New Practical Algebraic Public-Key Cryptosystem and Some Related Algebraic and Computational Aspects

DOI: 10.4236/am.2013.47142   PDF   HTML   XML   3,159 Downloads   4,423 Views   Citations

Abstract

The most popular present-day public-key cryptosystems are RSA and ElGamal cryptosystems. Some practical algebraic generalization of the ElGamal cryptosystem is considered-basic modular matrix cryptosystem (BMMC) over the modular matrix ring M2(Zn). An example of computation for an artificially small number n is presented. Some possible attacks on the cryptosystem and mathematical problems, the solution of which are necessary for implementing these attacks, are studied. For a small number n, computational time for compromising some present-day public-key cryptosystems such as RSA, ElGamal, and Rabin, is compared with the corresponding time for the ВММС. Finally, some open mathematical and computational problems are formulated.

Share and Cite:

S. Rososhek, "New Practical Algebraic Public-Key Cryptosystem and Some Related Algebraic and Computational Aspects," Applied Mathematics, Vol. 4 No. 7, 2013, pp. 1043-1049. doi: 10.4236/am.2013.47142.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] A. Menezes, P. van Ooshot and S. Vanstone, “Handbook of Applied Cryptography,” CRC Press, Waterloo, 1996. doi:10.1201/9781439821916
[2] P. W. Shor, “Algorithms for Quantum Computation: Discrete Logarithm and Factoring,” Proceedings of the IEEE 35th Communications Annual Symposium on Foundations of Computer Science, Santa Fe, 20-22 November 1994, pp. 124-134.
[3] S. K. Rososhek, “Cryptosystems in Automorphism Groups of Group Rings of Abelian Groups,” Fundamentalnaya I prikladnaya matematica, Vol. 13, No. 8, 2007, pp. 157-164 (in Russian).
[4] S. K. Rososhek, “Cryptosystems in Automorphism Groups of Group Rings of Abelian Groups,” Journal of Mathematical Sciences, Vol. 154, No. 3, 2008, pp. 386-391. doi:10.1007/s10958-008-9168-2
[5] A. N. Gribov, P. A. Zolotykh and A. V. Mikhalev, “A Construction of Algebraic Cryptosystem over the Quasigroup Ring,” Mathematical Aspects of Cryptography, Vol. 1, No. 4, 2010, pp. 23-32 (in Russian).
[6] K. N. Ponomarev, “Automorphically Rigid Group Alge bras I. Semisimple Algebras,” Algebra and Logic, Vol. 48, No. 5, 2009, pp. 654-674. doi:10.1007/s10469-009-9064-y
[7] K. N. Ponomarev, “Automorphically Rigid Group Alge bras II. Modular Algebras,” Algebra and Logic, Vol. 49, No. 2, 2010, pp. 216-237.
[8] K. N. Ponomarev, “Rigid Group Rings,” In: A. G. Pinus and K. N. Ponomarev, Eds., Algebra and Model Theory, 6, Novosobirsk Technical University Press, Novosibirsk, 2007, pp. 73-83 (in Russian). doi:10.1007/s10469-010-9086-5
[9] A. Popova and E. Poroshenko, “Units Group of Integral Group Rings of Finite Groups,” In: A. G. Pinus and K. N. Ponomarev, Eds., Algebra and Model Theory, 4, Novosi birsk Technical University Press, Novosibirsk, 2003, pp. 99-106 (in Russian).
[10] A. Dooms and E. Jespers, “Normal Complements of the Trivial Units in the Unit Group of Some Integral Group Rings,” Communications in Algebra, Vol. 31, No. 1, 2003, pp. 475-482. doi:10.1081/AGB-120016770
[11] Y. I. Merzlyakov, “Matrix Representations of Free Groups,” Doklady Akademii Nauk, Vol. 238, No. 3, 1978, pp. 527-533 (in Russian).
[12] A. Popova, “Group of Automorphisms of the Ring ,” In: A. G. Pinus and K. N. Ponomarev, Eds., Alge bra and Model Theory, 6, Novosibirsk Technical University Press, Novosibirsk, 2007, pp. 84-90 (in Russian).
[13] A. Mahalanobis, “A Simple Generalization of the ElGa mal Cryptosystem to Non-Abelian Groups,” Communications in Algebra, Vol. 36, No. 10, 2008, pp. 3878-3889. doi:10.1080/00927870802160883
[14] S.-H. Paeng, K.-C. Ha, J. N. Kim, S. Chee and C. Park, “New Public Key Cryptosystem Using Finite Non-Abelian Groups,” Proceedings of the Crypto 2001, Lecture Notes in Computer Sciences, Santa Barbara, 19-23 August 2001, pp. 470-485.
[15] M. I. Kargapolov and Y. I. Merzlyakov, “Foundations of Group Theory,” Nauka, Moscow, 1977 (in Russian).
[16] R. C. Lyndon and P. E. Schupp, “Combinatorial Group Theory,” Springer-Verlag, Berlin, Heidelberg, New York, 1977.

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.