Two Remarks Concerning the Goldwasser-Micali-Rivest Signature Scheme

  • Oded Goldreich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 263)


The focus of this note is the Goldwasser-Micali-Rivest Signature Scheme (presented in the 25th FOCS, 1984). The GMR scheme has the salient property that, unless factoring is easy, it is infeasible to forge any signature even through an adaptive chosen message attack. We present two technical contributions with respect to the GMR scheme:
  1. 1)

    The GMR scheme can be made totally “memoryless’: That is, the signature generated by the signer on message M does not depend on the previous signed messages. (In the original scheme, the signature to a message depends on the number of messages signed before.

  2. 2)

    The GMR scheme can be implemented almost as efficiently as the RSA: The original implementation of the GMR scheme based on factoring, can be speeded-up by a factor of |N|. Thus, both signing and verifying take time O(|N|3log2|N|). (Here N is the moduli.)



  1. [D]
    Damgard, I.B., “Collision Free Hash Functions and Public Key Signature Schemes”, manuscript, 1986.Google Scholar
  2. [DH]
    Diffie, W., and Hellman, M.E., “New Directions in Cryptography”, IEEE Trans. on Inform. Theory, Vol. IT-22, No. 6, November 1976, pp. 644–654.CrossRefMathSciNetGoogle Scholar
  3. [GGM]
    Goldreich, O., S. Goldwasser, and S. Micali, “How to Construct Random Functions”, Proc. of 25th Symp. on Foundation of Computer Science, 1984, pp. 464–479. To appear in Jour. of ACM.Google Scholar
  4. [GMR]
    Goldwasser, S., S. Micali, and R.L. Rivest, “A Paradoxical Solution to the Signature Problem”, Proc. of 25th Symp. on Foundation of Computer Science, 1984, pp. 441–448. A better version is available from the authors.Google Scholar
  5. [RSA]
    Rivest, R.L., Shamir, A., and Adleman, L., “A Method for Obtaining Digital Signatures and Public Key Cryptosystems”, Comm. of the ACM, Vol. 21, February 1978, pp. 120–126.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Oded Goldreich
    • 1
  1. 1.Computer Science DepartmentTechnionHaifaIsrael

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