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Practical Zero-Knowledge Proofs: Giving Hints and Using Deficiencies

  • Joan Boyar
  • Katalin Friedl
  • Carsten Lund
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 434)

Abstract

New practical zero-knowledge proofs are given for some number-theoretic problems. All of the problems are in NP, but the proofs given here are much more efficient than the previously known proofs. In addition, these proofs do not require the prover to be super-polynomial in power. A BPP prover with the appropriate trap-door knowledge is sufficient. The proofs are perfect or statistical zero-knowledge in all cases except one.

References

  1. [1]
    Adleman, L., and M.-D. Huang, Recognizing primes in random polynomial time, Proc. 19th ACM Symp. on Theory of Computing, 1987, pp. 462–469.Google Scholar
  2. [2]
    Adleman, L., K. Manders, and G. Miller, On taking roots in finite fields, Proc. 18th IEEE Symp. on Foundations of Computer Science, 1977, pp. 175–178.Google Scholar
  3. [3]
    Bach, E., How to generate factored random numbers, SIAM Journal on Computing, vol. 17, No. 2, April 1988, pp. 179–193.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Bellare, M., S. Micali and R. Ostrovsky, personal communication.Google Scholar
  5. [5]
    Benaloh, J., Cryptographic capsules: a disjunctive primitive for interactive protocols, Advances in Cryptology-Crypto’ 86 Proceedings, 1987, pp. 213–222.Google Scholar
  6. [6]
    Berlekamp, E. Factoring polynomials over large finite fields, Mathematics of Computations, vol. 24, 1970, pp. 713–735.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Brassard, G., and C. Crépeau, Non-transitive transfer of confidence: a perfect zero-knowledge interactive protocol for SAT and beyond, Proc. 27th IEEE Symp. on Foundations of Computer Science, 1986, pp. 188–195.Google Scholar
  8. [8]
    Brassard, G., C. Crépeau, and J.M. Robert, All-or-nothing disclosure of secrets, Advances in Cryptology-Crypto’ 86 Proceedings, 1987, pp. 234–238.Google Scholar
  9. [9]
    Chaum, D., Demonstrating that a public predicate can be satisfied without revealing any information about how, Advances in Cryptology-Crypto’ 86 Proceedings, 1987, pp. 195–199.Google Scholar
  10. [10]
    Chaum, D. J.-H. Evertse, J. van de Graaf, An improved protocol for demonstrating possession of discrete logarithms and some generalizations, Advances in Cryptology — EUROCRYPT’ 87 Proceedings, 1988, pp. 127–141.Google Scholar
  11. [11]
    Chaum, D., J.-H. Evertse, J. van de Graaf, and R. Peralta, Demonstrating possession of a discrete logarithm without revealing it, Advances in Cryptology-Crypto’ 86 Proceedings, 1987, pp. 200–212.Google Scholar
  12. [12]
    Davenport, H., Multiplicative Number Theory, Markham Publishing Company, 1967.Google Scholar
  13. [13]
    Feige, U., A. Fiat, and A. Shamir, Zero-knowledge proofs of identity, Journal of Cryptology, 1(2), 1988, pp. 77–94.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Goldreich, O., S. Micali, and A,. Wigderson, Proofs that yield nothing but their validity and a methodology of cryptographic protocol design, Proc. 27th IEEE Symp. on Foundations of Computer Science, 1986, pp. 174–187.Google Scholar
  15. [15]
    Goldreich, O., S. Micali, and A,. Wigderson, Proofs that yield nothing but their validity and a methodology of cryptographic protocol design, To appear.Google Scholar
  16. [16]
    Goldwasser, S., and S. Micali, Probabilistic encryption, Journal of Computer and System Sciences, vol. 28, 1984, pp. 270–299.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Goldwasser, S., S. Micali, and C. Rackoff, The knowledge complexity of interactive proof systems, SIAM Journal on Computing, vol. 18, 1989, pp. 186–208.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Van de Graaf, J., and R. Peralta, A simple and secure way to show the validity of your public key, Advances in Cryptology-Crypto’ 87 Proceedings, 1988, pp. 128–134.Google Scholar
  19. [19]
    Knuth, D. E. The Art of Computer Programming Vol 2, Addison-Wesley, 1969.Google Scholar
  20. [20]
    Oren, Y. On the Cunning Power of Cheating Verifiers: some Observations About Zero Knowledge Proofs, Proc. 28th IEEE Symp. on Foundations of Computer Science, 1987, pp. 462–471.Google Scholar
  21. [21]
    Rabin, M.O., Digitalized signatures and public-key functions as intractable as factorization, Technical Report MIT/LCS/TR-212, M.I.T., January 1979.Google Scholar
  22. [22]
    Rabin, M.O., Probabilistic algorithms in finite fields, SIAM Journal on Computing, vol. 9, 1980, pp. 273–280.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Rosser, J. B., and Schoenfeld, L., Approximate Formulas for some Functions of Prime Numbers, Illinois Journal of Math. vol. 6, 1962, pp. 64–94.zbMATHMathSciNetGoogle Scholar
  24. [24]
    Schwarz, W., in American Math. Monthly, vol. 73, 1966, pp. 426–427.CrossRefGoogle Scholar
  25. [25]
    Tompa, M., and H. Woll, Random self-reducibility and zero knowledge interactive proofs of possession of information, Proc. 28th IEEE Symp. on Foundations of Computer Science, 1987, pp. 472–482.Google Scholar
  26. [26]
    Wagstaff, S. S., Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Mathematics of Computation, vol. 33 no. 147, July 1979, pp. 1073–1080.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Joan Boyar
    • 1
  • Katalin Friedl
    • 1
  • Carsten Lund
    • 1
  1. 1.Computer Science DepartmentUniversity of ChicagoChicago

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