Zeta functions, one-way functions, and pseudorandom number generators
Michael Anshel and Dorian Goldfeld
Source: Duke Math. J. Volume 88, Number 2
(1997), 371-390.
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References
[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.
Zentralblatt MATH: 56.0173.02
[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.
Zentralblatt MATH: 0001.00801
[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.
Mathematical Reviews (MathSciNet): MR81m:10056
Zentralblatt MATH: 0412.10017
[4] M. Blum and S. Micali, How to generate cryptographically strong sequences of pseudorandom bits, SIAM J. Comput. 13 (1984), no. 4, 850–864.
Mathematical Reviews (MathSciNet): MR86a:68021
Zentralblatt MATH: 0547.68046
Digital Object Identifier: doi:10.1137/0213053
[5] H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993.
Mathematical Reviews (MathSciNet): MR94i:11105
Zentralblatt MATH: 0786.11071
[6] J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992.
Mathematical Reviews (MathSciNet): MR93m:11053
Zentralblatt MATH: 0758.14042
[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.
Mathematical Reviews (MathSciNet): MR91d:11155
Zentralblatt MATH: 0719.94013
[8] H. Davenport, Multiplicative Number Theory, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 1980.
Mathematical Reviews (MathSciNet): MR82m:10001
Zentralblatt MATH: 0453.10002
[9] L. E. Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. , Chelsea Publishing Co., New York, 1966.
Mathematical Reviews (MathSciNet): MR39:6807a
Zentralblatt MATH: 0958.11500
[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.
Mathematical Reviews (MathSciNet): MR94b:11037
Zentralblatt MATH: 0805.11040
[12] O. Goldreich, H. Krawczyk, and M. Luby, On the existence of pseudorandom generators, SIAM J. Comput. 22 (1993), no. 6, 1163–1175.
Mathematical Reviews (MathSciNet): MR95f:11054
Zentralblatt MATH: 0795.94011
Digital Object Identifier: doi:10.1137/0222069
[13] D. Husemoller, Elliptic Curves, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987.
Mathematical Reviews (MathSciNet): MR88h:11039
Zentralblatt MATH: 0605.14032
[14] H. Jacquet and R. Langlands, Automorphic Forms on $\rm GL(2)$, Lecture Notes in Math., vol. 278, Springer-Verlag, Berlin, 1970.
Mathematical Reviews (MathSciNet): MR53:5481
Zentralblatt MATH: 0236.12010
[15] S. Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970.
Mathematical Reviews (MathSciNet): MR44:181
Zentralblatt MATH: 0211.38404
[16] S. Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 37–75.
Mathematical Reviews (MathSciNet): MR90k:11032
Zentralblatt MATH: 0714.11034
Digital Object Identifier: doi:10.1090/S0273-0979-1990-15899-9
Project Euclid: euclid.bams/1183555717
[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.
Mathematical Reviews (MathSciNet): MR56:5506
Zentralblatt MATH: 0362.12011
[18] H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673.
Mathematical Reviews (MathSciNet): MR89g:11125
Zentralblatt MATH: 0629.10006
Digital Object Identifier: doi:10.2307/1971363
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet): MR56:5502
Zentralblatt MATH: 0359.12015
[20] A. Ogg, Modular Forms and Dirichlet Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
Mathematical Reviews (MathSciNet): MR41:1648
Zentralblatt MATH: 0191.38101
[21] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod $p$, Math. Comp. 44 (1985), no. 170, 483–494.
Mathematical Reviews (MathSciNet): MR86e:11122
Zentralblatt MATH: 0579.14025
Digital Object Identifier: doi:10.2307/2007968
JSTOR: links.jstor.org
[22] B. Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in $C$, 2d ed., John Wiley & Sons, New York, 1996.
Zentralblatt MATH: 0853.94001
[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.
Mathematical Reviews (MathSciNet): MR1220477
Zentralblatt MATH: 0787.11037
[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.
Mathematical Reviews (MathSciNet): MR27:133
Zentralblatt MATH: 0137.02601
[25] J. P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.
Mathematical Reviews (MathSciNet): MR52:8126
Zentralblatt MATH: 0235.14012
Digital Object Identifier: doi:10.1007/BF01405086
[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.
Mathematical Reviews (MathSciNet): MR1489242
[27] J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
Mathematical Reviews (MathSciNet): MR87g:11070
Zentralblatt MATH: 0585.14026
[28] J. Silverman and J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
Mathematical Reviews (MathSciNet): MR93g:11003
Zentralblatt MATH: 0752.14034
[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.
Mathematical Reviews (MathSciNet): MR36:3749
Zentralblatt MATH: 1179.11041
[31] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572.
Mathematical Reviews (MathSciNet): MR96d:11072
Zentralblatt MATH: 0823.11030
Digital Object Identifier: doi:10.2307/2118560
JSTOR: links.jstor.org
[32] A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.
Mathematical Reviews (MathSciNet): MR96d:11071
Zentralblatt MATH: 0823.11029
Digital Object Identifier: doi:10.2307/2118559
JSTOR: links.jstor.org
[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.
Mathematical Reviews (MathSciNet): MR780384
Digital Object Identifier: doi:10.1109/SFCS.1982.45
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