3 research outputs found
Maiorana-McFarland Functions with High Second-Order Nonlinearity
The second-order nonlinearity, and the best quadratic approximations, of Boolean functions are studied in this paper. We prove that cubic functions within the Maiorana-McFarland class achieve very high second order nonlinearity, which is close to an upper bound that was recently proved by Carlet et al., and much higher than the second order nonlinearity obtained by other known constructions. The structure of the cubic Boolean functions considered allows the efficient computation of (a subset of) their best quadratic approximations
Improved upper bound on root number of linearized polynomials and its application to nonlinearity estimation of Boolean functions
To determine the dimension of null space of any given linearized polynomial
is one of vital problems in finite field theory, with concern to design of
modern symmetric cryptosystems. But, the known general theory for this task is
much far from giving the exact dimension when applied to a specific linearized
polynomial. The first contribution of this paper is to give a better general
method to get more precise upper bound on the root number of any given
linearized polynomial. We anticipate this result would be applied as a useful
tool in many research branches of finite field and cryptography. Really we
apply this result to get tighter estimations of the lower bounds on the second
order nonlinearities of general cubic Boolean functions, which has been being
an active research problem during the past decade, with many examples showing
great improvements. Furthermore, this paper shows that by studying the
distribution of radicals of derivatives of a given Boolean functions one can
get a better lower bound of the second-order nonlinearity, through an example
of the monomial Boolean function over any
finite field \GF{n}