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Asymptotic enumeration of correlation-immune boolean functions

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Abstract

A boolean function of n boolean variables is correlation-immune of order k if the function value is uncorrelated with the values of any k of the arguments. Such functions are of considerable interest due to their cryptographic properties, and are also related to the orthogonal arrays of statistics and the balanced hypercube colourings of combinatorics. The weight of a boolean function is the number of argument values that produce a function value of 1. If this is exactly half the argument values, that is, 2n − 1 values, a correlation-immune function is called resilient. An asymptotic estimate of the number N(n,k) of n-variable correlation-immune boolean functions of order k was obtained in 1992 by Denisov for constant k. Denisov repudiated that estimate in 2000, but we show that the repudiation was a mistake. The main contribution of this paper is an asymptotic estimate of N(n,k) which holds if k increases with n within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of k = 1, our estimates are valid for all weights.

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Authors and Affiliations

  1. Department of Computer Science, University of Georgia, Athens, GA, 30602, USA

    E. Rodney Canfield & Robert W. Robinson

  2. School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada

    Zhicheng Gao

  3. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia, 2052

    Catherine Greenhill

  4. School of Computer Science, Australian National University, Canberra, ACT, Australia

    Brendan D. McKay

Authors
  1. E. Rodney Canfield
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  2. Zhicheng Gao
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  3. Catherine Greenhill
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  4. Brendan D. McKay
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  5. Robert W. Robinson
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Corresponding author

Correspondence to Catherine Greenhill.

Additional information

Research of E. Rodney Canfield supported by the NSA Mathematical Sciences Program.

Research of Brendan D. McKay supported by the Australian Research Council.

Research of Zhicheng Gao supported by NSERC.

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Canfield, E.R., Gao, Z., Greenhill, C. et al. Asymptotic enumeration of correlation-immune boolean functions. Cryptogr. Commun. 2, 111–126 (2010). https://doi.org/10.1007/s12095-010-0019-x

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  • DOI: https://doi.org/10.1007/s12095-010-0019-x

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