Efficient signature generation by smart cards

Abstract

We present a new public-key signature scheme and a corresponding authentication scheme that are based on discrete logarithms in a subgroup of units in ℤ p where p is a sufficiently large prime, e.g., p ≥ 2512. A key idea is to use for the base of the discrete logarithm an integer α in ℤ p such that the order of α is a sufficiently large prime q, e.g., q ≥ 2140. In this way we improve the ElGamal signature scheme in the speed of the procedures for the generation and the verification of signatures and also in the bit length of signatures. We present an efficient algorithm that preprocesses the exponentiation of a random residue modulo p.

References

  1. Beth, T.: Efficient Zero-Knowledge Identification Scheme for Smart Cards. Advances in Cryptology — Eurocrypt '88, Lecture Notes in Computer Science, Vol. 330 (1988), Springer-Verlag, Berlin, pp. 77–86.

    Google Scholar 

  2. Brickell, E. F., and McCurley, K. S.: An Interactive Identification Scheme Based on Discrete Logarithms and Factoring. Advances in Cryptology—Eurocrypt '90, Lecture Notes in Computer Science, Vol. 473 (1991), Springer-Verlag, Berlin, pp. 63–71.

    Google Scholar 

  3. Chaum, D., Evertse, J. H., and an de Graaf, J.: An Improved Protocol for Demonstrating Possession of Discrete Logarithms and Some Generalizations. Advances in Cryptology—Eurocrypt '87, Lecture Notes in Computer Science, Vol. 304 (1988), Springer-Verlag, Berlin, pp. 127–141.

    Google Scholar 

  4. Coppersmith, D., Odlyzko, A., and Schroeppel, R.: Discrete Logarithms in GF(p). Algorithmica, 1 (1986), 1–15.

    Google Scholar 

  5. ElGamal, T.: A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. IEEE Trans. Inform. Theory, 31 (1985), 469–472.

    Google Scholar 

  6. Even, S., Goldreich, O., and Micali, S.: On-Line/Off-Line Digital Signatures. Advances in Cryptology—Crypto '89. Lecture Notes in Computer Science, vol. 435 (1990), Springer-Verlag, Berlin, pp. 263–277.

    Google Scholar 

  7. Feige, U., Fiat, A. and Shamir, A.: Zero-Knowledge Proofs of Identity. Proceedings of STOC, 1987, pp. 210–217, and J. Cryptology, 1 (1988), 77–95.

  8. Fiat, A., and Shamir, A.: How To Prove Yourself: Practical Solutions of Identification and Signature Problems. Advances in Cryptology—Crypto '86, Lecture Notes in Computer Science, Vol. 263 (1987), Springer-Verlag, Berlin, pp. 186–194.

    Google Scholar 

  9. Girault, M.: An Identity-Based Identification Scheme Based on Discrete Logarithms. Advances in Cryptology—Eurocrypt '90, Lecture Notes in Computer Science, Vol. 473 (1991), Springer-Verlag, Berlin, pp. 481–486.

    Google Scholar 

  10. Girault, M.: Self-Certified Public Keys. Abstracts of Eurocrypt '91, Brighton, 8–11 April 1991, pp. 236–241.

  11. Goldwasser, S., Micali, S., and Rackoff, C: Knowledge Complexity of Interactive Proof Systems. Proceedings of STOC, 1985, pp. 291–304.

  12. Gordon, D.: Discrete Logarithms in GF(p) Using the Number Field Sieve. Technical Report, Sandia Laboratories (1990).

  13. Guillou, L. S., and Quisquater, J. J.: A Practical Zero-Knowledge Protocol Fitted to Security Microprocessor Minimizing both Transmission and Memory. Advances in Cryptology—Eurocrypt '88, Lecture Notes in Computer Sciences, Vol. 330 (1988), Springer-Verlag, Berlin, pp. 123–128.

    Google Scholar 

  14. Günther, C. G.: An Identity-Based Key-Exchange Protocol. Advances in Cryptology—Eurocrypt '89, Lecture Notes in Computer Science, Vol. 434 (1990). Springer-Verlag, Berlin, pp. 29–37.

    Google Scholar 

  15. Lenstra, A. K., Lenstra, H. W., Jr., Manasse, M. S., and Pollard, J. M.: The Number Field Sieve. Proceedings of STOC, 1990, pp. 564–572.

  16. Ong, H., and Schnorr, C. P.: Fast Signature Generation with a Fiat-Shamir-like Scheme. Advances in Cryptology—Eurocrypt '90, Lecture Notes in Computer Science, Vol. 473 (1991), Springer-Verlag, Berlin, pp. 432–440.

    Google Scholar 

  17. Pollard, J. M.: Monte Carlo Method for Index Computation (mod p). Math. Comp., 32 (1978), 918–924.

    Google Scholar 

  18. Pollard, J. M.: Some Algorithms in Number Theory. Technical Report, 15 pages, Feb. 1991.

  19. Rabin, M. O.: Digital Signatures and Public-Key Functions as Intractable as Factorization. Technical Report MIT/LCS/TR-212, Massachusetts Institute of Technology (1978).

  20. Rivest, R., Shamir, A., and Adleman, L.: A Method for Obtaining Digital Signatures and Public Key Cryptosystems. Comm. ACM, 21 (1978), 120–126.

    Google Scholar 

  21. de Rooij, P. J. N.: On the Security of the Schnorr Scheme Using Preprocessing. Proceedings Eurocrypt '91.

  22. Schnorr, C. P.: Efficient Identification and Signatures for Smart Cards. Advances in Cryptology—Crypto '89. Lecture Notes in Computer Science, Vol. 435 (1990), Springer-Verlag, Berlin, pp. 239–252.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

European patent application 89103290.6 from February 24, 1989. U.S. patent number 4995082 of February 19, 1991.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Schnorr, C.P. Efficient signature generation by smart cards. J. Cryptology 4, 161–174 (1991). https://doi.org/10.1007/BF00196725

Download citation

Key words

  • Digital signatures
  • Public-key signatures
  • Public-key authentication
  • ElGamal signatures
  • Discrete logarithm one-way function
  • Signatures with preprocessing
  • Random exponentiated residues