Abstract
In the present study the problem of efficient computation of the k-th root of the Discrete Logarithm is investigated. Lower bounds on the degree of interpolation polynomials of the root of the Discrete Logarithm for subsets of given data are obtained. These results support the assumption of hardness of the k-th root of the discrete logarithm.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ateniese, G., Tsudik, G.: Some open issues and new directions in group signatures. In: Franklin, M. (ed.) FCT 1999. LNCS, vol. 1684, pp. 196–211. Springer, Heidelberg (1999)
Adelmann, C., Winterhof, A.: Interpolation of functions related to the integer factoring problem. In: Ytrehus, Ø. (ed.) WCC 2005. LNCS, vol. 3969, pp. 144–154. Springer, Heidelberg (2006)
Aly, H., Winterhof, A.: Polynomial representations of the Lucas logarithm. Finite Fields Appl. 12(3), 413–424 (2006)
Bresson, E., Stern, J.: Efficient Revocation in Group Signatures. In: Kim, K.-c. (ed.) PKC 2001. LNCS, vol. 1992, pp. 190–206. Springer, Heidelberg (2001)
Bussard, L., Molva, R., Roudier, Y.: History-based signature or how to trust anonymous documents. In: Jensen, C., Poslad, S., Dimitrakos, T. (eds.) iTrust 2004. LNCS, vol. 2995, pp. 78–92. Springer, Heidelberg (2004)
Camenisch, J.L.: Group Signature Schemes and Payment Systems Based on the Discrete Logarithm Problem, Doctoral Dissertation, ZURICH (1998)
Camenisch, J.L., Stadler, M.A.: Efficient group signature schemes for large groups. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 410–424. Springer, Heidelberg (1997)
Coppersmith, D., Shparlinski, I.: On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping. J. Cryptology 13(3), 339–360 (2000)
El Mahassni, E., Shparlinski, I.E.: Polynomial representations of the Diffie-Hellman mapping. Bull. Austral. Math. Soc. 63, 467–473 (2001)
Jeong, I.R., Lee, D.-H.: Anonymity control in multi-bank E-cash system. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 104–116. Springer, Heidelberg (2000)
Konoma, C., Mambo, M., Shizuya, H.: The computational difficulty of solving cryptographic primitive problems related to the discrete logarithm problem. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E88-A(1), 81–88 (2005)
Lysyanskaya, A., Ramzan, Z.: Group blind digital signatures: A scalable solution to electronic cash. In: Hirschfeld, R. (ed.) FC 1998. LNCS, vol. 1465, pp. 184–197. Springer, Heidelberg (1998)
Meidl, W., Winterhof, A.: A polynomial representation of the Diffie-Hellman mapping. Appl. Algebra Engrg. Comm. Comput. 13, 313–318 (2002)
Meletiou, G.C.: Explicit form for the discrete logarithm over the field GF(p, k). Arch. Math (Brno) 29, 25–28 (1993)
Meletiou, G.C., Mullen, G.L.: A note on discrete logarithms in finite fields. Appl. Algebra Engrg. Comm. Comput. 3(1), 75–78 (1992)
Meletiou, G.C., Winterhof, A.: Interpolation of the double discrete logarithm. In: von zur Gathen, J., Imaña, J.L., Koç, Ç.K. (eds.) WAIFI 2008. LNCS, vol. 5130, pp. 1–10. Springer, Heidelberg (2008)
Mullen, G.L., White, D.: A polynomial representation for logarithms in GF(q). Acta Arith. 47(3), 255–261 (1986)
Niederreiter, H.: A short proof for explicit formulas for discrete logarithms in finite fields. Appl. Algebra Engrg. Comm. Comput. 1(1), 55–57 (1990)
Niederreiter, H., Winterhof, A.: Incomplete character sums and polynomial interpolation of the discrete logarithm. Finite Fields Appl. 8(2), 184–192 (2002)
Pavlovski, C., Boyd, C.: Attacks based on small factors in various group structures. In: Varadharajan, V., Mu, Y. (eds.) ACISP 2001. LNCS, vol. 2119, pp. 36–50. Springer, Heidelberg (2001)
Shparlinski, I.E.: Cryptographic Applications of Analytic Number Theory. Complexity Lower Bounds and Pseudorandomness. Progress in Computer Science and Applied Logic, vol. 22. Birkhauser Verlag, Basel (2003)
Stadler, M.A.: Publicly verifiable secret sharing. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 190–199. Springer, Heidelberg (1996)
Traoré, J.: Group signatures and their relevance to privacy protecting offline electronic cash systems. In: Pieprzyk, J.P., Safavi-Naini, R., Seberry, J. (eds.) ACISP 1999. LNCS, vol. 1587, pp. 228–243. Springer, Heidelberg (1999)
Winterhof, A.: A note on the interpolation of the Diffie-Hellman mapping. Bull. Austral. Math. Soc. 64, 475–477 (2001)
Winterhof, A.: Polynomial interpolation of the discrete logarithm. Des. Codes Cryptogr. 25, 63–72 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Meletiou, G.C. (2009). Polynomial Interpolation of the k-th Root of the Discrete Logarithm. In: Bozapalidis, S., Rahonis, G. (eds) Algebraic Informatics. CAI 2009. Lecture Notes in Computer Science, vol 5725. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03564-7_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-03564-7_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03563-0
Online ISBN: 978-3-642-03564-7
eBook Packages: Computer ScienceComputer Science (R0)