576 research outputs found
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}
On the Systematic Constructions of Rotation Symmetric Bent Functions with Any Possible Algebraic Degrees
In the literature, few constructions of -variable rotation symmetric bent
functions have been presented, which either have restriction on or have
algebraic degree no more than . In this paper, for any even integer
, a first systemic construction of -variable rotation symmetric
bent functions, with any possible algebraic degrees ranging from to , is
proposed
On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions
In this paper we consider cubic bent functions obtained by Leander and McGuire
(J. Comb. Th. Series A, 116 (2009) 960-970) which are
concatenations of quadratic Gold functions.
A lower bound of second-order nonlinearities of these
functions is obtained. This bound is compared with the lower
bounds of second-order nonlinearities obtained for functions
belonging to some other classes of functions which are recently
studied
Additive autocorrelation of some classes of cubic semi-bent Boolean functions
In this paper, we investigate the relation between the autocorrelation of a cubic Boolean function f\in \cB_n at a \in \BBF_{2^n} and the kernel of the bilinear form associated with , the derivative of at . Further, we apply this technique to obtain the tight upper bounds of absolute indicator and sum-of-squares indicator for avalanche characteristics of various classes of highly nonlinear non-bent cubic Boolean functions
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
The Good lower bound of Second-order nonlinearity of a class of Boolean function
In this paper we find the lower bound of second-order nonlinearity of Boolean function with , and . It is also demonstrated that the lower bound obtained in this paper is much better than the lower bound obtained by Iwata-Kurosawa \cite{c14}, and Gangopadhyay et al. (Theorem 1, \cite{c12})
The Lower Bounds on the Second Order Nonlinearity of Cubic Boolean Functions
It is a difficult task to compute the -th order nonlinearity of a
given function with algebraic degree strictly greater than .
Even the lower bounds on the second order nonlinearity is known only
for a few particular functions. We investigate the lower bounds on
the second order nonlinearity of cubic Boolean functions
, where , , and are
positive integers, . Especially, for a class of
Boolean functions , we
deduce a tighter lower bound on the second order nonlinearity of the
functions, where ,
, and
is a positive integer such that .
\\The lower bounds on
the second order nonlinearity of cubic monomial Boolean functions,
represented by , ,
and are positive integers such that , have recently
(2009) been obtained by Gode and Gangopadhvay. Our results have the
advantages over those of Gode and Gangopadhvay as follows. We first
extend the results from monomial Boolean functions to Boolean
functions with more trace terms. We further generalize and improve
the results to a wider range of . Also, our bounds are better
than those of Gode and Gangopadhvay for monomial functions
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