Abstract
XTR is a general method that can be applied to discrete logarithm based cryptosystems in extension fields of degree six, providing a compact representation of the elements involved. In this paper we present a precise formulation of the Brouwer-Pellikaan-Verheul conjecture, originally posed in [4], concerning the size of XTR-like representations of elements in extension fields of arbitrary degree. If true this conjecture would provide even more compact representations of elements than XTR in extension fields of degree thirty. We test the conjecture by experiment, showing that in fact it is unlikely that such a compact representation of elements can be achieved in extension fields of degree thirty.
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References
M. Adleman, J. DeMarrais A subexponentional algorithm over all finite fields, CRYPTO’ 93 Proc., Springer-Verlag, 147–158.
D. Bleichenbacher, W. Bosma, A.K. Lenstra, Some remarks on Lucas-Based Cryptosystems, CRYPTO’ 95 Proceedings, Springer-Verlag, pp. 386–396.
W. Bosma, J.J. Cannon, C. Playoust, The Magma Algebra System I: The User Language, Journal of Symbolic Computation 24 (1997), 235–265.
A.E. Brouwer, R. Pellikaan, E.R. Verheul, Doing more with fewer bits, Proceedings Asiacrypt99, LNCS 1716, Springer-Verlag 1999, 321–332.
D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Springer, 1992.
W. Diffie, M.E. Hellman, New directions in cryptography, IEEE Trans. on IT 22, 1976, 644–654.
G. Gong, L. Harn, Public key cryptosystems based on cubic finite field extensions, IEEE Trans. on I.T., November 1999.
S. Lang, Algebra, Addison-Welsey, 1993.
A.K. Lenstra, Using Cyclotomic Polynomials to Construct Efficient Discrete Logarithm Cryptosystems over Finite Fields, Information Security and Privacy-ACISP97 Proceedings (Sydney 1997), Lect. Notes in Comp. Sci. 1270, Springer-Verlag, pp. 127–138.
A.K. Lenstra, E.R. Verheul, The XTR public key system, Proceedings of Crypto 2000, LNCS 1880, Springer-Verlag, 2000, 1–19; available from http://www.ecstr.com.
A.K. Lenstra, E.R. Verheul, Key improvements to XTR, Proceedings of Asiacrypt 2000, LNCS 1976, Springer-Verlag, 2000, 220–223; available from http://www.ecstr.com.
A.K. Lenstra, E.R. Verheul, Fast irreducibility and subgroup membership testing in XTR, Proceedings of the 2001 Public Key Cryptography conference, LNCS 1992, Springer-Verlag, 2001, 73–86; available from http://www.ecstr.com.
A.K. Lenstra, E.R. Verheul, An overview of the XTR public key system, In: Public-Key Cryptography and Computational Number Theory, Walter de Gruyter, 2001, 151–180.
R. Lidl, W.B. Müller, Permutation Polynomials in RSA-cryptosystems, Crypto’ 83 Proceedings, Plemium Press, pp. 293–301.
R. Lidl, H. Niederreiter, Finite Fields, Addison-Wesley, 1983.
W.B. Müller, Polynomial functions in modern cryptology, Contributions to general Algebra 3, Proceedings of the Vienna Conference (1985), pp. 7–32. Proceedings, Springer-Verlag, pp. 50–61.
W.B. Müller, W. Nöbauer, Cryptanalysis of the Dickson-Scheme, Eurocrypt’ 85 Proceedings, Springer-Verlag, pp. 50–61.
W.K. Nicholson, Introduction to abstract algebra, PWS-Kent Publishing Company, Boston, 1993.
W. Nöbauer, Cryptanalysis of the Rédei Scheme, Contributions to general Algebra 3, Proceedings of the Vienna Conference (1985), pp. 255–264.
J.M. Pollard, Monte Carlo methods for index computation (mod p), Math. Comp., 32 (1978), 918–924.
C.P. Schnorr, Efficient signature generation by smart cards, Journal of Cryptology, 4 (1991), 161–174.
P. Smith, C. Skinner, A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms, Asiacrypt’ 94 proceedings, Springer-Verlag, pp. 357–364.
M. Stam, A.K. Lenstra, Speeding Up XTR, Proceedings of Asiacrypt 2001, LNCS 2248, Springer-Verlag, 2001, 125–143; available from http://www.ecstr.com.
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Bosma, W., Hutton, J., Verheul, E.R. (2002). Looking beyond XTR. In: Zheng, Y. (eds) Advances in Cryptology — ASIACRYPT 2002. ASIACRYPT 2002. Lecture Notes in Computer Science, vol 2501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36178-2_3
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DOI: https://doi.org/10.1007/3-540-36178-2_3
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