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A cubic analogue of the RSA cryptosystem

Published online by Cambridge University Press:  17 April 2009

Mohamad Rushdan Md Said  and
John Loxton
Mohamad Rushdan Md Said
Affiliation:
Mathematics Department, Universiti Putra Malaysia, 43400 UPM Serdant, Malaysia e-mail: mrushdan@fsas.upm.edu.my
John Loxton
Affiliation:
Deputy Vice-Chancellor, Macquarie University, Sydney NSW 2109, Australia e-mail: John.Loxton@mq.edu.au
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Abstract

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In this paper, we investigate a public key cryptosystem which is derived from a third order linear recurrence relation and is analogous to the RSA and LUC cryptosystems. The explicit formulation involves a generalisation of the rule for composition of powers and of the calculus of the Euler totient function which underlie the algebra of the RSA cryptosystem. The security of all these systems appears to be comparable and to depend on the intractability of factorization but the systems do not seem to be mathematically equivalent.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 1 , August 2003 , pp. 21 - 38
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Adams, W. and Shanks, D., ‘Strong primality tests that are not sufficient’, Math. Comp. 39 (1982), 255–300.CrossRefGoogle Scholar
[2]Bleichenbacher, D., Bosma, W. and Lenstra, A.K., ‘Some remarks on Lucas-based cryptosystems’, in Crypto '95, (Goos, G., Hartmanis, J. and van Leeuwen, J., Editors), Lecture Notes in Computer Science 963 (Springer-Verlag, New York, 1996), pp. 386–396.Google Scholar
[3]Bleichenbacher, D., Joyce, M. and Quisquater, J.J., ‘A new and optimal chosen-massage attack on RSA-type cryptosystems’, in Information and Communications Security, (Han, Y., Okamoto, T. and Qing, S., Editors), Lecture Notes in Computer Science 1334 (Springer-Verlag, New York, 1997), pp. 302–313.CrossRefGoogle Scholar
[4]Childs, L., A concrete introduction to higher algebra (Springer-Verlag, New York, 1979).CrossRefGoogle Scholar
[5]Diffie, W. and Hellman, M., ‘New directions in cryptography’, IEEE Trans. Inform. Theory IT-22 (1976), 644–654.CrossRefGoogle Scholar
[6]Knuth, D.E., The art of computer programming, Seminumerical Algorithms, Vol. 2 (Addison-Wesley, Reading, 1969).Google Scholar
[7]Lehmer, D.H., ‘An extended theory of Lucas' functions’, Annals Math. 31 (1930), 419–448.CrossRefGoogle Scholar
[8]Lucas, E.A., ‘Théorie des fonctions numériques simplement périodiques’, Amer. J. Math. 1 (1878), 184–239, 289–322.CrossRefGoogle Scholar
[9]Riesel, H., Prime numbers and computer methods for factorization, Progress in Mathematics 57 (Birkhauser, Boston, 1985).CrossRefGoogle Scholar
[10]Rivest, R., Shamir, A. and Adleman, L., ‘A method for obtaining digital signatures and public key cryptosystems’, Comm. ACM 21 (1978), 120–126.CrossRefGoogle Scholar
[11]Smith, P.J. and Lennon, M.J.J., ‘LUC: A new public key system’, in Proceedings of the ninth IFIP International Symposium on Computer Security(Elsevier Science Publications,Amsterdam,1994), pp. 103–117.Google Scholar
[12]Ward, M., ‘The characteristic number of a sequence of integers satisfying a linear recursion relation’, Trans. Amer. Math. Soc. 35 (1933), 153–165.Google Scholar
[13]Wiener, M.J., ‘Cryptanalysis of short RSA secret exponents’, IEEE Trans. Inform. Theory IT-36 (1990), 553–558.CrossRefGoogle Scholar
[14]Williams, H.C., ‘On a generalization of the Lucas Functions’, Acta Arith. 20 (1972), 33–51.CrossRefGoogle Scholar
[15]Williams, H.C., ‘A p + 1 method of factoring’, Math. Comp. 39 (1982), 225–234.Google Scholar
[16]Williams, H.C. and Zarnke, C.R., ‘Some algorithms for solving a cubic congruence modulo p’, Utilitas Math. 6 (1974), 285–306.Google Scholar