15,092 research outputs found
On Extracting Common Random Bits From Correlated Sources
Suppose Alice and Bob receive strings of unbiased independent but noisy bits from some random source. They wish to use their respective strings to extract a common sequence of random bits with high probability but without communicating. How many such bits can they extract? The trivial strategy of outputting the first k bits yields an agreement probability of (1-ε)k\u3c 2 −1.44kε, where ε is the amount of noise. We show that no strategy can achieve agreement probability better than 2−kε/(1−ε).
On the other hand, we show that when k ≥ 10 + 2(1 − ε)/ε, there exists a strategy which achieves an agreement probability of 0.003(kε)−1/2 · 2−kε/(1−ε)
On Extracting Common Random Bits From Correlated Sources on Large Alphabets
Suppose Alice and Bob receive strings X=(X1,...,Xn) and Y=(Y1,...,Yn) each uniformly random in [s]n, but so that X and Y are correlated. For each symbol i, we have that Yi=Xi with probability 1-ε and otherwise Yi is chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s]k, but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of (1-ε+ε/s)k. In a recent work by Bogdanov and Mossel, it was shown that in the binary case where s=2 and k=k(ε) is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability (kε)-1/2·2-kε/(2(1-ε/2)) using a random construction based on Hamming balls, and this is optimal up to lower order terms. In this paper, we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular, we show no strategy can achieve agreement probability better than (1-ε)k(1+δ(s))k where δ(s)→ 0 as s→∞. We also show that Hamming ball-based constructions have much lower agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities
Reconciliation of a Quantum-Distributed Gaussian Key
Two parties, Alice and Bob, wish to distill a binary secret key out of a list
of correlated variables that they share after running a quantum key
distribution protocol based on continuous-spectrum quantum carriers. We present
a novel construction that allows the legitimate parties to get equal bit
strings out of correlated variables by using a classical channel, with as few
leaked information as possible. This opens the way to securely correcting
non-binary key elements. In particular, the construction is refined to the case
of Gaussian variables as it applies directly to recent continuous-variable
protocols for quantum key distribution.Comment: 8 pages, 4 figures. Submitted to the IEEE for possible publication.
Revised version to improve its clarit
Trevisan's extractor in the presence of quantum side information
Randomness extraction involves the processing of purely classical information
and is therefore usually studied in the framework of classical probability
theory. However, such a classical treatment is generally too restrictive for
applications, where side information about the values taken by classical random
variables may be represented by the state of a quantum system. This is
particularly relevant in the context of cryptography, where an adversary may
make use of quantum devices. Here, we show that the well known construction
paradigm for extractors proposed by Trevisan is sound in the presence of
quantum side information.
We exploit the modularity of this paradigm to give several concrete extractor
constructions, which, e.g, extract all the conditional (smooth) min-entropy of
the source using a seed of length poly-logarithmic in the input, or only
require the seed to be weakly random.Comment: 20+10 pages; v2: extract more min-entropy, use weakly random seed;
v3: extended introduction, matches published version with sections somewhat
reordere
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