Algebraic Immunity Hierarchy of Boolean Functions

Ziran Tu; Yingpu Deng

Paper 2007/259

Algebraic Immunity Hierarchy of Boolean Functions

Ziran Tu and Yingpu Deng

Abstract

Algebraic immunity of Boolean functions is a very important concept in recently introduced algebraic attacks of stream cipher. For a $n$ -variable Boolean function $f$ , the algebraic immunity $A I_{n} (f)$ takes values in ${0, 1, \dots, ⌈ \frac{n}{2} ⌉}$ . For every $k$ in this range, denote the set of all -variable Boolean functions with algebraic immunity , and we know that is always non-empty. According to the algebraic immunity, we can form a hierarchy of Boolean functions. Trivially, . In general, about this integer sequence very few results are known. In this paper, we show an explicit formula for . That is, we obtain an exact formula for the number of Boolean functions with algebraic immunity one. This is the first exact formula for the terms in the above integer sequence. We also give a tight upper bound for nonlinearity of Boolean functions with algebraic immunity one.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Published elsewhere. Unknown where it was published
Keywords: Boolean functions algebraic attack algebraic immunity nonlinearity stream cipher
Contact author(s): dengyp @ amss ac cn
History: 2007-07-03: received
Short URL: https://ia.cr/2007/259
License: CC BY

BibTeX

@misc{cryptoeprint:2007/259,
      author = {Ziran Tu and Yingpu Deng},
      title = {Algebraic Immunity Hierarchy of Boolean Functions},
      howpublished = {Cryptology {ePrint} Archive, Paper 2007/259},
      year = {2007},
      url = {https://eprint.iacr.org/2007/259}
}