914 research outputs found
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Octal Bent Generalized Boolean Functions
In this paper we characterize (octal) bent generalized Boolean functions
defined on \BBZ_2^n with values in \BBZ_8. Moreover, we propose several
constructions of such generalized bent functions for both even and odd
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
On the normality of -ary bent functions
Depending on the parity of and the regularity of a bent function from
to , can be affine on a subspace of dimension
at most , or . We point out that many -ary bent
functions take on this bound, and it seems not easy to find examples for which
one can show a different behaviour. This resembles the situation for Boolean
bent functions of which many are (weakly) -normal, i.e. affine on a
-dimensional subspace. However applying an algorithm by Canteaut et.al.,
some Boolean bent functions were shown to be not - normal. We develop an
algorithm for testing normality for functions from to . Applying the algorithm, for some bent functions in small dimension we
show that they do not take on the bound on normality. Applying direct sum of
functions this yields bent functions with this property in infinitely many
dimensions.Comment: 13 page
Landscape Boolean Functions
In this paper we define a class of Boolean and generalized Boolean functions
defined on with values in (mostly, we consider
), which we call landscape functions (whose class containing generalized
bent, semibent, and plateaued) and find their complete characterization in
terms of their components. In particular, we show that the previously published
characterizations of generalized bent and plateaued Boolean functions are in
fact particular cases of this more general setting. Furthermore, we provide an
inductive construction of landscape functions, having any number of nonzero
Walsh-Hadamard coefficients. We also completely characterize generalized
plateaued functions in terms of the second derivatives and fourth moments.Comment: 19 page
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