42,629 research outputs found
Composition Theorems for CCA Cryptographic Security
We present two new theorems to analyze the indistinguishability of the composition of cryptographic permutations and the indistinguishability of the XOR of cryptographic functions. Using the H Coefficients technique of \cite{Patarin-2001}, for any two families of permutations and with CCA distinghuishability advantage and , we prove that the set of permutations has CCA distinguishability advantage . This simple composition result gives a CCA indistinguishability geometric gain when composing blockciphers (unlike previously known clasical composition theorems). As an example, we apply this new theorem to analyze and rounds Feistel schemes with and we improve previous best known bounds for a certain range of queries. Similarly, for any two families of functions and with distinghuishability advantage and , we prove that the set of functions has distinguishability advantage . As an example, we apply this new theorem to analyze the XOR of permutations and we improve the previous best known bounds for certain range of querie
Niceness theorems
Many things in mathematics seem lamost unreasonably nice. This includes
objects, counterexamples, proofs. In this preprint I discuss many examples of
this phenomenon with emphasis on the ring of polynomials in a countably
infinite number of variables in its many incarnations such as the representing
object of the Witt vectors, the direct sum of the rings of representations of
the symmetric groups, the free lambda ring on one generator, the homology and
cohomology of the classifying space BU, ... . In addition attention is paid to
the phenomenon that solutions to universal problems (adjoint functors) tend to
pick up extra structure.Comment: 52 page
The Lee-Yang and P\'olya-Schur Programs. II. Theory of Stable Polynomials and Applications
In the first part of this series we characterized all linear operators on
spaces of multivariate polynomials preserving the property of being
non-vanishing in products of open circular domains. For such sets this
completes the multivariate generalization of the classification program
initiated by P\'olya-Schur for univariate real polynomials. We build on these
classification theorems to develop here a theory of multivariate stable
polynomials. Applications and examples show that this theory provides a natural
framework for dealing in a uniform way with Lee-Yang type problems in
statistical mechanics, combinatorics, and geometric function theory in one or
several variables. In particular, we answer a question of Hinkkanen on
multivariate apolarity.Comment: 32 page
Compositions into Powers of : Asymptotic Enumeration and Parameters
For a fixed integer base , we consider the number of compositions of
into a given number of powers of and, related, the maximum number of
representations a positive integer can have as an ordered sum of powers of .
We study the asymptotic growth of those numbers and give precise asymptotic
formulae for them, thereby improving on earlier results of Molteni. Our
approach uses generating functions, which we obtain from infinite transfer
matrices. With the same techniques the distribution of the largest denominator
and the number of distinct parts are investigated
Analytic urns
This article describes a purely analytic approach to urn models of the
generalized or extended P\'olya-Eggenberger type, in the case of two types of
balls and constant ``balance,'' that is, constant row sum. The treatment starts
from a quasilinear first-order partial differential equation associated with a
combinatorial renormalization of the model and bases itself on elementary
conformal mapping arguments coupled with singularity analysis techniques.
Probabilistic consequences in the case of ``subtractive'' urns are new
representations for the probability distribution of the urn's composition at
any time n, structural information on the shape of moments of all orders,
estimates of the speed of convergence to the Gaussian limit and an explicit
determination of the associated large deviation function. In the general case,
analytic solutions involve Abelian integrals over the Fermat curve x^h+y^h=1.
Several urn models, including a classical one associated with balanced trees
(2-3 trees and fringe-balanced search trees) and related to a previous study of
Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a sporadic
urn of balance 3, are shown to admit of explicit representations in terms of
Weierstra\ss elliptic functions: these elliptic models appear precisely to
correspond to regular tessellations of the Euclidean plane.Comment: Published at http://dx.doi.org/10.1214/009117905000000026 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Multivariate Polya-Schur classification problems in the Weyl algebra
A multivariate polynomial is {\em stable} if it is nonvanishing whenever all
variables have positive imaginary parts. We classify all linear partial
differential operators in the Weyl algebra \A_n that preserve stability. An
important tool that we develop in the process is the higher dimensional
generalization of P\'olya-Schur's notion of multiplier sequence. We
characterize all multivariate multiplier sequences as well as those of finite
order. Next, we establish a multivariate extension of the Cauchy-Poincar\'e
interlacing theorem and prove a natural analog of the Lax conjecture for real
stable polynomials in two variables. Using the latter we describe all operators
in \A_1 that preserve univariate hyperbolic polynomials by means of
determinants and homogenized symbols. Our methods also yield homotopical
properties for symbols of linear stability preservers and a duality theorem
showing that an operator in \A_n preserves stability if and only if its
Fischer-Fock adjoint does. These are powerful multivariate extensions of the
classical Hermite-Poulain-Jensen theorem, P\'olya's curve theorem and
Schur-Mal\'o-Szeg\H{o} composition theorems. Examples and applications to
strict stability preservers are also discussed.Comment: To appear in Proc. London Math. Soc; 33 pages, 4 figures, LaTeX2
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
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