Paper 2017/432

Statistical and Linear Independence of Binary Random Variables

Kaisa Nyberg

Abstract

Linear cryptanalysis makes use of statistical models that consider linear approximations over practical and ideal block ciphers as binary random variables. Recently, more complex models have been proposed that take also into account the statistical behavior of correlations of linear approximations over the key space of the cipher and over the randomness of the ideal cipher. The goal of this ongoing work is to investigate independence properties of linear approximations and their relationships. In this third revised version we show that the assumptions of Proposition~1 of the previous version are contradictory and hence renders that result useless. In particular, we prove that linear and statistical independence of binary random variables are equivalent properties in a vector space of variables if and only if all non-zero variables in this vector space are balanced, that is, correspond to components of a permutation. This study is motivated by finding reasonable wrong-key hypotheses for linear cryptanalysis and its generalizations which will also be discussed.

Note: This is an updated version of ongoing work.

Metadata
Available format(s)
PDF
Publication info
Preprint.
Keywords
Xiao-Massey lemmablock cipherlinear cryptanalysislinear approximationrandom Boolean functionrandom vectorial Boolean functionmultidimensional linear cryptanalysiswrong-key hypothesis
Contact author(s)
kaisa nyberg @ aalto fi
History
2018-01-30: last of 2 revisions
2017-05-22: received
See all versions
Short URL
https://ia.cr/2017/432
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2017/432,
      author = {Kaisa Nyberg},
      title = {Statistical and Linear Independence of Binary Random Variables},
      howpublished = {Cryptology ePrint Archive, Paper 2017/432},
      year = {2017},
      note = {\url{https://eprint.iacr.org/2017/432}},
      url = {https://eprint.iacr.org/2017/432}
}
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