29,466 research outputs found
Quasirandom Permutations
Chung and Graham define quasirandom subsets of to be those
with any one of a large collection of equivalent random-like properties. We
weaken their definition and call a subset of -balanced
if its discrepancy on each interval is bounded by . A quasirandom
permutation, then, is one which maps each interval to a highly balanced set. In
the spirit of previous studies of quasirandomness, we exhibit several
random-like properties which are equivalent to this one, including the property
of containing (approximately) the expected number of subsequences of each
order-type. We provide a few applications of these results, present a
construction for a family of strongly quasirandom permutations, and prove that
this construction is essentially optimal, using a result of W. Schmidt on the
discrepancy of sequences of real numbers.Comment: 30 pages, 2 figures, submitted to JCT
On the superimposition of Christoffel words
Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be
reformulated as follows: for a -letter alphabet A, with a fixed ,
there exists a unique balanced infinite word, up to letter permutations and
shifts, that has mutually distinct letter frequencies. Motivated by the
Fraenkel conjecture, we study in this paper whether two Christoffel words can
be superimposed. Following from previous works on this conjecture using Beatty
sequences, we give a necessary and sufficient condition for the superimposition
of two Christoffel words having same length, and more generally, of two
arbitrary Christoffel words. Moreover, for any two superimposable Christoffel
words, we give the number of different possible superimpositions and we prove
that there exists a superimposition that works for any two superimposable
Christoffel words. Finally, some new properties of Christoffel words are
obtained as well as a geometric proof of a classic result concerning the money
problem, using Christoffel words
Block Sensitivity of Minterm-Transitive Functions
Boolean functions with symmetry properties are interesting from a complexity
theory perspective; extensive research has shown that these functions, if
nonconstant, must have high `complexity' according to various measures.
In recent work of this type, Sun gave bounds on the block sensitivity of
nonconstant Boolean functions invariant under a transitive permutation group.
Sun showed that all such functions satisfy bs(f) = Omega(N^{1/3}), and that
there exists such a function for which bs(f) = O(N^{3/7}ln N). His example
function belongs to a subclass of transitively invariant functions called the
minterm-transitive functions (defined in earlier work by Chakraborty).
We extend these results in two ways. First, we show that nonconstant
minterm-transitive functions satisfy bs(f) = Omega(N^{3/7}). Thus Sun's example
function has nearly minimal block sensitivity for this subclass. Second, we
give an improved example: a minterm-transitive function for which bs(f) =
O(N^{3/7}ln^{1/7}N).Comment: 10 page
Loiss: A Byte-Oriented Stream Cipher
This paper presents a byte-oriented stream cipher -- Loiss, which takes a 128-bit initial key and a 128-bit initial vector as inputs, and outputs a key stream of bytes. The algorithm is based on a linear feedback shift register, and uses a structure called BOMM in the filter generator, which has good property on resisting against algebraic attacks, linear distinguishing attacks and fast correlation attacks. In order for BOMM to be balanced, the S-boxes in BOMM must be orthomorphic permutations. To further improve the capability in resisting against those attacks, the S-boxes in BOMM must also possess some good cryptographic properties, for example, high algebraic immunity, high nonlinearity, and so on. However current researches on orthomorphic permutations pay little attention on their cryptographic properties, and we believe that Loiss not only enriches applications of orthomorphic permutations in cryptography, but also motivates the research on a variety of cryptographic properties of orthomorphic permutations
A natural generalization of Balanced Tableaux
We introduce the notion of "type" of a tableau, that allows us to define new
families of tableaux including both balanced and standard Young tableaux. We
use these new objects to describe the set of reduced decompositions of any
permutation. We then generalize the work of Fomin \emph{et al.} by giving,
among other things, a new proof of the fact that balanced and standard tableaux
are equinumerous, and by exhibiting many new families of tableaux having
similar combinatorial properties to those of balanced tableaux.Comment: This new version cointains several major changes in order to take new
results into accoun
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