Synthesizers and their application to the parallel construction of pseudo-random functions

Journal of Computer and System Sciences

Volumn 58, Issue 2, 1999, Pages 336-375

  Naor, Moni ^a   Reingold, Omer ^a  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

FUNCTIONS; RANDOM PROCESSES;

COMPUTATIONAL LEARNING THEORY; DIFFIE-HELLMAN ASSUMPTION; PSEUDO-RANDOM FUNCTIONS;

CRYPTOGRAPHY;

EID: 0032647187     PISSN: 00220000     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcss.1998.1618     Document Type: Article

References (67)

1. M. Ajtai and A. Wigderson, "Deterministic Simulations of Probabilistic Constant Depth Circuits, ADVCR: Advances in Computing Research," Vol. 5, 1989. [Preliminary version: "Proc. 26th Symp. on Foundations of Computer Science, 1985," pp. 11-19]
2. W. B. Alexi, B. Chor, O. Goldreich, and C. P. Schnorr, RSA and Rabin functions: Certain parts are as hard as the whole, SIAM J. Comput. 17, No. 2 (1988), 194-209.
3. D. Angluin and M. Kharitonov, When won't membership queries help?, J. Comput. System Sci. 50 (1995), 336-355.
4. L. Babai, N. Nisan, and M. Szegedy, Multiparty protocols, pseudo-random generators for logspace, and time-space tradeoffs, J. Comput. System Sci. 45, No. 2 (1992), 204-232.
5. E. Bach, How to generate factored random numbers, SIAM J. Comput. 17 No. 2 (1988), 179-193.
6. P. W. Beame, S. A. Cook, and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput. 15 (1986), 994-1003.
7. M. Bellare, O. Goldreich, and S. Goldwasser, Incremental cryptography: The case of hashing and signing, in "Advances in Cryptology - CRYPTO '94," Lecture Notes in Computer Science, Vol. 839, pp. 216-233, Springer-Verlag, New York/Berlin, 1994.
8. M. Bellare, O. Goldreich, and S. Goldwasser, Incremental cryptography with application to virus protection, in "Proc. 27th Ann. ACM Symp. on Theory of Computing, 1995," pp. 45-56.
9. M. Bellare and S. Goldwasser, New paradigms for digital signatures and message authentication based on non-interactive zero knowledge proofs, in "Advances in Cryptology - CRYPTO '89," Lecture Notes in Computer Science, Vol. 435, pp. 194-211, Springer-Verlag, 1990.
10. E. Biham, D. Boneh, and O. Reingold, Generalized Diffie-Hellman modulo a composite is not weaker than factoring, in "Theory of Cryptography Library," Record 97-14 at http://theory.lcs.mit.edu/t ̃cryptol/homepage.html.
11. L. Blum, M. Blum, and M. Shub, A simple secure unpredictable pseudo-random number generator, SIAM J. Comput. 15 (1986), 364-383.
12. M. Blum, W. Evans, P. Gemmell, S. Kannan, and M. Naor, Checking the correctness of memories, in "Algorithmica, 1994," pp. 225-244. [Preliminary version: "Proc. 31st Symp. on Foundations of Computer Science, 1990"]
13. A. Blum, M. Furst, M. Kearns, and R. J. Lipton, Cryptographic primitives based on hard learning problems, in "Advances in Cryptology - CRYPTO '93," Lecture Notes in Computer Science, Vol. 773, pp. 278-291, Springer-Verlag, 1994.
14. M. Blum and S. Micali, How to generate cryptographically strong sequence of pseudo-random bits, SIAM J. Comput. 13 (1984), 850-864.
15. D. Boneh and R. Lipton, Algorithms for Black-Box fields and their application to cryptography, in "Advances in Cryptology - CRYPTO '96," LNCS, Vol. 1109, pp. 283-297, Springer, 1996.
16. G. Brassard, "Modern Cryptology," Lecture Notes in Computer Science, Vol. 325, Springer-Verlag, New York/Berlin, 1988.
17. R. Cleve, "Methodologies for Designing Block Ciphers and Cryptographic Protocols, Part I," Ph.D. thesis, University of Toronto, 1989.
18. R. Cleve, Complexity theoretic issues concerning block ciphers related to D.E.S., in "Advances in Cryptology - CRYPTO '90," Lecture Notes in Computer Science, Vol. 537, pp. 530-544, Springer-Verlag, 1991.
19. B. Chor, A. Fiat, and M. Naor, Tracing traitors, in "Advances in Cryptology - CRYPTO '94," pp. 257-270, Springer-Verlag, 1994.
20. B. Chor and O. Goldreich, Unbiased bits from sources of weak randomness and probabilistic communication complexity, SIAM J. Comput. 17 (1988), 230-261.
21. [Preliminary version in "Proc. 26th IEEE Symp. on Foundations of Computer Science, 1985," pp. 429-442]
22. W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory 22, No. 6 (1976), 644-654.
23. R. Fischlin and C. P. Schnorr, Stronger security proofs for RSA and Rabin bits, in "Advances in Cryptology - EUROCRYPT '97," Lecture Notes in Computer Science, Vol. 1233, pp. 267-279, Springer-Verlag, 1997.
24. O. Goldreich, Two remarks concerning the Goldwasser-Micali-Rivest signature scheme, in "Advances in Cryptology - CRYPTO '86," Lecture Notes in Computer Science, Vol. 263, pp. 104-110, Springer-Verlag, 1987.
25. O. Goldreich, Towards a theory of software protection, in "Proc. 19th Ann. ACM Symp. on Theory of Computing, 1987," pp. 182-194.
26. O. Goldreich, "Foundations of Cryptography (Fragments of a Book)," 1995. Electronic publication: http://www.eccc.uni-trier.de/eccc/info/ECCC-Books/eccc-books.html (Electronic Colloquium on Computational Complexity).
27. O. Goldreich, S. Goldwasser, and S. Micali, How to construct random functions, J. Assoc. Comput. Mach. 33 (1986), 792-807.
28. O. Goldreich, S. Goldwasser, and S. Micali, On the cryptographic applications of random functions, in "Advances in Cryptology - CRYPTO '84," Lecture Notes in Computer Science, Vol. 196, pp. 276-288, Springer-Verlag, 1985.
29. O. Goldreich and L. Levin, A hard-core predicate for all one-way functions, in "Proc. 21st Ann. ACM Symp. on Theory of Computing, 1989," pp. 25-32.
30. S. Goldwasser and S. Micali, Probabilistic encryption, J. Comput. System Sci. 28, No. 2 (1984), 270-299.
31. J. Hastad, A. W. Schrift, and A. Shamir, The discrete logarithm modula a composite hides O(n) bits, J. of Computer and System Sciences 47 (1993), 376-404.
32. J. Hastad, R. Impagliazzo, L. A. Levin, and M. Luby, Construction of a pseudo-random generator from any one-way function, SIAM J. Comput., to appear in [Preliminary versions by Impagliazzo et al. in "21st STOC 1989" and Hastad in "22nd STOC, 1990"]
33. A. Herzberg and M. Luby, Public randomness in cryptography, in "Advances in Cryptology - CRYPTO '92," Lecture Notes in Computer Science, Vol. 740, pp. 421-432, Springer-Verlag, 1992.
34. R. Impagliazzo and L. Levin, No better ways to generate hard NP instances than picking uniformly at random, in "Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990," pp. 812-821.
35. R. Impagliazzo and M. Naor, Efficient cryptographic schemes probably secure as subset sum, J. Cryptology 9 (1996), 199-216.
36. R. Impagliazzo and D. Zuckerman, Recycling random bits, in "Proc. 30th IEEE Symposium on Foundations of Computer Science, 1989," pp. 248-253.
37. R. M. Karp and V. Ramachandran, Parallel algorithms for shared-memory machines, in "Handbook of Theoretical Computer Science, Vol. A" (J. van Leeuwen, Ed.), pp. 869-941, MIT Press, Cambridge, MA, 1990.
38. M. Kearns and L. Valiant, Cryptographic limitations on learning Boolean formulae and finite automata, J. Assoc. Comput. Mach. 41, No. 1 (1994), 67-95.
39. M. Kharitonov, Cryptographic hardness of distribution-specific learning, in "Proc. 25th ACM Symp. on Theory of Computing, 1993," pp. 372-381.
40. N. Linial, Y. Mansour, and N. Nisan, Constant depth circuits, Fourier transform, and learnability, J. Assoc. Comput. Mach. 40, No. 3 (1993), 607-620.
41. M. Luby, "Pseudo-Randomness and Applications," Princeton Univ. Press, Princeton, NJ, 1996.
42. M. Luby and C. Rackoff, How to construct pseudo-random permutations and pseudo-random functions, SIAM J. Comput. 17 (1988), 373-386.
43. L. A. Levin, One-way function and pseudo-random generators, in "Proc. 17th Ann. ACM Symp. on Theory of Computing, 1985," pp. 363-365.
44. U. Maurer, Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms, in "Advances in Cryptology - CRYPTO '94," LNCS, Vol.740, pp. 271-281, Springer, 1994.
45. K. McCurley, A key distribution system equivalent to factoring, J. of Cryptology 1 (1988), 95-105.
46. M. Naor and O. Reingold, On the construction of pseudo-random permutations: Luby-Rackoff revisited, J. of Cryptology 12 (1999), 29-66.
47. [Preliminary version in "Proc. 29th Ann. ACM Symp. on Theory of Computing, 1997," pp. 189-199]
48. M. Naor and O. Reingold, Number-Theoretic constructions of efficient pseudo-random functions, in "Proc. 38th IEEE Symp. on Foundations of Computer Science, 1997."
49. N. Nisan, Pseudo-random generators for space-bounded computation, Combinatorica 12, No. 4 (1992), 449-461.
50. N. Nisan, RL ⊆ SC, in "Proc. 24th Ann. ACM Symp. on Theory of Computing, 1992," pp. 619-623.
51. N. Nisan and A. Wigderson, Hardness vs. randomness, J. Comput. System Sci. 49, No. 2 (1994), 149-167.
52. [Preliminary version in "Proc. 29th IEEE Symp. on Foundations of Computer Science, 1988," pp. 2-12]
53. N. Nisan and D. Zuckerman, Randomness is linear in space, J. Comput. System Sci. 52 (1996), 43-52.
54. [Preliminary version: More deterministic simulations in logspace "Proc. 25th Ann. ACM Symp. on Theory of Computing, 1993," pp. 235-244]
55. R. Ostrovsky, An efficient software protection scheme, in "Proc. 22nd Ann. ACM Symp. on Theory of Computing, 1990," pp. 514-523.
56. M. Ram Murty, Artin's conjecture for primitive roots, in "The Mathematical Intelligencer," Vol. 10, No. 4, pp. 59-67, Springer-Verlag, 1988.
57. A. Razborov and S. Rudich, Natural proofs, J. of Computer and System Sciences 55, No. 1 (1997), 24-35. [
58. Preliminary version: "Proc. 26th Ann. ACM Symp. on Theory of Computing, 1994," pp. 204-213]
59. J. H. Reif and J. D. Tygar, Efficient parallel pseudo-random number generation, SIAM J. Comput. 17, No. 2 (1988), 404-411.
60. R. L. Rivest, A. Shamir, and L. M. Adleman, A method for obtaining digital signature and public key cryptosystems, Comm. ACM 21 (1978), 120-126.
61. A. Shamir, On the generation of cryptographically strong pseudo-random number sequences, in "ACM Trans. Comput. Sys., 1983," pp. 38-44.
62. V. Shoup, Lower bounds for discrete logarithms and related problems, in "Proc. Advances in Cryptology - EUROCRYPT '97," Lecture Notes in Computer Science, pp. 256-266, Springer-Verlag, 1997.
63. Z. Shmuely, "Composite Diffie-Hellman Public-Key Generating Systems Are Hard to Break," Technical Report No. 356, Computer Science Department, Technion, Israel, 1985.
64. L. G. Valiant, A theory of the learnable, Comm. ACM 21 (1984), 1134-1142.
65. U. V. Vazirani, Strong communication complexity or generating quasi-random sequences from two communicating semi-random sources, Combinatorica 7 (1987).
66. [Preliminary version in "Proc. 17th. Ann. ACM Symp. on Theory of Computing, 1985," pp. 366-378]
67. A. C. Yao, Theory and applications of trapdoor functions, in "Proc. 23rd IEEE Symp. on Foundations of Computer Science, 1982," pp. 80-91.