Duke Mathematical Journal

Zeta functions, one-way functions, and pseudorandom number generators

Michael Anshel and Dorian Goldfeld

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J., Volume 88, Number 2 (1997), 371-390.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11K45: Pseudo-random numbers; Monte Carlo methods 94A60: Cryptography [See also 11T71, 14G50, 68P25, 81P94]


Anshel, Michael; Goldfeld, Dorian. Zeta functions, one-way functions, and pseudorandom number generators. Duke Math. J. 88 (1997), no. 2, 371--390. doi:10.1215/S0012-7094-97-08815-3. https://projecteuclid.org/euclid.dmj/1077241583

Export citation


  • [1] E. Artin, Zur Theorie der $L$-Reihen mit allgemeinen Grupencharakteren, Hamb. Abh. 8 (1930), 292–306, Collected Papers, Addison-Wesley, Reading, Mass., 1965, no. 8.
  • [2] E. Artin, Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahl-körper, J. Reine Angew. Math. 164 (1931), 1–11, Collected Papers, Addison-Wesley, Reading, Mass., 1965, no. 9.
  • [3] A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61.
  • [4] M. Blum and S. Micali, How to generate cryptographically strong sequences of pseudorandom bits, SIAM J. Comput. 13 (1984), no. 4, 850–864.
  • [5] H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993.
  • [6] J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992.
  • [7] I. B. Damgȧrd, On the randomness of Legendre and Jacobi sequences, Advances in cryptology—CRYPTO '88 (Santa Barbara, CA, 1988), Lecture Notes in Comput. Sci., vol. 403, Springer, Berlin, 1990, pp. 163–172.
  • [8] H. Davenport, Multiplicative Number Theory, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 1980.
  • [9] L. E. Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. , Chelsea Publishing Co., New York, 1966.
  • [10] J. von zur Gathen, M. Karpinski, and I. Shparlinski, Counting curves and their projections, Proceedings of the 25th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1993, pp. 805–812.
  • [11] D. Goldfeld and J. Hoffstein, On the number of Fourier coefficients that determine a modular form, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 385–393.
  • [12] O. Goldreich, H. Krawczyk, and M. Luby, On the existence of pseudorandom generators, SIAM J. Comput. 22 (1993), no. 6, 1163–1175.
  • [13] D. Husemoller, Elliptic Curves, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987.
  • [14] H. Jacquet and R. Langlands, Automorphic Forms on $\rm GL(2)$, Lecture Notes in Math., vol. 278, Springer-Verlag, Berlin, 1970.
  • [15] S. Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970.
  • [16] S. Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 37–75.
  • [17] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 409–464.
  • [18] H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673.
  • [19] J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 1–87.
  • [20] A. Ogg, Modular Forms and Dirichlet Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
  • [21] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod $p$, Math. Comp. 44 (1985), no. 170, 483–494.
  • [22] B. Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in $C$, 2d ed., John Wiley & Sons, New York, 1996.
  • [23] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Collected Papers, Vol. 2, Springer-Verlag, Berlin, 1991, pp. 47–63.
  • [24] J. P. Serre, Corps locaux, Publications de l'Institut de Mathématique de l'Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962.
  • [25] J. P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.
  • [26] P. Shor, Algorithms for quantum computation: discrete logarithms and factoring, 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994), IEEE Comput. Soc. Press, Los Alamitos, CA, 1994, pp. 124–134.
  • [27] J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
  • [28] J. Silverman and J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
  • [29] H. J. S. Smith, Report on the Theory of Numbers, Chelsea, New York, 1965, 88–92.
  • [30] J. Tate, Global class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162–203.
  • [31] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572.
  • [32] A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.
  • [33] A. C. Yao, Theory and applications of trapdoor functions, 23rd annual symposium on foundations of computer science (Chicago, Ill., 1982), IEEE, New York, 1982, pp. 80–91.