21,401 research outputs found
Asymptotic information leakage under one-try attacks
We study the asymptotic behaviour of (a) information leakage and (b) adversary’s error probability in information hiding systems modelled as noisy channels. Specifically, we assume the attacker can make a single guess after observing n independent executions of the system, throughout which the secret information is kept fixed. We show that the asymptotic behaviour of quantities (a) and (b) can be determined in a simple way from the channel matrix. Moreover, simple and tight bounds on them as functions of n show that the convergence is exponential. We also discuss feasible methods to evaluate the rate of convergence. Our results cover both the Bayesian case, where a prior probability distribution on the secrets is assumed known to the attacker, and the maximum-likelihood case, where the attacker does not know such distribution. In the Bayesian case, we identify the distributions that maximize the leakage. We consider both the min-entropy setting studied by Smith and the additive form recently proposed by Braun et al., and show the two forms do agree asymptotically. Next, we extend these results to a more sophisticated eavesdropping scenario, where the attacker can perform a (noisy) observation at each state of the computation and the systems are modelled as hidden Markov models
Tight polynomial worst-case bounds for loop programs
In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid
Forbidden subposet problems for traces of set families
In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets form a
copy of a poset , if there exists a bijection such that for any the relation implies
. A family of sets is \textit{-free} if
it does not contain any copy of . The trace of a family on a
set is .
We introduce the following notions: is
-trace -free if for any -subset , the family
is -free and is trace -free if it is
-trace -free for all . As the first instances of these problems
we determine the maximum size of trace -free families, where is the
butterfly poset on four elements with and determine the
asymptotics of the maximum size of -trace -free families for
. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems
Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak
This is a survey of several exciting recent results in which techniques
originating in the area known as additive combinatorics have been applied to
give results in other areas, such as group theory, number theory and
theoretical computer science. We begin with a discussion of the notion of an
approximate group and also that of an approximate field, describing key results
of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure
of such objects is elucidated. We then move on to the applications. In
particular we will look at the work of Bourgain and Gamburd on expansion
properties of Cayley graphs on SL_2(F_p) and at its application in the work of
Bourgain, Gamburd and Sarnak on nonlinear sieving problems.Comment: 25 pages. Survey article to accompany my forthcoming talk at the
Current Events Bulletin of the AMS, 2010. A reference added and a few small
changes mad
On the cyclicity of the rational points group of abelian varieties over finite fields
We propose a simple criterion to know if an abelian variety defined over
a finite field is cyclic, i.e., it has a cyclic group of
rational points; this criterion is based on the endomorphism ring
End. We also provide a criterion to know if an isogeny
class is cyclic, i.e., all its varieties are cyclic; this criterion is based on
the characteristic polynomial of the isogeny class. We find some asymptotic
lower bounds on the fraction of cyclic -isogeny classes among
certain families of them, when tends to infinity. Some of these bounds
require an additional hypothesis. In the case of surfaces, we prove that this
hypothesis is achieved and, over all -isogeny classes with
endomorphism algebra being a field and where is an even power of a prime,
we prove that the one with maximal number of rational points is cyclic and
ordinary.Comment: 13 pages, this is a preliminary version, comments are welcom
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