694 research outputs found
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
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}
Many-body quantum magic
Magic (non-stabilizerness) is a necessary but "expensive" kind of "fuel" to
drive universal fault-tolerant quantum computation. To properly study and
characterize the origin of quantum "complexity" in computation as well as
physics, it is crucial to develop a rigorous understanding of the
quantification of magic. Previous studies of magic mostly focused on small
systems and largely relied on the discrete Wigner formalism (which is only well
behaved in odd prime power dimensions). Here we present an initiatory study of
the magic of genuinely many-body quantum states (with focus on the important
case of many qubits) at a quantitative level. We first address the basic
question of how magical a many-body state can be, and show that the maximum and
typical magic of an -qubit state is essentially , simultaneously for a
range of natural resource measures. As a corollary, we show that the resource
theory of magic with stabilizer-preserving free operations is asymptotically
reversible. In the quest for explicit, scalable cases of highly entangled
states whose magic can be understood, we connect the magic of hypergraph states
with the second-order nonlinearity of their underlying Boolean functions. Next,
we go on and investigate many-body magic in practical and physical contexts. We
first consider Pauli measurement-based quantum computation, in which magic is a
necessary feature of the initial resource state. We show that -qubit states
with nearly magic, or indeed almost all states, cannot supply nontrivial
speedups over classical computers. We then present an example of analyzing the
magic of "natural" condensed matter systems. We apply the Boolean function
techniques to derive explicit bounds on the magic of the ground states of
certain 2D symmetry-protected topological (SPT) phases, and comment on possible
further connections between magic and the quantum complexity of matter.Comment: 15 pages, 3 figure
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
Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review
The paper characterizes classes of functions for which deep learning can be
exponentially better than shallow learning. Deep convolutional networks are a
special case of these conditions, though weight sharing is not the main reason
for their exponential advantage
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
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