86 research outputs found
Enumeration of 3-letter patterns in compositions
Let A be any set of positive integers and n a positive integer. A composition
of n with parts in A is an ordered collection of one or more elements in A
whose sum is n. We derive generating functions for the number of compositions
of n with m parts in A that have r occurrences of 3-letter patterns formed by
two (adjacent) instances of levels, rises and drops. We also derive asymptotics
for the number of compositions of n that avoid a given pattern. Finally, we
obtain the generating function for the number of k-ary words of length m which
contain a prescribed number of occurrences of a given pattern as a special case
of our results.Comment: 20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer
Conferenc
Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness
We define the notion of asymptotically free for locally restricted
compositions, which means roughly that large parts can often be replaced by any
larger parts. Two well-known examples are Carlitz and alternating compositions.
We show that large parts have asymptotically geometric distributions. This
leads to asymptotically independent Poisson variables for numbers of various
large parts. Based on this we obtain asymptotic formulas for the probability of
being gap free and for the expected values of the largest part, number of
distinct parts and number of parts of multiplicity k, all accurate to o(1).Comment: 28 page
Congruence successions in compositions
A \emph{composition} is a sequence of positive integers, called \emph{parts},
having a fixed sum. By an \emph{-congruence succession}, we will mean a pair
of adjacent parts and within a composition such that . Here, we consider the problem of counting the compositions of
size according to the number of -congruence successions, extending
recent results concerning successions on subsets and permutations. A general
formula is obtained, which reduces in the limiting case to the known generating
function formula for the number of Carlitz compositions. Special attention is
paid to the case , where further enumerative results may be obtained by
means of combinatorial arguments. Finally, an asymptotic estimate is provided
for the number of compositions of size having no -congruence
successions
Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding
Secure computation often benefits from the use of correlated randomness to
achieve fast, non-cryptographic online protocols. A recent paradigm put forth
by Boyle (CCS 2018, Crypto 2019) showed how pseudorandom
correlation generators (PCG) can be used to generate large amounts of useful
forms of correlated (pseudo)randomness, using minimal interactions followed
solely by local computations, yielding silent secure two-party computation
protocols (protocols where the preprocessing phase requires almost no
communication). An additional property called programmability allows to extend
this to build N-party protocols. However, known constructions for programmable
PCG's can only produce OLE's over large fields, and use rather new splittable
Ring-LPN assumption.
In this work, we overcome both limitations. To this end, we introduce the
quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which
generalises the well-established quasi-cyclic syndrome decoding assumption.
Building upon QA-SD, we construct new programmable PCG's for OLE's over any
field with . Our analysis also sheds light on the security
of the ring-LPN assumption used in Boyle (Crypto 2020). Using
our new PCG's, we obtain the first efficient N-party silent secure computation
protocols for computing general arithmetic circuit over for any
.Comment: This is a long version of a paper accepted at CRYPTO'2
Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module
Motivated by finding analogues of elliptic curve point counting techniques,
we introduce one deterministic and two new Monte Carlo randomized algorithms to
compute the characteristic polynomial of a finite rank-two Drinfeld module. We
compare their asymptotic complexity to that of previous algorithms given by
Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In
particular, we find that all three approaches represent either an improvement
in complexity or an expansion of the parameter space over which the algorithm
may be applied. Some experimental results are also presented
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
Point Counting On Genus 2 Curves
For cryptographic purposes, counting points on the jacobian variety of a given hyperelliptic curve is of great importance. There has been several approaches to obtain the cardinality of such a group, specially for hyperelliptic curves of genus 2. The best known algorithm for counting points on genus 2 curves over prime fields of large characteristic is a variant of Schoof’s genus 1 algorithm. Following a recent work of Gaudry and Schost, we show how to speed up the current state of the art genus 2 point counting algorithm by proposing various computational improvements to its basic arithmetical ingredients
Symmetries in algebraic Property Testing
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D
Sociality of sable island horses: population, group, and individual interactions
The social structure of feral horses (Equus ferus caballus) is strongly influenced by their female-defense polygynous mating system, and in turn, social structure is known to impact individual fitness. What regulates structure of social groups is not as well understood. I examine how population density and adult sex ratio (ASR) influence sociality in the feral horse population of Sable Island, (Nova Scotia, Canada), and characterize the duration of associations and the differences in social characteristics between reproductive and non-reproductive adult members of the population. I also explore how concentrations of testosterone and cortisol recovered from tail hair relate to physiological and sociological correlates in this free-living population.
Density and ASR influence a number of social characteristics in this population. High local density is associated with larger harems and increased probability of adult females switching between social groups. Male ability to compete for sexual opportunities is reached 2–3 years after females leading to a typically female biased ASR. However, as ASR becomes more neutral or male-biased group size decreases and adult females become less likely to change bands. Foal production decreased with increasingly male-biased ASR while the same conditions improved foal survival. Female reproductive success increased when they maintained long-term associations with specific males and minimized their overall associations. While several studies suggest that female–female associations in the harem are important, I present evidence to suggest that could be a side-effect of females attempting to remain in association with the same male and to avoid antagonistic interactions associated with establishing their position in the social hierarchy of a new group. Investigation of hair cortisol concentrations (HCC) also revealed that the endocrine response to particular stressors may be different between males and females. Top models describing male HCC included only physiological factors of age and body condition and also year. While the best model describing female HCC included body condition, age, and presence of a foal, they also included social variables of harem size and abundance of males in vicinity. Males in reproductive roles as dominant harem stallions had higher hair testosterone concentrations than non-reproductive males. This research provides a better understanding of the interactions between population level processes and indirect actions on individual fitness, opportunities for sexual selection, and endocrinology as they relate to the social and mating structure of an island-bound population of feral horses.
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