Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers

Kannan, R.; Lenstra, A. K.; Lovász, L.

doi:10.1090/S0025-5718-1988-0917831-4

Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers

Authors: R. Kannan, A. K. Lenstra and L. Lovász
Journal: Math. Comp. 50 (1988), 235-250
MSC: Primary 68Q20; Secondary 11A51, 11A63, 11J99, 11Y16, 68Q25
DOI: https://doi.org/10.1090/S0025-5718-1988-0917831-4
MathSciNet review: 917831
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the binary expansions of algebraic numbers do not form secure pseudorandom sequences; given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables us to devise a simple algorithm to factor polynomials with rational coefficients. All algorithms work in polynomial time.

[1] A. Baker, Linear forms in the logarithms of algebraic numbers. I, II, III, Mathematika 13 (1966), 204-216; ibid. 14 (1967), 102-107; ibid. 14 (1967), 220–228. MR 0220680
[2] L. Blum, M. Blum & M. Shub, A Simple Secure Pseudo Random Number Generator, Proceedings of Crypto 82.
[3] Manuel Blum and Silvio Micali, How to generate cryptographically strong sequences of pseudorandom bits, 23rd annual symposium on foundations of computer science (Chicago, Ill., 1982) IEEE, New York, 1982, pp. 112–117. MR 780388
[4] É. Borel, Leçons sur la Théorie des Fonctions, 2nd ed., 1914, pp. 182-216.
[5] A. J. Brentjes, Multidimensional continued fraction algorithms, Computational methods in number theory, Part II, Math. Centre Tracts, vol. 155, Math. Centrum, Amsterdam, 1982, pp. 287–319. MR 702520
[6] Arthur H. Copeland and Paul Erdös, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857–860. MR 0017743, https://doi.org/10.1090/S0002-9904-1946-08657-7
[7] D. G. Champernowne, "The construction of decimals normal in the scale of ten," J. London Math. Soc., v. 8, 1933, pp. 254-260.
[8] O. Goldreich, S. Goldwasser, and S. Micali, How to construct random functions, Theory of algorithms (Pécs, 1984) Colloq. Math. Soc. János Bolyai, vol. 44, North-Holland, Amsterdam, 1985, pp. 161–189. MR 872307
[9] Shafi Goldwasser, Silvio Micali, and Po Tong, Why and how to establish a private code on a public network, 23rd annual symposium on foundations of computer science (Chicago, Ill., 1982) IEEE, New York, 1982, pp. 134–144. MR 780391
[10] Peter Henrici, Applied and computational complex analysis, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros; Pure and Applied Mathematics. MR 0372162
[11] I. N. Herstein, Topics in algebra, 2nd ed., Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. MR 0356988
[12] M.-P. Van der Hulst & A. K. Lenstra, Polynomial Factorization by Transcendental Evaluation, Proceedings Eurocal 85.
[13] R. Kannan, A. K. Lenstra & L. Lovász, Polynomial Factorization and Nonrandomness of Bits of Algebraic and Some Transcendental Numbers, Proc. 16th Annual ACM Symposium on Theory of Computing, 1984, pp. 191-200.
[14] Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR 633878
[15] S. Landau & G. Miller, Solvability by Radicals is in Polynomial Time, Proc. 15th Annual ACM Symposium on Theory of Computing, 1983, pp. 140-151.
[16] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, https://doi.org/10.1007/BF01457454
[17] R. Loos, Computing in algebraic extensions, Computer algebra, Springer, Vienna, 1983, pp. 173–187. MR 728972, https://doi.org/10.1007/978-3-7091-7551-4_12
[18] M. Mignotte, An inequality about factors of polynomials, Math. Comp. 28 (1974), 1153–1157. MR 0354624, https://doi.org/10.1090/S0025-5718-1974-0354624-3
[19] Michael O. Rabin, Probabilistic algorithms in finite fields, SIAM J. Comput. 9 (1980), no. 2, 273–280. MR 568814, https://doi.org/10.1137/0209024
[20] C. P. Schnorr, "A more efficient algorithm for lattice basis reduction," manuscript, 1985.
[21] A. Schönhage, The Fundamental Theorem of Algebra in Terms of Computational Complexity, Preliminary report, Math. Inst. Univ. Tübingen, 1982.
[22] Arnold Schönhage, Factorization of univariate integer polynomials by Diophantine approximation and an improved basis reduction algorithm, Automata, languages and programming (Antwerp, 1984) Lecture Notes in Comput. Sci., vol. 172, Springer, Berlin, 1984, pp. 436–447. MR 784270, https://doi.org/10.1007/3-540-13345-3_40
[23] A. Shamir, On the Generation of Cryptographically Strong Pseudo-Random Sequences, Proc. 8th International Colloquium on Automata, Languages, and Programming, 1981.
[24] B. Trager, Algebraic Factoring and Rational Function Integration, Proc. SYMSAC 76, pp. 219-226.
[25] Andrew 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. MR 780384