22 research outputs found
Short proofs of the Quantum Substate Theorem
The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives
us a powerful operational interpretation of relative entropy, in fact, of the
observational divergence of two quantum states, a quantity that is related to
their relative entropy. Informally, the theorem states that if the
observational divergence between two quantum states rho, sigma is small, then
there is a quantum state rho' close to rho in trace distance, such that rho'
when scaled down by a small factor becomes a substate of sigma. We present new
proofs of this theorem. The resulting statement is optimal up to a constant
factor in its dependence on observational divergence. In addition, the proofs
are both conceptually simpler and significantly shorter than the earlier proof.Comment: 11 pages. Rewritten; included new references; presented the results
in terms of smooth relative min-entropy; stronger results; included converse
and proof using SDP dualit
Distributed Hypothesis Testing with Privacy Constraints
We revisit the distributed hypothesis testing (or hypothesis testing with
communication constraints) problem from the viewpoint of privacy. Instead of
observing the raw data directly, the transmitter observes a sanitized or
randomized version of it. We impose an upper bound on the mutual information
between the raw and randomized data. Under this scenario, the receiver, which
is also provided with side information, is required to make a decision on
whether the null or alternative hypothesis is in effect. We first provide a
general lower bound on the type-II exponent for an arbitrary pair of
hypotheses. Next, we show that if the distribution under the alternative
hypothesis is the product of the marginals of the distribution under the null
(i.e., testing against independence), then the exponent is known exactly.
Moreover, we show that the strong converse property holds. Using ideas from
Euclidean information theory, we also provide an approximate expression for the
exponent when the communication rate is low and the privacy level is high.
Finally, we illustrate our results with a binary and a Gaussian example
Distributed Private Heavy Hitters
In this paper, we give efficient algorithms and lower bounds for solving the
heavy hitters problem while preserving differential privacy in the fully
distributed local model. In this model, there are n parties, each of which
possesses a single element from a universe of size N. The heavy hitters problem
is to find the identity of the most common element shared amongst the n
parties. In the local model, there is no trusted database administrator, and so
the algorithm must interact with each of the parties separately, using a
differentially private protocol. We give tight information-theoretic upper and
lower bounds on the accuracy to which this problem can be solved in the local
model (giving a separation between the local model and the more common
centralized model of privacy), as well as computationally efficient algorithms
even in the case where the data universe N may be exponentially large
Reduce to the Max: A Simple Approach for Massive-Scale Privacy-Preserving Collaborative Network Measurements (Extended Version)
Privacy-preserving techniques for distributed computation have been proposed
recently as a promising framework in collaborative inter-domain network
monitoring. Several different approaches exist to solve such class of problems,
e.g., Homomorphic Encryption (HE) and Secure Multiparty Computation (SMC) based
on Shamir's Secret Sharing algorithm (SSS). Such techniques are complete from a
computation-theoretic perspective: given a set of private inputs, it is
possible to perform arbitrary computation tasks without revealing any of the
intermediate results. In fact, HE and SSS can operate also on secret inputs
and/or provide secret outputs. However, they are computationally expensive and
do not scale well in the number of players and/or in the rate of computation
tasks. In this paper we advocate the use of "elementary" (as opposite to
"complete") Secure Multiparty Computation (E-SMC) procedures for traffic
monitoring. E-SMC supports only simple computations with private input and
public output, i.e., it can not handle secret input nor secret (intermediate)
output. Such a simplification brings a dramatic reduction in complexity and
enables massive-scale implementation with acceptable delay and overhead.
Notwithstanding its simplicity, we claim that an E-SMC scheme is sufficient to
perform a great variety of computation tasks of practical relevance to
collaborative network monitoring, including, e.g., anonymous publishing and set
operations. This is achieved by combining a E-SMC scheme with data structures
like Bloom Filters and bitmap strings.Comment: This is an extended version of the paper presented at the Third
International Workshop on Traffic Monitoring and Analysis (TMA'11), Vienna,
27 April 201