kiev, Institute of Mathematics pp. 125-152.Search in Google Scholar

[18] LAZEBNIK, F.-USTIMENKO, V. A.: Some algebraic constractions of dense graphs of large girth and of large size. In: Expanding graphs, Princeton, NJ, 1992, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 10, 1993, Amer. Math. Soc., Providence, RI, pp. 75-93.10.1090/dimacs/010/07Search in Google Scholar

[19] LAZEBNIK, F.-USTIMENKO, V. A. -WOLDAR, A. J.: New series of dense graphs of high girth, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 73-79.10.1090/S0273-0979-1995-00569-0Search in Google Scholar

[20] USTIMENKO, V.- ROMANCZUK, U.: On dynamical systems of large girth or cycle indicator and their applications to multivariate cryptography. In: Artificial Intelligence, Evolutionary Computing and Metaheuristics, In the footsteps of Alan Turing Series: Studies in Computational Intelligence, Vol. 427, Springer-Verlag, 2013, pp. 257-285.10.1007/978-3-642-29694-9_10Search in Google Scholar

[21] USTIMENKO, V.-WROBLEWSKA, A.: Dynamical systems as the main instrument for the constructions of new quadratic families and their usage in cryptography, Ann. Univ. Mariae Curie-Skodowska Sect. AI-Inform. 12 (2012), no. 3, 65-74.10.2478/v10065-012-0030-2Search in Google Scholar

[22] KLISOWSKI, M.: Zwi¸eszenie Bezpiecze´nstwa Kryptograficznych Algorytm´owielu Zmiennych Bazuj¸acych na Algebraicznej Teorii Graf´ow. PhD Thesis, Cz¸estochowa, 2014.Search in Google Scholar

[23] KLISOWSKI, M.-USTIMENKO, V.: Graph based cubical multivariate maps and their cryptographical applications. In: Advances on Superelliptic curves and their Applications, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., Vol. 41, IOS, Amsterdam, 2015 pp. 305-327.Search in Google Scholar

[24] WROBLEWSKA, A.-USTIMENKO, V.: On new examples of families of multivariate stable maps and their cryptographical applications, Ann. Univ. Mariae Curie-Sk_lodowska Sect. AI-Inform. 14 (2014), no. 1, 19-35.10.2478/umcsinfo-2014-0004Search in Google Scholar

[25] , USTIMENKO, V.- WOLDAR, A.: A geometric approach to orbital recognition in Chevalley-type coherent configurations and association schemes, Australas. J. Combin. 67 (2017), no. 2, 166-202.Search in Google Scholar

eISSN:
1210-3195
Language:
English
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Journal Subjects:
Mathematics, General Mathematics