Highly nonlinear resilient functions through disjoint codes
in projective spaces
Pascale Charpin and Enes Pasalic
INRIA, projet CODES
Designs, Codes and Cryptography, 37, 319-346, 2005.
Functions which map n-bits to m-bits
are important cryptographic
sub-primitives in the design of
additive stream ciphers. We construct highly nonlinear t-resilient
such functions ((n,m,t) functions)
using a class of binary disjoint codes,
a construction which was recently introduced by Johansson and
Our main contribution
concerns the generation of suitable sets of such disjoint codes.
We propose a deterministic method for finding disjoint codes
of length um by considering the points of
We then obtain some lower bounds on the number of disjoint codes,
by fixing some
Through these sets, we deduce
in certain cases the existence of resilient functions
with very high
nonlinearity values. We show how, thanks to our method,
the degree and the differential
of (n,m,t) functions can be improved.
Boolean function, $n$-input $m$-output function,
resilient function, nonlinearity, propagation characteristic,
symmetric cryptography, stream cipher, linear code,
projective space, complete weight enumerator.