27,290 research outputs found
The combinatorics of biased riffle shuffles
This paper studies biased riffle shuffles, first defined by Diaconis, Fill,
and Pitman. These shuffles generalize the well-studied Gilbert-Shannon-Reeds
shuffle and convolve nicely. An upper bound is given for the time for these
shuffles to converge to the uniform distribution; this matches lower bounds of
Lalley. A careful version of a bijection of Gessel leads to a generating
function for cycle structure after one of these shuffles and gives new results
about descents in random permutations. Results are also obtained about the
inversion and descent structure of a permutation after one of these shuffles.Comment: 11 page
Statistics of Random Permutations and the Cryptanalysis Of Periodic Block Ciphers
A block cipher is intended to be computationally indistinguishable from a
random permutation of appropriate domain and range. But what are the properties
of a random permutation? By the aid of exponential and ordinary generating
functions, we derive a series of collolaries of interest to the cryptographic
community. These follow from the Strong Cycle Structure Theorem of
permutations, and are useful in rendering rigorous two attacks on Keeloq, a
block cipher in wide-spread use. These attacks formerly had heuristic
approximations of their probability of success. Moreover, we delineate an
attack against the (roughly) millionth-fold iteration of a random permutation.
In particular, we create a distinguishing attack, whereby the iteration of a
cipher a number of times equal to a particularly chosen highly-composite number
is breakable, but merely one fewer round is considerably more secure. We then
extend this to a key-recovery attack in a "Triple-DES" style construction, but
using AES-256 and iterating the middle cipher (roughly) a million-fold. It is
hoped that these results will showcase the utility of exponential and ordinary
generating functions and will encourage their use in cryptanalytic research.Comment: 20 page
Large cycles and a functional central limit theorem for generalized weighted random permutations
The objects of our interest are the so-called -permutations, which are
permutations whose cycle length lie in a fixed set . They have been
extensively studied with respect to the uniform or the Ewens measure. In this
paper, we extend some classical results to a more general weighted probability
measure which is a natural extension of the Ewens measure and which in
particular allows to consider sets depending on the degree of the
permutation. By means of complex analysis arguments and under reasonable
conditions on generating functions we study the asymptotic behaviour of
classical statistics. More precisely, we generalize results concerning large
cycles of random permutations by Vershik, Shmidt and Kingman, namely the weak
convergence of the size ordered cycle length to a Poisson-Dirichlet
distribution. Furthermore, we apply our tools to the cycle counts and obtain a
Brownian motion central limit theorem which extends results by DeLaurentis,
Pittel and Hansen.Comment: 24 pages, 3 Figure
Random permutation matrices under the generalized Ewens measure
We consider a generalization of the Ewens measure for the symmetric group,
calculating moments of the characteristic polynomial and similar multiplicative
statistics. In addition, we study the asymptotic behavior of linear statistics
(such as the trace of a permutation matrix or of a wreath product) under this
new measure.Comment: Published in at http://dx.doi.org/10.1214/12-AAP862 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Random and exhaustive generation of permutations and cycles
In 1986 S. Sattolo introduced a simple algorithm for uniform random
generation of cyclic permutations on a fixed number of symbols. This algorithm
is very similar to the standard method for generating a random permutation, but
is less well known.
We consider both methods in a unified way, and discuss their relation with
exhaustive generation methods. We analyse several random variables associated
with the algorithms and find their grand probability generating functions,
which gives easy access to moments and limit laws.Comment: 9 page
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