Dynamical systems generated by rational functions

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 2643, Issue , 2003, Pages 6-17

  Niederreiter, Harald ^a   Shparlinski, Igor E ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b MACQUARIE UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; DYNAMICAL SYSTEMS; FINITE ELEMENT METHOD;

FUNCTIONS OVER FINITE FIELDS; RESIDUE CLASS RING;

RATIONAL FUNCTIONS;

EID: 33947419513     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-44828-4_2     Document Type: Article

References (71)

1. L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
2. G. Barat, T. Downarowicz and P. Liardet, Dynamiques associées à une échelle de numération, Acta Arith. 103, 41-78 (2002).
3. V. Berthé and H. Nakada, On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Expositiones Math. 18, 257-284 (2000).
4. L. Blum, M. Blum and M. Shub, A simple unpredictable pseudo-random number generator, SIAM J. Computing 15, 364-383 (1986).
5. T. Cochrane, Exponential sums modulo prime powers, Acta Arith. 101, 131-149 (2002).
6. T. Cochrane, C. Li and Z.Y. Zheng, Upper bounds on character sums with rational function entries, Acta Math. Sinica (English Ser.) 19, 1-11 (2003).
7. T. Cochrane and Z.Y. Zheng, Pure and mixed exponential sums, Acta Arith. 91, 249-278 (1999).
8. T. Cochrane and Z.Y. Zheng, Exponential sums with rational function entries, Acta Arith. 95, 67-95 (2000).
9. T. Cochrane and Z.Y. Zheng, On upper bounds of Chalk and Hua for exponential sums, Proc. Amer. Math. Soc. 129, 2505-2516 (2001).
10. T. Cochrane and Z.Y. Zheng, A survey on pure and mixed exponential sums modulo prime powers, Number Theory for the Millennium (M.A. Bennett et al., eds.), vol. 1, pp. 271-300, A.K. Peters, Natick, MA, 2002.
11. S.D. Cohen, H. Niederreiter, I.E. Shparlinski and M. Zieve, Incomplete character sums and a special class of permutations, J. Théorie des Nombres Bordeaux 13, 53-63 (2001).
12. R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag, New York, 2001.
13. J. Eichenauer-Herrmann, E. Herrmann and S. Wegenkittl, A survey of quadratic and inversive congruential pseudorandom numbers, Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter et al., eds.), Lect. Notes in Stat., vol. 127, pp. 66-97, Springer-Verlag, New York, 1998.
14. G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer-Verlag, London, 1999.
15. M. Flahive and H. Niederreiter, On inversive congruential generators for pseudorandom numbers, Finite Fields, Coding Theory, and Advances in Communications and Computing (G.L. Mullen and P.J.-S. Shiue, eds.), pp. 75-80, Marcel Dekker, New York, 1993.
16. P. Flajolet and A.M. Odlyzko, Random mapping statistics, Advances in Cryptology - EUROCRYPT '89 (J.-J. Quisquater and J. Vandewalle, eds.), Lect. Notes in Comp. Sci., vol. 434, pp. 329-354, Springer-Verlag, Berlin, 1990.
17. L. Flatto, Z-numbers and β-transformations, Symbolic Dynamics and Its Applications, Contemporary Math., vol. 135, pp. 181-201, Amer. Math. Soc., Providence, RI, 1992.
18. L. Flatto, J.C. Lagarias and B. Poonen, The zeta function of the beta transformation, Ergodic Theory Dynam. Systems 14, 237-266 (1994).
19. J.B. Friedlander, J. Hansen and I.E. Shparlinski, Character sums with exponential functions, Mathematika 47, 75-85 (2000).
20. J.B. Friedlander, J. Hansen, and I.E. Shparlinski, On the distribution of the power generator modulo a prime power, Proc. DIMACS Workshop on Unusual Applications of Number Theory 2000, Amer. Math. Soc., Providence, RI (to appear).
21. m, Acta Arith. 105, 349-370 (2002).
22. J.B. Friedlander, D. Lieman and I.E. Shparlinski, On the distribution of the RSA generator, Sequences and Their Applications (C. Ding, T. Helleseth and H. Niederreiter, eds.), pp. 205-212, Springer-Verlag, London, 1999.
23. J.B. Friedlander, C. Pomerance and I.E. Shparlinski, Period of the power generator and small values of Carmichael's function, Math. Comp. 70, 1591-1605 (2001).
24. J.B. Friedlander and I.E. Shparlinski, On the distribution of the power generator, Math. Comp. 70, 1575-1589 (2001).
25. S. Gao, Elements of provable high orders in finite fields, Proc. Amer. Math. Soc. 127, 1615-1623 (1999).
26. F. Griffin, H. Niederreiter and I.E. Shparlinski, On the distribution of nonlinear recursive congruential pseudorandom numbers of higher orders, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (M. Fossorier et al., eds.), Lect. Notes in Comp. Sci., vol. 1719, pp. 87-93, Springer-Verlag, Berlin, 1999.
27. J. Gutierrez and D. Gomez-Perez, Iterations of multivariate polynomials and discrepancy of pseudorandom numbers, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (S. Boztaş and I.E. Shparlinski, eds.), Lect. Notes in Comp. Sci., vol. 2227, pp. 192-199, Springer-Verlag, Berlin, 2001.
28. J. Gutierrez, H. Niederreiter and I.E. Shparlinski, On the multidimensional distribution of inversive congruential pseudorandom numbers in parts of the period, Monatsh. Math. 129, 31-36 (2000).
29. K. Huber, On the period length of generalized inversive pseudorandom number generators, Applicable Algebra Engrg. Comm. Comput. 5, 255-260 (1994).
30. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
31. n), Finite Fields, Coding Theory, and Advances in Communications and Computing (G.L. Mullen and P.J.-S. Shiue, eds.), pp. 430-431, Marcel Dekker, New York, 1993.
32. D.E. Knuth, The Art of Computer Programming, vol. 2, 3rd ed., Addison-Wesley, Reading, MA, 1998.
33. J.C. Lagarias, Pseudorandom number generators in cryptography and number theory, Cryptology and Computational Number Theory (C. Pomerance, ed.), Proc. Symp. in Appl. Math., vol. 42, pp. 115-143, Amer. Math. Soc., Providence, RI, 1990.
34. J.C. Lagarias, Number theory and dynamical systems, The Unreasonable Effectiveness of Number Theory, Proc. Symp. in Appl. Math., vol. 46, pp. 35-72, Amer. Math. Soc., Providence, RI, 1992.
35. J.C. Lagarias, H.A. Porta and K.B. Stolarsky, Asymmetric tent map expansions. I. Eventually periodic points, J. London Math. Soc. 47, 542-556 (1993).
36. J.C. Lagarias, H.A. Porta and K.B. Stolarsky, Asymmetric tent map expansions. II. Purely periodic points, Illinois J. Math. 38, 574-588 (1994).
37. R. Lidl, G.L. Mullen and G. Turnwald, Dickson Polynomials, Longman, Harlow, 1993.
38. R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.
39. C. J. Moreno and O. Moreno, Exponential sums and Goppa codes: I, Proc. Amer. Math. Soc. 111, 523-531 (1991).
40. P. Morton, Periods of maps on irreducible polynomials over finite fields, Finite Fields Appl. 3, 11-24 (1997).
41. P. Morton, Arithmetic properties of periodic points of quadratic maps. II, Acta Arith. 87, 89-102 (1998).
42. P. Morton and J.H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 2, 97-110 (1994).
43. P. Morton and J.H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461, 81-122 (1995).
44. P. Morton and F. Vivaldi, Bifurcations and discriminants for polynomial maps, Nonlinearity 8, 571-584 (1995).
45. W. Narkiewicz, Polynomial Mappings, Lect. Notes in Math., vol. 1600, Springer-Verlag, Berlin, 1995.
46. H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84, 957-1041 (1978).
47. H. Niederreiter, The probabilistic theory of linear complexity, Advances in Cryptology - EUROCRYPT '88 (C.G. Günther, ed.), Lect. Notes in Comp. Sci., vol. 330, pp. 191-209, Springer-Verlag, Berlin, 1988.
48. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.
49. H. Niederreiter, New developments in uniform pseudorandom number and vector generation, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P.J.-S. Shiue, eds.), Lect. Notes in Stat., vol. 106, pp. 87-120, Springer-Verlag, New York, 1995.
50. H. Niederreiter, Design and analysis of nonlinear pseudorandom number generators, Monte Carlo Simulation (G.I. Schuëller and P.D. Spanos, eds.), pp. 3-9, A.A. Balkema Publishers, Rotterdam, 2001.
51. H. Niederreiter and I.E. Shparlinski, On the distribution and lattice structure of nonlinear congruential pseudorandom numbers, Finite Fields Appl. 5, 246-253 (1999).
52. H. Niederreiter and I.E. Shparlinski, On the distribution of pseudorandom numbers and vectors generated by inversive methods, Applicable Algebra Engrg. Comm. Comput. 10, 189-202 (2000).
53. H. Niederreiter and I.E. Shparlinski, Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus, Acta. Arith. 92, 89-98 (2000).
54. H. Niederreiter and I.E. Shparlinski, On the distribution of inversive congruential pseudorandom numbers in parts of the period, Math. Comp. 70, 1569-1574 (2001).
55. H. Niederreiter and I.E. Shparlinski, Recent advances in the theory of nonlinear pseudorandom number generators, Monte Carlo and Quasi-Monte Carlo Methods 2000 (K.-T. Fang, F.J. Hickernell and H. Niederreiter, eds.), pp. 86-102, Springer-Verlag, Berlin, 2002.
56. H. Niederreiter and I.E. Shparlinski, On the average distribution of inversive pseudorandom numbers, Finite Fields Appl. 8, 491-503 (2002).
57. H. Niederreiter and I.E. Shparlinski, On the distribution of power residues and primitive elements in some nonlinear recurring sequences, Bull. Lond. Math. Soc. (to appear).
58. H. Niederreiter and A. Winterhof, Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number generators, Acta Arith. 93, 387-399 (2000).
59. H. Niederreiter and A. Winterhof, On the distribution of compound inversive congruential pseudorandom numbers, Monatsh. Math. 132, 35-48 (2001).
60. H. Niederreiter and A. Winterhof, On the lattice structure of pseudorandom numbers generated over arbitrary finite fields, Applicable Algebra Engrg. Comm. Comput. 12, 265-272 (2001).
61. H. Niederreiter and A. Winterhof, On a new class of inversive pseudorandom numbers for parallelized simulation methods, Periodica Math. Hungarica 42, 77-87 (2001).
62. n), preprint, 2002.
63. A.G. Postnikov, Ergodic Problems in the Theory of Congruences and of Diophantine Approximations, Proc. Steklov Inst. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1967.
64. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1999.
65. J. Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory Dynam. Systems 17, 675-694 (1997).
66. K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Math., vol. 128, Birkhäuser Verlag, Basel, 1995.
67. I.E. Shparlinski, Exponential sums in coding theory, cryptology and algorithms, Coding Theory and Cryptology (H. Niederreiter, ed.), pp. 323-383, World Scientific Publ., Singapore, 2002.
68. 1, Compositio Math. 98, 269-304 (1995).
69. 1, Duke Math. J. 94, 41-77 (1998).
70. S.B. Stečkin, An estimate of a complete rational trigonometric sum (Russian), Trudy Mat. Inst. Steklov. 143, 188-207 (1977).
71. G.J. Wirsching, The Dynamical System Generated by the 3n + 1 Function, Lect. Notes in Math., vol. 1681, Springer-Verlag, Berlin, 1998.