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Permutations of finite fields for check digit systems

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Abstract

Let q be a prime power. For a divisor n of q − 1 we prove an asymptotic formula for the number of polynomials of the form

$$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_q[X]$$

such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over \({\mathbb{F}_q}\) . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Altenberger Straße 69, 4040, Linz, Austria

    Rasha Shaheen & Arne Winterhof

  2. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    Rasha Shaheen

Authors
  1. Rasha Shaheen
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  2. Arne Winterhof
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Correspondence to Arne Winterhof.

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Communicated by D. Hachenberger.

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Shaheen, R., Winterhof, A. Permutations of finite fields for check digit systems. Des. Codes Cryptogr. 57, 361–371 (2010). https://doi.org/10.1007/s10623-010-9374-z

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  • DOI: https://doi.org/10.1007/s10623-010-9374-z

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