Rényi entropy | EPFL Graph Search

In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. The Rényi entropy is important in ecology and statistics as index of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of α can be calculated explicitly because it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context of randomness extractors. Definition The

The Chaining Lemma and Its Application

Sebastian Faust

We present a new information-theoretic result which we call the Chaining Lemma. It considers a so-called "chain" of random variables, defined by a source distribution X-(0) with high min-entropy and a number (say, t in total) of arbitrary functions (T-1,...,T-t) which are applied in succession to that source to generate the chain X-(0) (sic) X-(1) (sic) X-(2)...(sic) X-(t). Intuitively, the Chaining Lemma guarantees that, if the chain is not too long, then either (i) the entire chain is "highly random", in that every variable has high min-entropy; or (ii) it is possible to find a point j (1

Springer-Verlag Berlin2015

Affine-malleable Extractors, Spectrum Doubling, and Application to Privacy Amplification

Divesh Aggarwal

The study of seeded randomness extractors is a major line of research in theoretical computer science. The goal is to construct deterministic algorithms which can take a weak random source x with min-entropy k and a uniformly random seed Y of length d, and outputs a string of length close to k that is close to uniform and independent of Y. Dodis and Wichs [DW09] introduced a generalization of randomness extractors called non-malleable extractors (nmExt) where nmExt(X, Y) is close to uniform and independent of Y and nmExt(X, f(Y)) for any function f with no fixed points. We relax the notion of a non-malleable extractor and introduce what we call an affine-malleable extractor (AmExt : Fn x Fd -> F) where AmExt(X, Y ) is close to uniform and independent of Y and has some limited dependence of AmExt(X, f(Y )) - that conditioned on Y , (AmExt(X, Y ), AmExt(X, f(Y ))) is epsilon-close to (U, A U + B) where U is uniformly distributed in F and A, B is an element of F are random variables independent of U. We show that the inner-product function (,) : FnxFn -> F is an affine-malleable extractor for min-entropy k = n/2 + Omega(log(1/epsilon)). Moreover, under a plausible conjecture in additive combinatorics (called the Spectrum Doubling Conjecture), we show that this holds for k = Omega(log n log(1/epsilon)). As a modest justification of the conjecture, we show that a weaker version of the conjecture is implied by the widely believed Polynomial Freiman-Ruzsa conjecture. We also study the classical problem of privacy amplification, where two parties Alice and Bob share a weak secret X of min-entropy k, and wish to agree on secret key R of length m over a public communication channel completely controlled by a computationally unbounded attacker Eve. The main application of non-malleable extractors and their many variants has been in constructing secure privacy amplification protocols. We show that affine-malleable extractors along with affine-evasive sets can also be used to construct efficient privacy amplification protocols. This gives a much simpler protocol for min-entropy k = n/2 + Omega(log(1/epsilon)), and additionally, under the Spectrum Doubling Conjecture, achieves near optimal parameters and achieves additional security properties like source privacy that have been the focus of some recent results in privacy amplification.

Ieee2016

Affine-malleable Extractors, Spectrum Doubling, and Application to Privacy Amplification

Divesh Aggarwal

Ieee2016