Cybernetics and Systems Analysis

, Volume 43, Issue 6, pp 830–837 | Cite as

Applying fast simulation to find the number of good permutations

  • N. Yu. Kuznetsov


A permutation (s0, s1,…, sN − 1) of symbols 0, 1,…, N − 1, is called good if the set (t0, t1,…, tN − 1) formed according to the rule ti = i + si (mod N), i = 0, 1, … N − 1, is also a permutation. A fast simulation method is proposed. It allows the number of good permutations to be evaluated with high accuracy for large N (in particular, N > 100). Empirical upper and lower bounds for the number of good permutations are presented and verified against numerical data.


permutation fast simulation method unbiased estimate sample variance relative error 


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  1. 1.
    V. N. Sachkov, Introduction to Combinatorial Methods of Discrete Mathematics [in Russian], Nauka, Moscow (1982).MATHGoogle Scholar
  2. 2.
    V. N. Sachkov, “Markov chains of iterative systems of transforms,” Tr. Discret. Mat., 6, 165–183 (2002).Google Scholar
  3. 3.
    C. Cooper, R. Gilchrist, I. N. Kovalenko, and D. Novakovic, “Estimation of the number of “good” permutations, with application to cryptography,” Cybern. Syst. Analysis, 35, No. 5, 688–693 (1999).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. Konheim, Cryptography: A Primer, Wiley, Chichester (1991).Google Scholar
  5. 5.
    C. Cooper and I. N. Kovalenko, “An upper bound for the number of complete mappings,” Probability Theory and Math. Statistics, 53, 69–75 (1995).MATHGoogle Scholar
  6. 6.
    I. N. Kovalenko, “Upper bound on the number of complete maps,” Cybern. Syst. Analysis, 32, No. 1, 65–68 (1996).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. A. Levitskaya, “A combinatorial problem in a class of permutations over a ring Z n of odd modulo n residues,” J. Autom. Inform. Sci., 28, No. 5, 99–108 (1996).MathSciNetGoogle Scholar
  8. 8.
    N. Yu. Kuznetsov, “Computing the permanent by importance sampling method,” Cybern. Syst. Analysis, 32, No. 6, 749–755 (1996).MATHCrossRefGoogle Scholar
  9. 9.
    N. Yu. Kuznetsov, “Using stratified sampling to solve the knapsack problem,” Cybern. Syst. Analysis, 34, No. 1, 61–68 (1998).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

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