85 research outputs found
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
Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis
For fixed , we consider the class of representations of as sum of
unit fractions whose denominators are powers of or equivalently the class
of canonical compact -ary Huffman codes or equivalently rooted -ary plane
"canonical" trees. We study the probabilistic behaviour of the height (limit
distribution is shown to be normal), the number of distinct summands (normal
distribution), the path length (normal distribution), the width (main term of
the expectation and concentration property) and the number of leaves at maximum
distance from the root (discrete distribution)
Microwave cavity-enhanced transduction for plug and play nanomechanics at room temperature
Nanomechanical resonators with increasingly high quality factors are enabled
following recent insights into energy storage and loss mechanisms in
nanoelectromechanical systems (NEMS). Consequently, efficient, non-dissipative
transduction schemes are required to avoid the dominating influence of coupling
losses. We present an integrated NEMS transducer based on a microwave cavity
dielectrically coupled to an array of doubly-clamped pre-stressed silicon
nitride beam resonators. This cavity-enhanced detection scheme allows resolving
the resonators' Brownian motion at room temperature while preserving their high
mechanical quality factor of 290,000 at 6.6 MHz. Furthermore, our approach
constitutes an "opto"mechanical system in which backaction effects of the
microwave field are employed to alter the effective damping of the resonators.
In particular, cavity-pumped self-oscillation yields a linewidth of only 5 Hz.
Thereby, an adjustement-free, all-integrated and self-driven
nanoelectromechanical resonator array interfaced by just two microwave
connectors is realised, potentially useful for applications in sensing and
signal processing
A counterexample to the chain rule for conditional HILL entropy
Most entropy notions H(.) like Shannon or min-entropy satisfy a chain rule stating that for random variables X,Z, and A we have H(X|Z,A)≥H(X|Z)−|A|. That is, by conditioning on A the entropy of X can decrease by at most the bitlength |A| of A. Such chain rules are known to hold for some computational entropy notions like Yao’s and unpredictability-entropy. For HILL entropy, the computational analogue of min-entropy, the chain rule is of special interest and has found many applications, including leakage-resilient cryptography, deterministic encryption, and memory delegation. These applications rely on restricted special cases of the chain rule. Whether the chain rule for conditional HILL entropy holds in general was an open problem for which we give a strong negative answer: we construct joint distributions (X,Z,A), where A is a distribution over a single bit, such that the HILL entropy H HILL (X|Z) is large but H HILL (X|Z,A) is basically zero.
Our counterexample just makes the minimal assumption that NP⊈P/poly. Under the stronger assumption that injective one-way function exist, we can make all the distributions efficiently samplable.
Finally, we show that some more sophisticated cryptographic objects like lossy functions can be used to sample a distribution constituting a counterexample to the chain rule making only a single invocation to the underlying object
LNCS
A chain rule for an entropy notion H(.) states that the entropy H(X) of a variable X decreases by at most l if conditioned on an l-bit string A, i.e., H(X|A)>= H(X)-l. More generally, it satisfies a chain rule for conditional entropy if H(X|Y,A)>= H(X|Y)-l.
All natural information theoretic entropy notions we are aware of (like Shannon or min-entropy) satisfy some kind of chain rule for conditional entropy. Moreover, many computational entropy notions (like Yao entropy, unpredictability entropy and several variants of HILL entropy) satisfy the chain rule for conditional entropy, though here not only the quantity decreases by l, but also the quality of the entropy decreases exponentially in l. However, for
the standard notion of conditional HILL entropy (the computational equivalent of min-entropy) the existence of such a rule was unknown so far.
In this paper, we prove that for conditional HILL entropy no meaningful chain rule exists, assuming the existence of one-way permutations: there exist distributions X,Y,A, where A is a distribution over a single bit, but H(X|Y)>>H(X|Y,A), even if we simultaneously allow for a massive degradation in the quality of the entropy.
The idea underlying our construction is based on a surprising connection between the chain rule for HILL entropy and deniable encryption
Stronger Security for Sanitizable Signatures
Sanitizable signature schemes (SSS) enable a designated party (called the sanitizer ) to alter admissible
blocks of a signed message. This primitive can be used to remove or alter sensitive data from already signed
messages without involvement of the original signer.
Current state-of-the-art security definitions of SSSs only dene a \weak form of security. Namely, the unforgeability,
accountability and transparency definitions are not strong enough to be meaningful in certain use-cases. We
identify some of these use-cases, close this gap by introducing stronger definitions, and show how to alter an
existing construction to meet our desired security level. Moreover, we clarify a small yet important detail in the
state-of-the-art privacy definition. Our work allows to deploy this primitive in more and different scenarios
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