An unconditional study of computational zero knowledge

SIAM Journal on Computing

Volumn 36, Issue 4, 2006, Pages 1160-1214

  Vadhan, Salil P ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

Auxiliary input one way functions; Computational complexity; Cryptography; Language dependent commitment schemes; Pseudoentropy; Zero knowledge proofs

Indexed keywords

COMPUTATIONAL COMPLEXITY; FUNCTION EVALUATION; PROBLEM SOLVING; STATISTICAL METHODS; THEOREM PROVING;

AUXILIARY-INPUT ONE-WAY FUNCTIONS; BLACK-BOX SIMULATORS; LANGUAGE-DEPENDENT COMMITMENT SCHEMES; PSEUDOENTROPY; ZERO-KNOWLEDGE PROOFS;

PUBLIC KEY CRYPTOGRAPHY;

EID: 34547839124     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539705447207     Document Type: Article

References (60)

1. W. AIELLO AND J. HÅSTAD, Statistical zero-knowledge languages can be recognized in two rounds, J. Comput. System Sci., 42 (1991), pp. 327-345.
2. L. BABAI AND S. MORAN, Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes, J. Comput. System Sci., 36 (1988), pp. 254-276.
3. B. BARAK, How to go beyond the black-box simulation barrier, in Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV), 2001, pp. 106-115.
4. B. BARAK, Y. LINDELL, AND S. VADHAN, Lower bounds for non-black-box zero knowledge, J. Comput. System Sci., 72 (2006), pp. 321-391.
5. M. BELLARE, O. GOLDREICH, AND E. PETRANK, Uniform generation of NP-witnesses using an NP-oracle, Inform. and Comput., 163 (2000), pp. 510-526.
6. M. BELLARE, S. MICALI, AND R. OSTROVSKY, Perfect zero-knowledge in constant rounds, in Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing (Baltimore, MD), 1990, pp. 482-493.
7. M. BEN-OR, O. GOLDREICH, S. GOLDWASSER, J. HÅSTAD, J. KILIAN, S. MICALI, AND P. ROGAWAY, Everything provable is provable in zero-knowledge, in Advances in Cryptology - CRYPTO '88, Lecture Notes in Comput. Sci. 403, S. Goldwasser, ed., Springer-Verlag, Berlin, 1990, pp. 37-56.
8. C. H. BENNETT, G. BRASSARD, AND J.-M. ROBERT, Privacy amplification by public discussion, SIAM J. Comput., 17 (1988), pp. 210-229.
9. M. BLUM, How to prove a theorem so no one else can claim it, in Proceedings of the International Congress of Mathematicians (Berkeley, CA), AMS, Providence, RI, 1987, pp. 1444-1451.
10. G. BRASSARD, D. CHAUM, AND C. CRÉPEAU, Minimum disclosure proofs of knowledge, J. Comput. System Sci., 37 (1988), pp. 156-189.
11. T. M. COVER AND J. A. THOMAS, Elements of Information Theory, 2nd ed., Wiley Series in Telecommunications, John Wiley & Sons, New York, 1991.
12. I. B. DAMGÅRD, On the existence of bit-commitment schemes and zero-knowledge proofs, in Advances in Cryptology-CRYPTO '89, Lecture Notes in Comput. Sci. 435, G. Brassard, ed., Springer-Verlag, Berlin, 1990, pp. 17-29.
13. I. B. DAMGÅRD, Interactive hashing can simplify zero-knowledge protocol design without computational assumptions (extended abstract), in Advances in Cryptology-CRYPTO '93, Lecture Notes in Comput. Sci. 773, D. R. Stinson, ed., Springer-Verlag, Berlin, pp. 100-109.
14. A. DE SANTIS, G. DI CRESCENZO, G. PERSIANO, AND M. YUNG, Image density is complete for non-interactive-SZK, in Proceedings of the 25th International Colloquium on Automata, Languages and Programming (Aalborg, Denmark), Lecture Notes in Comput. Sci., Springer-Verlag, Berlin, pp. 784-795.
15. G. DI CRESCENZO, T. OKAMOTO, AND M. YUNG, Keeping the SZK-verifier honest unconditionally, in Advances in Cryptology-CRYPTO '97, Lecture Notes in Comput. Sci. 1294, B. S. Kaliski, Jr., ed., Springer-Verlag, Berlin, pp. 31-45.
16. C. DWORK, M. NAOR, AND A. SAHAI, Concurrent zero-knowledge, J. ACM, 51 (2004), pp. 851-898.
17. S. EVEN, A. L. SELMAN, AND Y. YACOBI, The complexity of promise problems with applications to public-key cryptography, Inform. Control, 61 (1984), pp. 159-173.
18. L. FORTNOW, The complexity of perfect zero-knowledge, in Advances in Computing Research, Vol. 5, S. Micali, ed., JAI Press, Greenwich, CT, 1989, pp. 327-343.
19. M. FÜRER, O. GOLDREICH, Y. MANSOUR, M. SIPSER, AND S. ZACHOS, On completeness and soundness in interactive proof systems, in Advances in Computing Research, Vol. 5, S. Micali, ed., JAI Press, Greenwich, CT, 1989, pp. 429-442.
20. O. GOLDREICH, A note on computational indistinguishability, Inform. Process. Lett., 34 (1990), pp. 277-281.
21. O. GOLDREICH, A uniform-complexity treatment of encryption and zero-knowledge, J. Cryptology, 6 (1993), pp. 21-53.
22. O. GOLDREICH, Modern Cryptography, Probabilistic Proofs and Pseudorandomness, Algorithms and Combin. 17, Springer-Verlag, Berlin, 1999.
23. O. GOLDREICH, Foundations of Cryptography: Basic Tools, Cambridge University Press, Cambridge, UK, 2001.
24. O. GOLDREICH, On Promise Problems (A Survey in Memory of Shimon Even [1935-2004]), Tech. Report TR05-018, Electronic Colloquium on Computational Complexity, 2005. Available online at http://eccc.hpi-web.de/.
25. O. GOLDREICH AND A. KAHAN, How to construct constant-round zero-knowledge proof systems for NP, J. Cryptology, 9 (1996), pp. 167-190.
26. O. GOLDREICH AND H. KRAWCZYK, Sparse pseudorandom distributions, Random Structures Algorithms, 3 (1992), pp. 163-174.
27. O. GOLDREICH AND H. KRAWCZYK, On the composition of zero-knowledge proof systems, SIAM J. Comput., 25 (1996), pp. 169-192.
28. O. GOLDREICH AND E. KUSHILEVITZ, A perfect zero-knowledge proof system for a problem equivalent to the discrete logarithm, J. Cryptology, 6 (1993), pp. 97-116.
29. O. GOLDREICH, S. MICALI, AND A. WIGDERSON, Proofs that yield nothing but their validity, or all languages in NP have zero-knowledge proof systems, J. Assoc. Comput. Mach., 38 (1991), pp. 691-729.
30. O. GOLDREICH AND Y. OREN, Definitions and properties of zero-knowledge proof systems, J. Cryptology, 7 (1994), pp. 1-32.
31. O. GOLDREICH AND E. PETRANK, Quantifying knowledge complexity, Comput. Complex., 8 (1999), pp. 50-98.
32. O. GOLDREICH, A. SAHAI, AND S. VADHAN, Honest verifier statistical zero-knowledge equals general statistical zero-knowledge, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing (Dallas, TX), 1998, pp. 399-408.
33. O. GOLDREICH, A. SAHAI, AND S. VADHAN, Can statistical zero-knowledge be made non-interactive?, or On the relationship of SZK and NISZK, in Advances in Cryptology - CRYPTO '99, Lecture Notes in Comput. Sci. 1666, M. Wiener, ed., Springer-Verlag, Berlin, pp. 467-484.
34. O. GOLDREICH AND S. VADHAN, Comparing entropies in statistical zero-knowledge with applications to the structure of SZK, in Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity (Atlanta, GA), 1999, pp. 54-73.
35. S. GOLDWASSER, S. MICALI, AND C. RACKOFF, The knowledge complexity of interactive proof systems, SIAM J. Comput., 18 (1989), pp. 186-208.
36. S. GOLDWASSER AND M. SIPSER, Private coins versus public coins in interactive proof systems, in Advances in Computing Research, Vol. 5, S. Micali, ed., JAI Press, Greenwich, CT, 1989, pp. 73-90.
37. J. HASTAD, R. IMPAGLIAZZO, L. A. LEVIN, AND M. LUBY, A pseudorandom generator from any one-way function, SIAM J. Comput., 28 (1999), pp. 1364-1396.
38. R. IMPAGLIAZZO, L. A. LEVIN, AND M. LUBY, Pseudo-random generation from one-way functions (extended abstract), in Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing (Seattle, WA), 1989, pp. 12-24.
39. R. IMPAGLIAZZO AND M. YUNG, Direct minimum-knowledge computations (extended abstract), in Advances in Cryptology-CRYPTO '87 Lecture Notes in Comput. Sci. 293, C. Pomerance, ed., Springer-Verlag, Berlin, 1988, pp. 40-51.
40. T. ITOH, Y. OHTA, AND H. SHIZUYA, A language-dependent cryptographic primitive, J. Cryptology, 10 (1997), pp. 37-49.
41. M. R. JERRUM, L. G. VALIANT, AND V. V. VAZIRANI, Random generation of combinatorial structures from a uniform distribution, Theoret. Comput. Sci., 43 (1986), pp. 169-188.
42. C. LUND, L. FORTNOW, H. KARLOFF, AND N. NISAN, Algebraic methods for interactive proof systems, J. Assoc. Comput. Mach., 39 (1992), pp. 859-868.
43. D. MICCIANCIO, S. J. ONG, A. SAHAI, AND S. VADHAN, Concurrent zero knowledge without complexity assumptions, in Proceedings of the Third Theory of Cryptography Conference (TCC '06), Lecture Notes in Comput. Sci. 3876, S. Halevi and T. Rabin, eds., Springer-Verlag, Berlin, pp. 1-20.
44. D. MICCIANCIO AND S. VADHAN, Statistical zero-knowledge proofs with efficient provers: Lattice problems and more, in Advances in Cryptology-CRYPTO '03, Lecture Notes in Comput. Sci. 2729, D. Boneh, ed., Springer-Verlag, Berlin, pp. 282-298.
45. M. NAOR, Bit commitment using pseudorandomness, J. Cryptology, 4 (1991), pp. 151-158.
46. M. NGUYEN AND S. VADHAN, Zero knowledge with efficient provers, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC '06), Seattle, WA, 2006, pp. 287-295.
47. N. NISAN AND A. TA-SHMA, Extracting randomness: A survey and new constructions, J. Comput. System Sci., 58 (1999), pp. 148-173.
48. T. OKAMOTO, On relationships between statistical zero-knowledge proofs, J. Comput. System Sci., 60 (2000), pp. 47-108.
49. R. OSTROVSKY, One-way functions, hard on average problems, and statistical zero-knowledge proofs, in Proceedings of the Sixth Annual Structure in Complexity Theory Conference (Chicago, IL), 1991, pp. 133-138.
50. R. OSTROVSKY AND A. WIGDERSON, One-way functions are essential for non-trivial zero-knowledge, in Proceedings of the Second Israel Symposium on Theory of Computing and Systems, IEEE Press, Los Alamitos, CA, 1993, pp. 3-17.
51. E. PETRANK AND G. TARDOS, On the knowledge complexity of NP, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT), 1996, pp. 494-503.
52. A. SAHAI AND S. VADHAN, A complete problem for statistical zero knowledge, J. ACM, 50 (2003), pp. 196-249.
53. R. SHALTIEL, Recent developments in explicit constructions of extractors, in Current Trends in Theoretical Computer Science: The Challenge of the New Century, Vol. I: Algorithms and Complexity, G. Paun, G. Rozenberg, and A. Salomaa, eds., World Scientific, River Edge, NJ, 2004, pp. 189-228.
54. A. SHAMIR, IP = PSPACE, J. Assoc. Comput. Mach., 39 (1992), pp. 869-877.
55. M. SIPSER, A complexity theoretic approach to randomness, in Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (Boston, MA), 1983, pp. 330-335.
56. L. STOCKMEYER, On approximation algorithms for #P, SIAM J. Comput., 14 (1985), pp. 849-861.
57. S. P. VADHAN, A Study of Statistical Zero-Knowledge Proofs, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1999. Available from author's Web page, http://eccs.harvard.edu/~salil.
58. S. P. VADHAN, An unconditional study of computational zero knowledge, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (Rome, Italy), 2004, pp. 176-185.
59. S. P. VADHAN, An Unconditional Study of Computational Zero Knowledge, Tech. Report TR06-056, Electronic Colloquium on Computational Complexity, 2006. Available online from http://eccc.hpi-web.de/.
60. A. C. YAO, Theory and applications of trapdoor functions (extended abstract), in Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science (Chicago, IL), 1982, pp. 80-91.