**Four Measures of Nonlinearity***Joan Boyar, Magnus Find, René Peralta*

CIAC 2013.

Cryptographic applications, such as hashing, block ciphers
and stream ciphers,
make use of functions which are simple by some criteria (such as
circuit implementations), yet hard to invert almost everywhere.
A necessary condition for the latter property is to be ``sufficiently
distant'' from linear, and
cryptographers have proposed several
measures for this distance.
In this paper, we show that four common measures, * nonlinearity,
algebraic degree, annihilator immunity*, and * multiplicative complexity
*,
are incomparable in the
sense that for
each pair of measures, μ_{1},μ_{2}, there exist
functions *f*_{1},*f*_{2} with μ_{1}(*f*_{1}) > μ_{1}(*f*_{2})
but μ_{2}(*f*_{1}) < μ_{2}(*f*_{2}).
We also present new connections
between two of these measures.
Additionally, we give a lower bound on the multiplicative complexity
of collision-free functions.

Last modified: Mon Apr 9 13:59:00 CEST 2012