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Correlation immune functions with respect to the q-transform


Correlation immunity is a measure of resistance to Siegenthaler’s divide and conquer attack on nonlinear combiners. In this work, we study functions with regard to the q-transform, a generalization of the Walsh-Hadamard transform, that measures the proximity of a function to the set of functions obtained from a function q(x) by linear base change. We propose two notions of q-correlation immune functions and study their relationship between them. We also analyze certain properties of these functions and present some techniques to design these functions.

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  1. Of course, there maybe several coset leaders for a coset. We only need one of them.

  2. The minimum distance of a linear code is the minimum Hamming weight of its nonzero code words.


  1. Carlet, C., Guillot, P., Mesnager, S.: On immunity profile of boolean functions. In: Gong, G., Helleseth, T., Song, H.-Y., Yang, K. (eds.) Sequences and Their Applications–SETA 2006, Lecture Notes in Computer Science, vol. 4086, pp 364–375. Springer, Berlin (2006)

  2. Golomb, S.: Shift Register Sequences. Aegean Park Press, Laguna Hills (1982)

    MATH  Google Scholar 

  3. Klapper, A.: A new transform related to distance from a boolean function (extended abstract). In: Schmidt, K.-U., Winterhof, A. (eds.) Sequences and Their Applications–SETA 2014, Lecture Notes in Computer Science, vol. 8865, pp 47–59. Springer International Publishing (2014)

  4. Klapper, A.: A new transform related to distance from a boolean function. IEEE Trans. Inf. Theory 62.5, 2798–2812 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  5. Neyman, J., Pearson, E. S.: On the problem of the most efficient tests of statistical hypotheses. In: Kotz, S., Johnson, N. (eds.) Breakthroughs in Statistics, pp 73–108. Springer, New York (1992)

  6. Siegenthaler, T.: Decrypting a class of stream ciphers using ciphertext only. IEEE Trans. Comput. 100.1, 81–85 (1985)

    Article  Google Scholar 

  7. Wu, C. -K., Dawson, E.: Construction of correlation immune boolean functions. Aust. J. Commun. 21, 141–166 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Xiao, G., Massey, J.: A spectral characterization of correlation-immune combining functions. IEEE Trans. Inf. Theory 34.3, 569–571 (1988)

    MathSciNet  Article  MATH  Google Scholar 

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Parts of this work were written during a very pleasant visit of the second author to University of Kentucky in Lexington, USA, in 2014 - 2015. He wishes to thank his hosts for their hospitality.

This material is based upon work supported by the National Science Foundation under Grant No. CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Zhixiong Chen was partially supported by the National Natural Science Foundation of China under grant No. 61772292 and China Scholarship Council.

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Correspondence to Ting Gu.

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This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

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Gu, T., Chen, Z. & Klapper, A. Correlation immune functions with respect to the q-transform. Cryptogr. Commun. 10, 1063–1073 (2018).

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  • Correlation immunity
  • q-transform
  • q-correlation immune functions