Kaave Hosseini

Structure of Protocols for XOR Functions

Let f be a boolean function on n variables. Its associated XOR function is the two-party function F(x, y) = f(x xor y). We show that, up to polynomial factors, the deterministic communication complexity of F is equal to the parity decision tree complexity of f. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.

Affine-malleable extractors, spectrum doubling, and application to privacy amplification

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 ε-close to (U, A · U + B) where U is uniformly distributed in F and A, B E F are random variables independent of U. We show that the inner-product function (·, ·) : Fn×Fn → F is an affine-malleable extractor for min-entropy k = n/2 + Ω(log(1/ε)). Moreover, under a plausible conjecture in additive combinatorics (called the Spectrum Doubling Conjecture), we show that this holds for k = Ω(log n log(1/ε)). 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 + Ω(log(1/ε)), 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.

An Improved Lower Bound for Arithmetic Regularity

The arithmetic regularity lemma due to Green (GAFA 2005) is an analogue of the famous Szemeredi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → (0,1), there exists a subgroup H ≤ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are s Quantitatively, if one wishes to obtain that for 1−ǫ fraction of the cosets, the nontrivial Fourier coefficients are bounded by ǫ, then Green shows that |G/H| is bounded by a tower of twos of height 1/ǫ 3 . He also gives an example showing that a tower of height (log1/ǫ) is necessary. Here, we give an improved example, showing that a tower of height (1/ǫ) is necessary.

Pseudorandom generators from polarizing random walks

We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1, 1]n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1, 1]n that converges to {-1, 1}n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.