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

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- by R. Kannan, A. K. Lenstra and L. Lovász PDF
- Math. Comp.
**50**(1988), 235-250 Request permission

## 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.## References

- 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**220680**, DOI 10.1112/s0025579300003843
L. Blum, M. Blum & M. Shub, - 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**
É. Borel, - 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** - Arthur H. Copeland and Paul Erdös,
*Note on normal numbers*, Bull. Amer. Math. Soc.**52**(1946), 857–860. MR**17743**, DOI 10.1090/S0002-9904-1946-08657-7
D. G. Champernowne, "The construction of decimals normal in the scale of ten," - 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** - 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** - Peter Henrici,
*Applied and computational complex analysis*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR**0372162** - I. N. Herstein,
*Topics in algebra*, 2nd ed., Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. MR**0356988**
M.-P. Van der Hulst & A. K. Lenstra, - Donald E. Knuth,
*The art of computer programming. Vol. 2*, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR**633878**
S. Landau & G. Miller, - 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**, DOI 10.1007/BF01457454 - R. Loos,
*Computing in algebraic extensions*, Computer algebra, Springer, Vienna, 1983, pp. 173–187. MR**728972**, DOI 10.1007/978-3-7091-7551-4_{1}2 - M. Mignotte,
*An inequality about factors of polynomials*, Math. Comp.**28**(1974), 1153–1157. MR**354624**, DOI 10.1090/S0025-5718-1974-0354624-3 - Michael O. Rabin,
*Probabilistic algorithms in finite fields*, SIAM J. Comput.**9**(1980), no. 2, 273–280. MR**568814**, DOI 10.1137/0209024
C. P. Schnorr, "A more efficient algorithm for lattice basis reduction," manuscript, 1985.
A. Schönhage, - 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**, DOI 10.1007/3-540-13345-3_{4}0
A. Shamir, - 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**

*A Simple Secure Pseudo Random Number Generator*, Proceedings of Crypto 82.

*Leçons sur la Théorie des Fonctions*, 2nd ed., 1914, pp. 182-216.

*J. London Math. Soc.*, v. 8, 1933, pp. 254-260.

*Polynomial Factorization by Transcendental Evaluation*, Proceedings Eurocal 85. 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.

*Solvability by Radicals is in Polynomial Time*, Proc. 15th Annual ACM Symposium on Theory of Computing, 1983, pp. 140-151.

*The Fundamental Theorem of Algebra in Terms of Computational Complexity*, Preliminary report, Math. Inst. Univ. Tübingen, 1982.

*On the Generation of Cryptographically Strong Pseudo-Random Sequences*, Proc. 8th International Colloquium on Automata, Languages, and Programming, 1981. B. Trager,

*Algebraic Factoring and Rational Function Integration*, Proc. SYMSAC 76, pp. 219-226.

## Additional Information

- © Copyright 1988 American Mathematical Society
- 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