On the power of commutativity in cryptography

  • Adi Shamir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 85)


Every field needs some unifying ideas which are applicable to a wide variety of situations. In cryptography, the notion of commutativity seems to play such a role. This paper surveys its potential applications, such as the generation of common keys, challenge-and-response identification, signature generation and verification, key-less communication and remote game playing.


Commutative Diagram Unify Idea Encryption Function Modular Exponentiation Root Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    B. Blakley and G. Blakely [1978], "Security of Number Theoretic Public-Key Cryptosystems Against Random Attack," Cryptologia, October 1978.Google Scholar
  2. 2.
    W. Diffie and M. Hellman [1976], "New Directions in Cryptography", IEEE Trans. Info. Theory, November 1976.Google Scholar
  3. 3.
    S. Even and Y. Yacobi [1980], "Cryptocomplexity and NP-Completeness", Seventh ICALP, July 1980.Google Scholar
  4. 4.
    D. Knuth [1969], The Art of Computer Programming, Vol 2, Addison-Wesley, 1969.Google Scholar
  5. 5.
    S. Pohlig and M. Hellman [1978], "An Improved Algorithm for Computing Logarithms Over GF(P) and Its Cryptographic Significance", IEEE Trans. Info. Theory, January 1978.Google Scholar
  6. 6.
    R. Rivest, A. Shamir and L. Adleman [1978], "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems", CACM, February 1978.Google Scholar
  7. 7.
    A. Shamir [1979], "On the Cryptocomplexity of Knapsack Systems", Proc. Eleventh ACM Symposium on the Theory of Computing, May 1979.Google Scholar
  8. 8.
    A. Shamir [1980], "A Pseudo-Random Sequence Generator Whose Cryptocomplexity is Provably Equivalent to that of the RSA", in preparation.Google Scholar
  9. 9.
    A. Shamir, R. Rivest and L. Adleman [1979], "Mental Poker", MIT/LCS/TM-125, February 1979.Google Scholar
  10. 10.
    C. Shannon [1948], "The Mathematical Theory of Communication", Bell System Technical Journal, July and October 1948.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • Adi Shamir
    • 1
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  2. 2.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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