Guest column: from randomness extraction to rotating needles
ACM SIGACT News
High-rate codes with sublinear-time decoding
Proceedings of the forty-third annual ACM symposium on Theory of computing
An introduction to randomness extractors
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Short Seed Extractors against Quantum Storage
SIAM Journal on Computing
Kakeya Sets, New Mergers, and Old Extractors
SIAM Journal on Computing
Better short-seed quantum-proof extractors
Theoretical Computer Science
Design extractors, non-malleable condensers and privacy amplification
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
New affine-invariant codes from lifting
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Weighted Reed---Muller codes revisited
Designs, Codes and Cryptography
Hi-index | 0.00 |
We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. (A) We show that every Kakeya set (a set of points that contains a line in every direction) in $\F_q^n$ must be of size at least $q^n/2^n$. This bound is tight to within a $2 + o(1)$ factor for every $n$ as $q \to \infty$, compared to previous bounds that were off by exponential factors in $n$. (B) We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input $\Lambda$ (possibly correlated) random variables in $\{0,1\}^N$ and a short random seed and output a single random variable in $\{0,1\}^N$ that is statistically close to having entropy $(1-\delta) \cdot N$ when one of the $\Lambda$ input variables is distributed uniformly. The seed we require is only $(1/\delta)\cdot \log \Lambda$-bits long, which significantly improves upon previous construction of mergers. (C) Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting $1 - o(1)$ fraction of the min-entropy of the source. The "method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {\em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {\em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses.