Efficient Signature Schemes Based on Birational Permutations

  • Adi Shamir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 773)


Many public key cryptographic schemes (such as cubic RSA) are based on low degree polynomials whose inverses are high degree polynomials. These functions are very easy to compute but time consuming to invert even by their legitimate users. To overcome this problem, it is natural to consider the class of birational permutations ƒ over k-tuples of numbers, in which both ƒ and ƒ−1 are low degree rational functions. In this paper we develop two new families of birational permutations, and discuss their cryptographic applications.


  1. 1.
    D. Coppersmith J. Stern and S. Vaudenay [1993]: “Attacks on the Birational Permutation Signature Schemes”, Proceedings of CRYPTO 93 (this volume).Google Scholar
  2. 2.
    W. Diffie and M. Hellman [1976]: “New Directions in Cryptography”, IEEE Trans. Information Theory, Vol IT-22, No 6, pp 644–654.CrossRefMathSciNetGoogle Scholar
  3. 3.
    DSS [1991]: “Specifications for a Digital Signature Standard”, US Federal Register Vol 56 No 169, August 30 1991.Google Scholar
  4. 4.
    H. Fell and W. Diffie [1985]: “Analysis of a Public Key Approach Based on Polynomial Substitution”, Proceedings of CRYPTO 85, Springer-Verlag Vol 218, pp 340–349.MathSciNetGoogle Scholar
  5. 5.
    A. Fiat and A. Shamir [1986]: “How to Prove Yourself: Practical Solutions to Identification and Signature Problems”, Proceedings of CRYPTO 86, Springer-Verlag Vol 263, pp 186–194.CrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Merkle and M. Hellman [1978]: “Hiding Information and Signatures in Trapdoor Knapsacks”, IEEE Trans. Information Theory, Vol IT-24, No 5, pp 525–530.CrossRefGoogle Scholar
  7. 7.
    R. Rivest, A. Shamir and L. Adleman [1978]: “A Method for Obtaining Digital Signatures and Public Key Cryptosystems”, Comm. ACM, Vol 21. No 2, pp 120–126.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    A. Shamir [1993]: “On the Generation of Multivariate Polynomials Which Are Hard To Factor”, Proceedings of STOC 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Adi Shamir
    • 1
  1. 1.Dept. Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

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