Advertisement

On the Inversive Pseudorandom Number Generator

  • Wilfried Meidl
  • Alev Topuzoğlu

Abstract

The inversive generator was introduced by J. Eichenauer and J. Lehn in 1986. A large number of papers on this generator have appeared in the last three decades, some investigating its properties, some generalizing it. It has been shown that the generated sequence and its variants behave very favorably with respect to most measures of randomness.

In this survey article we present a comprehensive overview of results on the inversive generator, its generalizations and variants. As regards to recent work, our emphasis is on a particular generalization, focusing on the underlying permutation P(x)=ax p−2+b of \(\mathbb{F}_{p}\).

Keywords

Linear Complexity Pseudorandom Number Turbo Code Pseudorandom Number Generator Cycle Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aksoy, E., Çeşmelioğlu, A., Meidl, W., Topuzoğlu, A.: On the Carlitz rank of permutation polynomials. Finite Fields Appl. 15, 418–440 (2009) CrossRefGoogle Scholar
  2. Avenancio-Leon, C.: Analysis of some properties of interleavers for Turbo codes. In: Proc. of NCUR, Lexington, USA (2005) Google Scholar
  3. Beck, I.: Cycle decomposition of transpositions. J. Comb. Theory, Ser. A 23, 198–207 (1977) MATHCrossRefGoogle Scholar
  4. Blackburn, S., Gomez-Perez, D., Gutierrez, J., Shparlinski, I.: In: Predicting the inversive generator. Lecture Notes in Computer Science, vol. 2898, pp. 264–275. Springer, Berlin (2003) Google Scholar
  5. Carlitz, L.: Permutations in a finite field. Proc. Am. Math. Soc. 4, 538 (1953) CrossRefGoogle Scholar
  6. Çeşmelioğlu, A.: Personal communication (2009) Google Scholar
  7. Çeşmelioğlu, A., Meidl, W., Topuzoğlu, A.: On the cycle structure of permutation polynomials. Finite Fields Appl. 14, 593–614 (2008a) MATHCrossRefMathSciNetGoogle Scholar
  8. Çeşmelioğlu, A., Meidl, W., Topuzoğlu, A.: Enumeration of a class of sequences generated by inversions. In: Li, Y.Q., et al. (eds.) Proceedings of the Int. Workshop on Coding and Cryptology, Fujian, China, June 2007, pp. 44–57 (2008b) Google Scholar
  9. Çeşmelioğlu, A., Meidl, W., Topuzoğlu, A.: On a class of APN permutation polynomials. Preprint (2009) Google Scholar
  10. Chou, W.-S.: On inversive maximal period polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 6, 245–250 (1995a) MATHCrossRefGoogle Scholar
  11. Chou, W.-S.: The period lengths of inversive pseudorandom vector generations. Finite Fields Appl. 1, 126–132 (1995b) MATHCrossRefMathSciNetGoogle Scholar
  12. Comtet, L.: Advanced Combinatorics, the Art of Finite and Infinite Expansions. Reidel, Dordrecht (1974) MATHGoogle Scholar
  13. Corrada-Bravo, C.J., Rubio, I.M.: Deterministic interleavers for Turbo codes with random-like performance and simple implementation. In: Proc. of the 3rd International Symposium on Turbo Codes and Related Topics, Brest, France, pp. 555–558 (2003) Google Scholar
  14. Dorfer, G., Winterhof, A.: Lattice structure and linear complexity profile of nonlinear pseudorandom number generators. Appl. Algebra Eng. Commun. Comput. 13, 499–508 (2003) MATHCrossRefMathSciNetGoogle Scholar
  15. Drmota, M., Tichy, R.F.: Sequences, Descrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997) Google Scholar
  16. Eichenauer, J., Lehn, J.: A non-linear congruential pseudorandom number generator. Stat. Hefte 27, 315–326 (1986) MATHCrossRefMathSciNetGoogle Scholar
  17. Eichenauer, J., Niederreiter, H.: On Marsaglia’s lattice test for pseudorandom numbers. Manuscr. Math. 62, 245–248 (1988) MATHCrossRefMathSciNetGoogle Scholar
  18. Eichenauer, J., Grothe, H., Lehn, J., Topuzoğlu, A.: A multiple recursive nonlinear congruential pseudo random number generator. Manuscr. Math. 59, 331–346 (1987) MATHCrossRefGoogle Scholar
  19. Eichenauer, J., Grothe, H., Lehn, J.: Marsaglia’s lattice test and non-linear congruential pseudo random number generators. Metrika 35, 241–250 (1988a) MATHCrossRefGoogle Scholar
  20. Eichenauer, J., Lehn, J., Topuzoğlu, A.: A nonlinear congruential pseudorandom number generator with power of two modulus. Math. Comput. 51, 757–759 (1988b) MATHCrossRefGoogle Scholar
  21. Eichenauer, J., Grothe, H., Lehn, J.: On the period length of pseudorandom vector sequences generated by matrix generators. Math. Comput. 52, 145–148 (1989) MATHCrossRefGoogle Scholar
  22. Eichenauer-Herrmann, J.: Inversive congruential pseudorandom numbers avoid the planes. Math. Comput. 56, 297–301 (1991) MATHCrossRefMathSciNetGoogle Scholar
  23. Eichenauer-Herrmann, J.: Inversive congruential pseudorandom numbers: a tutorial. Int. Stat. Rev. 60, 167–176 (1992a) MATHCrossRefGoogle Scholar
  24. Eichenauer-Herrmann, J.: On the autocorrelation structure of inversive congruential pseudorandom number sequences. Stat. Pap. 33, 261–268 (1992b) MATHCrossRefMathSciNetGoogle Scholar
  25. Eichenauer-Herrmann, J.: Construction of inversive congruential pseudorandom number generators with maximal period length. J. Comput. Appl. Math. 40, 345–349 (1992c) MATHCrossRefMathSciNetGoogle Scholar
  26. Eichenauer-Herrmann, J.: Statistical independence of a new class of inversive congruential pseudorandom numbers. Math. Comput. 60, 375–384 (1993) MATHCrossRefMathSciNetGoogle Scholar
  27. Eichenauer-Herrmann, J.: Pseudorandom number generation by nonlinear methods. Int. Stat. Rev. 63, 245–255 (1995) Google Scholar
  28. Eichenauer-Herrmann, J., Emmerich, F.: Compound inversive congruential pseudorandom numbers: an average-case analysis. Math. Comput. 65, 215–225 (1996) MATHCrossRefMathSciNetGoogle Scholar
  29. Eichenauer-Herrmann, J., Grothe, H.: A new inversive congruential pseudorandom number generator with power of two modulus. ACM Trans. Model. Comput. Simul. 2, 1–11 (1992) MATHCrossRefGoogle Scholar
  30. Eichenauer-Herrmann, J., Ickstadt, K.: Explicit inversive congruential pseudorandom numbers with power of two modulus. Math. Comput. 62, 787–797 (1994) MATHCrossRefMathSciNetGoogle Scholar
  31. Eichenauer-Herrmann, J., Niederreiter, H.: Digital inversive pseudorandom numbers. ACM Trans. Model. Comput. Simul. 4, 339–349 (1994) MATHCrossRefGoogle Scholar
  32. Eichenauer-Herrmann, J., Topuzoğlu, A.: On the period length of congruential pseudorandom number sequences generated by inversions. J. Comput. Appl. Math. 31, 87–96 (1990) MATHCrossRefMathSciNetGoogle Scholar
  33. Eichenauer-Herrmann, J., Herrmann, E., Wegenkittl, S.: A survey of quadratic and inversive congruential pseudorandom numbers. In: Niederreiter, H., et al. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 1996. Lecture Notes in Statistics, vol. 127, pp. 66–97. Springer, New York (1998) Google Scholar
  34. Emmerich, F.: Pseudorandom number and vector generation by compound inversive methods. PhD thesis. Technische Hochschule Darmstadt (1996) Google Scholar
  35. Flahive, M., Niederreiter, H.: On inversive congruential generators for pseudorandom numbers. In: Mullen, G.L., Shiue, P.J.-S. (eds.) Finite Fields, Coding Theory, and Advances in Communications and Computing, Las Vegas, NV, 1991. Lecture Notes in Pure and Appl. Math., vol. 141, pp. 75–80. Marcel Dekker, New York (1993) Google Scholar
  36. Golomb, S.W., Gong, G.: The status of Costas arrays. IEEE Trans. Inf. Theory 53, 4260–4265 (2007) CrossRefMathSciNetGoogle Scholar
  37. Gutierrez, J., Shparlinski, I., Winterhof, A.: On the linear and nonlinear complexity profile of nonlinear pseudorandom number-generators. IEEE Trans. Inf. Theory 49, 60–64 (2003) MATHCrossRefMathSciNetGoogle Scholar
  38. Heegard, C., Wicker, S.B.: Turbo Coding. Kluwer Academic, Dordrecht (1999) MATHGoogle Scholar
  39. Hellekalek, P.: On the assessment of random and quasi-random point sets. In: Hellekalek, P., Larcher, G. (eds.) Random and Quasi-Random Point Sets. Lecture Notes in Statistics, vol. 138, pp. 49–108. Springer, Berlin (1998) Google Scholar
  40. Larcher, G., Wolf, R., Eichenauer-Herrmann, J.: On the average discrepancy of successive tuples of pseudo-random numbers over parts of the period. Monatshefte Math. 127, 141–154 (1999) MATHCrossRefMathSciNetGoogle Scholar
  41. L’Ecuyer, P., Hellekalek, P.: Random number generators: selection criteria and testing. In: Hellekalek, P., Larcher, G. (eds.) Random and Quasi-Random Point Sets. Lecture Notes in Statistics, vol. 138, pp. 223–265. Springer, Berlin (1998) Google Scholar
  42. Marsaglia, G.: Random numbers fall mainly in the planes. Proc. Natl. Acad. Sci. USA 61, 25–28 (1968) MATHCrossRefMathSciNetGoogle Scholar
  43. Massey, J.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15, 122–127 (1969) MATHCrossRefMathSciNetGoogle Scholar
  44. Mauduit, C., Sarközi, A.: On finite pseudorandom binary sequences, I: measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377 (1997) MATHMathSciNetGoogle Scholar
  45. Meidl, W., Winterhof, A.: On the linear complexity profile of explicit nonlinear pseudorandom numbers. Inf. Process. Lett. 85, 13–18 (2003) MATHCrossRefMathSciNetGoogle Scholar
  46. Niederreiter, H.: Pseudo-random numbers and optimal coefficients. Adv. Math. 26, 99–181 (1977) MATHCrossRefMathSciNetGoogle Scholar
  47. Niederreiter, H.: Remarks on nonlinear congruential pseudorandom numbers. Metrika 35, 321–328 (1988) MATHCrossRefMathSciNetGoogle Scholar
  48. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992) MATHGoogle Scholar
  49. Niederreiter, H.: Pseudorandom vector generation by the inversive method. ACM Trans. Model. Comput. Simul. 4, 191–212 (1994) MATHCrossRefGoogle Scholar
  50. Niederreiter, H.: Linear complexity and related complexity measures for sequences. In: Johansson, T., Maitra, S. (eds.) Progress in Cryptology (INDOCRYPT 2003). Lecture Notes in Computer Science, vol. 2904, pp. 1–17. Springer, Berlin (2003) Google Scholar
  51. Niederreiter, H., Rivat, J.: On the correlation of pseudorandom numbers generated by inversive methods. Monatshefte Math. 153, 251–264 (2008) MATHCrossRefMathSciNetGoogle Scholar
  52. Niederreiter, H., Shparlinski, I.: On the distribution and lattice structure of nonlinear congruential pseudorandom numbers. Finite Fields Appl. 5, 246–253 (1999) MATHCrossRefMathSciNetGoogle Scholar
  53. Niederreiter, H., Shparlinski, I.: On the distribution of inversive congruential pseudorandom numbers in parts of the period. Math. Comput. 70, 1569–1574 (2001) MATHMathSciNetGoogle Scholar
  54. Niederreiter, H., Shparlinski, I.: Recent advances in the theory of nonlinear pseudorandom number generators. In: Fang, K.T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and quasi-Monte Carlo methods, 2000, pp. 86–102. Springer, Berlin (2002a) Google Scholar
  55. Niederreiter, H., Shparlinski, I.: On the average distribution of inversive pseudorandom numbers. Finite Fields Appl. 8, 86–102 (2002b) MathSciNetGoogle Scholar
  56. Niederreiter, H., Winterhof, A.: Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number generators. Acta Arith. 93, 387–300 (2000) MATHMathSciNetGoogle Scholar
  57. Niederreiter, H., Winterhof, A.: On the distribution of some new explicit nonlinear congruential pseudorandom numbers. In: Helleseth, T., et al. (eds.) Proceedings of SETA 2004. Lecture Notes in Computer Science, vol. 3486, pp. 266–274. Springer, Berlin (2005) Google Scholar
  58. Rubio, I.M., Corrada-Bravo, C.J.: Cyclic decomposition of permutations of finite fields obtained using monomials. In: Poli, A., Stichtenoth, H. (eds.) Proceedings of \(\mathbb{F}_{q}\)7. Lecture Notes in Computer Science, vol. 2948, pp. 254–261. Springer, Berlin (2004) Google Scholar
  59. Rubio, I.M., Mullen, G.L., Corrada, C.J., Castro, F.N.: Dickson permutation polynomials that decompose in cycles of the same length. In: Mullen, G.L., Panario, D., Shparlinski, I. (eds.) Proceedings of \(\mathbb{F}_{q}\)8. Contemp. Math., vol. 461, pp. 229–239 (2008) Google Scholar
  60. Shparlinski, I.: Cryptographic Applications of Analytic Number Theory. Progress in Computer Science and Applied Logic, vol. 22. Birkhäuser, Basel (2003) MATHGoogle Scholar
  61. Sloane, N.J.: On-line Encyclopedia of integer sequences. Published electronically at http://www.research.att.com/~njas/sequences
  62. Sole, P., Zinoviev, D.: Inversive pseudorandom numbers over Galois rings. Eur. J. Comb. 30, 458–467 (2009) MATHCrossRefMathSciNetGoogle Scholar
  63. Topuzoğlu, A., Winterhof, A.: On the linear complexity profile of nonlinear congruential pseudorandom number generators of higher orders. Appl. Algebra Eng. Commun. Comput. 16, 219–228 (2005) MATHCrossRefGoogle Scholar
  64. Topuzoğlu, A., Winterhof, A.: Pseudorandom sequences. In: Garcia, A., Stichtenoth, H. (eds.) Topics in Geometry, Coding Theory and Cryptography. Algebra and Applications, vol. 6, pp. 135–166. Springer, Berlin (2007) CrossRefGoogle Scholar
  65. Winterhof, A.: On the distribution of some new explicit inversive pseudorandom numbers and vectors. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 487–499. Springer, Berlin (2006) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.MDBFSabancı UniversityTuzlaTurkey

Personalised recommendations