3 research outputs found
On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions
The -th order nonlinearity of a Boolean function is an important
cryptographic criterion in analyzing the security of stream as well
as block ciphers. It is also important in coding theory as it is
related to the covering radius of the Reed-Muller code .
In this paper we deduce the lower bounds of the second order nonlinearity
of the two classes of Boolean functions of the form
\begin{enumerate}
\item
with
and where .
\item
where and
is an integer such that , .
\end{enumerate}
For some , the first class gives bent functions whereas
Boolean functions of the second class are all bent, i.e., they achieve
optimum first order nonlinearity
On lower bounds on second--order nonliearities of bent functions obtained by using Niho power functions
In this paper we find a lower bound of the second-order nonlinearities
of Boolean bent functions of the form ,where and are Niho exponents. A lower bound of the second-order nonlinearities of these Boolean functions can also be obtained by using a result proved by Li, Hu and Gao (eprint.iacr.org/2010 /009.pdf). It is demonstrated that for large values of the lower bound obtained in this paper are better than the lower bound obtained by Li, Hu and Gao
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}