Advertisement

Factor-4 and 6 (De)Compression for Values of Pairings Using Trace Maps

  • Tomoko Yonemura
  • Taichi Isogai
  • Hirofumi Muratani
  • Yoshikazu Hanatani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7708)

Abstract

The security of pairing-based cryptosystems relies on the hardness of the discrete logarithm problems in elliptic curves and in finite fields related to the curves, namely, their embedding fields. Public keys and ciphertexts in the pairing-based cryptosystems are composed of points on the curves or values of pairings. Although the values of the pairings belong to the embedding fields, the representation of the field is inefficient in size because the size of the embedding fields is usually larger than the size of the elliptic curves. We show factor-4 and 6 compression and decompression for the values of the pairings with the supersingular elliptic curves of embedding degrees 4 and 6, respectively. For compression, we use the fact that the values of the pairings belong to algebraic tori that are multiplicative subgroups of the embedding fields. The algebraic tori can be expressed by the affine representation or the trace representation. Although the affine representation allows decompression maps, decompression maps for the trace representation has not been known. In this paper, we propose a trace representation with decompression maps for the characteristics 2 and 3. We first construct efficient decompression maps for trace maps by adding extra information to the trace representation. Our decompressible trace representation with additional information is as efficient as the affine representation is in terms of the costs of compression, decompression and exponentiation, and the size.

Keywords

public-key cryptosystems the discrete logarithm problem algebraic tori compression decompression 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adleman, L.M.: The Function Field Sieve. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 108–121. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  2. 2.
    Barker, E., Barker, W., Burr, W., Polk, W., Smid, M.: Recommendation for Key Management - Part 1: Genaral (Revised). Special Publication 800/57, NIST (2007)Google Scholar
  3. 3.
    Boneh, D., Franklin, M.: Identity-Based Encryption from the Weil Pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–220. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Bosma, W., Hutton, J., Verheul, E.R.: Looking beyond XTR. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 46–63. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Giuliani, K.J., Gong, G.: Efficient Key Agreement and Signature Schemes Using Compact Representations in GF(p 10). In: ISIT 2004, p. 13. IEEE (2004)Google Scholar
  6. 6.
    Gong, G., Harn, L.: Public-key Cryptosystems besed on Cubic Finite Field Extensions. IEEE Trans. Inform. Theory 45, 2601–2605 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gordon, D.: Discrete Logarithms in GF(p) Using the Number Field Sieve. SIAM J. on Discrete Math. 6, 124–138 (1993)zbMATHCrossRefGoogle Scholar
  8. 8.
    Itoh, T., Tsujii, S.: A Fast Algorithm for Computing Multiplicative Inverses in GF(2m) Using Normal Bases. Information and Computation 78(3), 171–177 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Joux, A.: A One Round Protocol for Tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–393. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Joux, A., Lercier, R.: The Function Field Sieve in the Medium Prime Case. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 254–270. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Joux, A., Lercier, R., Smart, N.P., Vercauteren, F.: The Number Field Sieve in the Medium Prime Case. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 326–344. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Karabina, K.: Factor-4 and 6 Compression of Cyclotomic Subgroups. J. of Mathematical Cryptology 4(1), 1–42 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Karabina, K.: Torus-based Compression by Factor 4 and 6. Cryptology ePrint Archive, Report 2010/525 (2010)Google Scholar
  14. 14.
    Lenstra, A.K., Verheul, E.R.: The XTR Public Key System. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 1–19. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Miyaji, A., Nakabayashi, M., Takano, S.: New Explicit Conditions of Elliptic Curve Traces for FR-Reduction. IEICE Trans. E84-A(5), 1234–1243 (2001)Google Scholar
  16. 16.
    Rubin, K., Silverberg, A.: Torus-Based Cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Sakai, R., Ohgishi, K., Kasahara, M.: Cryptsystems based on Pairing. In: SCIS 2000 (2000)Google Scholar
  18. 18.
    Shirase, M., Han, D., Hibino, Y., Kim, H., Takagi, T.: A More Compact Representation of XTR Cryptosystem. IEICE Trans. E91-A(10), 2843–2850 (2008)CrossRefGoogle Scholar
  19. 19.
    Smith, P., Skinner, C.: A Public-key Cryptosystem and a Digital Signature Based on the Lucas Function Analogue to Discrete Logarithms. In: Safavi-Naini, R., Pieprzyk, J.P. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 357–364. Springer, Heidelberg (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tomoko Yonemura
    • 1
  • Taichi Isogai
    • 1
  • Hirofumi Muratani
    • 1
  • Yoshikazu Hanatani
    • 1
  1. 1.Toshiba CorporationSaiwai-kuJapan

Personalised recommendations