Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity

Sihem Mesnager

Paper 2007/117

Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity

Sihem Mesnager

Abstract

The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudo-random generators using it as a nonlinear filtering or combining function. Very few results have been found concerning its relation with the other cryptographic parameters or with the $r$-th order nonlinearity. As recalled by Carlet at Crypto'06, many papers have illustrated the importance of the $r$th-order nonlinearity profile (which includes the first-order nonlinearity). The role of this parameter relatively to the currently known attacks has been also shown for block ciphers. Recently, two lower bounds involving the algebraic immunity on the $r$th-order nonlinearity have been shown by Carlet et \emph{al}. None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on the $r$th-order nonlinearity profile of Boolean functions, given their algebraic immunity, that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders.

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Metadata

Available format(s): PDF
Category: Foundations
Publication info: Published elsewhere. Unknown where it was published
Keywords: stream cipher block cipher algebraic attack Boolean function algebraic immunity algebraic degree higher order nonlinearity annihilator
Contact author(s): hachai @ math jussieu fr
History: 2007-08-03: revised; 2007-04-03: received; See all versions
Short URL: https://ia.cr/2007/117
License: CC BY

BibTeX

@misc{cryptoeprint:2007/117,
      author = {Sihem Mesnager},
      title = {Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity},
      howpublished = {Cryptology ePrint Archive, Paper 2007/117},
      year = {2007},
      note = {\url{https://eprint.iacr.org/2007/117}},
      url = {https://eprint.iacr.org/2007/117}
}