On the statistical properties of Diffie-Hellman distributions

Israel Journal of Mathematics

Volumn 120, Issue , 2000, Pages 23-46

  Canetti, Ran ^a   Friedlander, John ^b   Konyagin, Sergei ^c   Larsen, Michael ^d   Lieman, Daniel ^d   Shparlinski, Igor ^e  

^a IBM T J WATSON RESEARCH CENTER   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

^c MOSCOW STATE UNIVERSITY   (Russian Federation)

^d UNIVERSITY OF MISSOURI   (United States)

^e MACQUARIE UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002107440     PISSN: 00212172     EISSN: 15658511     Source Type: Journal    
DOI: 10.1007/s11856-000-1270-1     Document Type: Article

References (41)

1. R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), 331-361.
2. D. Boneh and R. Lipton, Algorithms for Black-Box fields and their application to cryptography, in CRYPTO '96, Lecture Notes in Computer Science 1109, Springer-Verlag, Berlin, 1996, pp. 283-297.
3. D. Boneh and R. Venkatessan, Hardness of computing the most significant bits of secret keys in Diffie-Hellman and related schemes, in CRYPTO '96, Lecture Notes in Computer Science 1109, Springer-Verlag, Berlin, 1996, pp. 129-142.
4. S. Brands, An efficient off-line electronic cash system based on the representation problem, CWI TR CS-R9323, 1993.
5. R. Canetti, Towards realizing random oracles: Hash functions that hide all partial information, in CRYPTO '97, Lecture Notes in Computer Science 1294, Springer-Verlag, Berlin, 1997, pp. 455-469.
6. R. Canetti, J. B. Friedlander and I. E. Shparlinski, On certain exponential sums and the distribution of Diffie-Hellman triples , Journal of the London Mathematical Society (to appear).
7. J. H. H. Chalk, The Vinogradov-Mordell-Tietäväinen inequalities, Indagationes Mathematicae 42 (1980), 367-374.
8. D. Chaum and H. van Antwerpen, Undeniable signatures, in CRYPTO '89, Lecture Notes in Computer Science 435, Springer-Verlag, Berlin, 1990, pp. 212-216.
9. D. Coppersmith and I. E. Shparlinski, On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping, Journal of Cryptology (to appear).
10. W. Diffie and M. Hellman, New directions in cryptography, IEEE Transactions on Information Theory 22 (1976), 644-654.
11. W. Diffie, P. van Oorscot and M. Wiener, Authentication and authenticated key exchanges, Designs, Codes and Cryptography 2 (1992), 107-125.
12. T. El-Gamal, Cryptography and logarithms over finite fields, Ph.D. Thesis, Stanford University, 1984.
13. P. Feldman, A practical scheme for non-interactive verifiable secret sharing, Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, 1987, pp. 427-437.
14. J. B. Friedlander, M. Larsen, D. Lieman and I. E. Shparlinski, On the correlation of binary M-sequences, Designs, Codes and Cryptography (to appear).
15. J. B. Friedlander, D. Lieman and I. E. Shparlinski, On the distribution of the RSA generator, preprint, 1998, pp. 1-10.
16. O. Goldreich, Foundations of Cryptography (fragments of a book), Weizmann Institute of Science, 1995. (Available at http://theory.lcs.mit.edu/∼tcryptol/).
17. S. Goldwasser and S. Micali, Probabilistic encryption, Journal of Computer and System Sciences 28 (1984), 270-299.
18. H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.
19. D. R. Heath-Brown, Artin's conjecture for primitive roots, Quarterly Journal of Mathematics 37 (1986), 27-38.
20. S. Konyagin and I. E. Shparlinski, Character Sums with Exponential Functions and their Applications, Cambridge University Press, Cambridge (to appear).
21. N. M. Korobov, On the distribution of digits in periodic fractions, Mathematics of the USSR - Sbornik 18 (1972), 659-676.
22. N. M. Korobov, Exponential sums and their applications, Kluwer Academic Publ., Dordrecht, 1992.
23. R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.
24. U. Maurer, Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms, in CRYPTO '94, Lecture Notes in Computer Science 740, Springer-Verlag, Berlin, 1994, pp. 271-281.
25. U. M. Maurer and S. Wolf, On the complexity of breaking the Diffie-Hellman protocol, preprint, 1996, pp. 1-30.
26. H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1994.
27. M. Naor and O. Reingold, Number-theoretic constructions of efficient pseudo-random functions and other cryptographic primitives, Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, 1997.
28. H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bulletin of the American Mathematical Society 84 (1978), 957-1041.
29. T. P. Pedersen, Distributed provers with applications to undeniable signatures, in EUROCRYPT '91, Lecture Notes in Computer Science 547, Springer-Verlag, Berlin, 1991, pp. 221-242.
30. A. J. van der Poorten and I. E. Shparlinski, On zeros of exponential polynomials and related questions, Bulletin of the Australian Mathematical Society 46 (1992), 399-410.
31. A. W. Schrift and A. Shamir, The discrete log is very discreet, in Proceedings of the Twenty Second Annual ACM Symposium on Theory of Computing (STOC), 1990, pp. 405-415.
32. V. Shoup, Lower bounds for discrete logarithms and related problems, in EUROCRYPT '97, Lecture Notes in Computer Science 1233, Springer-Verlag, Berlin, 1997, pp. 256-266.
33. I. E. Shparlinski, On prime divisors of recurrence sequences, Izvestija Vysshih Uchebnyh Zavedenii, Seriya Matematika (1980), no. 1, 100-103 (in Russian).
34. I. E. Shparlinski, Finite Fields: Theory and Computation, Kluwer Academic Publ., Dordrecht, 1999 (to appear).
35. I. E. Shparlinski, Number Theoretic Methods in Cryptography: Complexity Lower Bounds, Birkhäuser, 1999 (to appear).
36. I. E. Shparlinski and S. A. Stepanov, On construction of a primitive normal basis of a finite field, Mathematics of the USSR - Sbornik 67 (1990), 527-533.
37. I. E. Shparlinski and S. A. Stepanov, On construction of primitive elements and primitive normal bases in a finite field, in Proceedings of a Colloquium on Computational Number Theory (Hungary, 1989), De Gruyter, Berlin, 1991, pp. 1-24.
38. Y. S. Tsiounis and M. Yung, The Semantic Security of El Gamal Encryption is Equivatent to the Decision Diffie-Herman, TM-0992-05-97-582 GTE Laboratories, 1997.
39. I. M. Vinogradov, Elements of Number Theory, Dover Publ., New York, 1954.
40. H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Mathematische Annalen 77 (1916), 313-352.
41. A. Yao, Theory and applications of trapdoor functions, Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982, pp. 80-91.