On Cryptographic Schemes Based on Discrete Logarithms and Factoring

  • Marc Joye
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5888)


At CRYPTO 2003, Rubin and Silverberg introduced the concept of torus-based cryptography over a finite field. We extend their setting to the ring of integers modulo N. We so obtain compact representations for cryptographic systems that base their security on the discrete logarithm problem and the factoring problem. This results in smaller key sizes and substantial savings in memory and bandwidth. But unlike the case of finite fields, analogous trace-based compression methods cannot be adapted to accommodate our extended setting when the underlying systems require more than a mere exponentiation. As an application, we present an improved, torus-based implementation of the ACJT group signature scheme.


Torus-based cryptography ring ℤN discrete logarithm problem factoring problem compression ACJT group signatures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ateniese, G., Camenisch, J., Joye, M., Tsudik, G.: A practical and provably secure coalition-resistant group signature scheme. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 255–270. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Barić, N., Pfitzmann, B.: Collision-free accumulators and fail-stop signature schemes without trees. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 480–494. Springer, Heidelberg (1997)Google Scholar
  3. 3.
    Boneh, D.: The decision Diffie-Hellman problem. In: Buhler, J. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 48–63. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Camenisch, J., Groth, J.: Group signatures: Better efficiency and new theoretical aspects. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 120–133. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Chaum, D., van Heyst, E.: Group signatures. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 257–265. Springer, Heidelberg (1991)Google Scholar
  6. 6.
    Déchène, I.: Generalized Jacobians in Cryptography. PhD thesis, McGill University, Montreal, Canada (2005)Google Scholar
  7. 7.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    van Dijk, M., Granger, R., Page, D., Rubin, K., Silverberg, A., Stam, M., Woodruff, D.P.: Practical cryptography in high dimensional tori. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 234–250. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    van Dijk, M., Woodruff, D.: Asymptotically optimal communication for torus-based cryptography. In: Franklin, M.K. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 157–178. Springer, Heidelberg (2004)Google Scholar
  10. 10.
    Fujisaki, E., Okamoto, T.: Statistical zero-knowledge protocols to prove modular polynomial equations. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 16–30. Springer, Heidelberg (1997)Google Scholar
  11. 11.
    Granger, R., Page, D., Stam, M.: A comparison of CEILIDH and XTR. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 235–249. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    Granger, R., Vercauteren, F.: On the discrete logarithm problem on algebraic tori. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 66–85. Springer, Heidelberg (2005)Google Scholar
  13. 13.
    Joye, M., Yen, S.-M.: The Montgomery powering ladder. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 291–302. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Lemmermeyer, F.: Higher descent on Pell conics, III (2003) (preprint)Google Scholar
  15. 15.
    Lenstra, A.K.: Using cyclotomic polynomials to construct efficient discrete logarithm cryptosystems over finite fields. In: Mu, Y., Pieprzyk, J., Varadharajan, V. (eds.) ACISP 1997. LNCS, vol. 1270, pp. 127–138. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  16. 16.
    Lenstra, A.K., Verheul, E.R.: The XTR public key system. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, p. 119. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    McCurley, K.S.: A key distribution system equivalent to factoring. Journal of Cryptology 1(2), 95–105 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Menezes, A.J.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, Dordrecht (1993)zbMATHGoogle Scholar
  19. 19.
    Menezes, A.J., van Oorchot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)zbMATHGoogle Scholar
  20. 20.
    Niewenglowski, B.: Note sur les équations x 2 − ay 2 = 1 et x 2 − ay 2 = − 1. Bulletin de la Société Mathématique de France 35, 126–131 (1907)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Rubin, K., Silverberg, A.: Torus-based cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)Google Scholar
  22. 22.
    Rubin, K., Silverberg, A.: Using primitive subgroups to do more with fewer bits. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 18–41. Springer, Heidelberg (2004)Google Scholar
  23. 23.
    Rubin, K., Silverberg, A.: Compression in finite fields and torus-based cryptography. SIAM Journal on Computing 37(5), 1401–1428 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Schnorr, C.-P.: Efficient signature generation by smart cards. Journal of Cryptology 4(3), 161–174 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Shmuely, Z.: Composite Diffie-Hellman public key generating systems hard to break. Technical Report 356, Israel Institute of Technology, Computer Science Department, Technion (February 1985)Google Scholar
  26. 26.
    Smith, P., Skinner, C.: A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms. In: Safavi-Naini, R., Pieprzyk, J. (eds.) ASIACRYPT 1994. LNCS, vol. 917, pp. 357–364. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  27. 27.
    Smith, P.J., Lennon, M.J.J.: LUC: A new public key system. In: Dougall, E.G. (ed.) 9th International Conference on Information Security (IFIP/Sec 1993). IFIP Transactions, vol. A-37, pp. 103–117. North-Holland, Amsterdam (1993)Google Scholar
  28. 28.
    Stam, M., Lenstra, A.K.: Speeding up XTR. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 125–143. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  29. 29.
    Trusted Computing Group. TCG TPM specification 1.2 (2003),

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Marc Joye
    • 1
  1. 1.Thomson R&DSecurity Competence CenterCesson-Sévigné CedexFrance

Personalised recommendations