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We show new and improved list decoding properties of folded Reed--Solomon (RS) codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the source of recent advances in coding theory: folded RS codes were the first known explicit construction of capacity-achieving list decodable codes [V. Guruswami and A. Rudra, IEEE Trans. Inform. Theory, 54 (2008), pp. 135--150], and multiplicity codes were the first construction of high-rate locally decodable codes [S. Kopparty, S. Saraf, and S. Yekhanin, J. ACM, 61 (2014), 28]. In this work, we show that folded RS codes and multiplicity codes are in fact better than previously known in the context of list decoding and local list decoding. Our first main result shows that folded RS codes achieve list decoding capacity with constant list sizes, independent of the block length. Prior work with constant list sizes first obtained list sizes that are polynomial in the block length and relied on pre-encoding with subspace evasive sets to reduce the list sizes to a constant [V. Guruswami and C. Wang, IEEE Trans. Inform. Theory, 59 (2013), pp. 3257--3268], [Z. Dvir and S. Lovett, Proc. 44th STOC, ACM, 2012, 351--358]. The list size we obtain is (1/\varepsilon)O(1/\varepsilon) where \varepsilon is the gap to capacity, which matches the list size obtained by pre-encoding with subspace evasive sets. For our second main result, we observe that univariate multiplicity codes exhibit similar behavior, and we use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance. By known reductions, this gives, in turn, capacity-achieving locally list decodable codes with query complexity exp(\~ O((logN)5/6)). This improves on the tensor-based construction of [B. Hemenway, N. Ron-Zewi, and M. Wootters, SIAM J. Comput., 49 (2019), pp. 157--195], which gave capacity-achieving locally list decodable codes of query complexity N \~ O(1/\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g}N), and is close to the best known query complexity of exp(\~ O(\surd logN)) for high-rate locally (uniquely) decodable codes [S. Kopparty et al., J. ACM, 64 (2017), 11]. [ABSTRACT FROM AUTHOR]

Fuzzy extractors (Dodis et al., SIAM J. Computing 2008) convert repeated noisy readings of a high-entropy secret into the same uniformly distributed key. A minimum condition for the security of the key is the hardness of guessing a value that is similar to the secret, because the fuzzy extractor converts such a guess to the key. We quantify this property in a new notion called fuzzy min-entropy. We ask: is fuzzy min-entropy sufficient to build fuzzy extractors? We provide two answers for different settings. 1) If the construction is provided a description of the probability distribution $W$ that defines the noisy source then fuzzy min-entropy is a sufficient condition for information-theoretic key extraction from $W$. 2) A more ambitious goal is to design a single extractor that works for all possible sources. This more ambitious goal is impossible: there is a family of sources with high fuzzy min-entropy for which no single fuzzy extractor is secure. This is true in three settings: a) for standard fuzzy extractors, b) for fuzzy extractors that are allowed to sometimes be wrong, c) and for secure sketches, which are the main ingredient of most fuzzy extractor constructions. [ABSTRACT FROM AUTHOR]

Fuzzy extractors derive strong keys from noisy sources. Their security is usually defined information-theoretically, with gaps between known negative results, existential constructions, and polynomial-time constructions. We ask whether using computational security can close these gaps. We show the following: • Negative result: Noise tolerance in fuzzy extractors is usually achieved using an information reconciliation component called a secure sketch. We show that secure sketches defined using pseudoentropy (Håstad et al., SIAM J. Comput. 1999) instead of information-theoretic security are still subject to upper bounds from coding theory. • Positive result: We show that our negative result can be avoided by constructing and analyzing a computational fuzzy extractor directly. We modify the code-offset construction (Juels and Wattenberg, CCS 1999) to use random linear codes. Security is based on the Learning with Errors problem and holds when the noisy source is uniform or symbol-fixing (that is, each dimension is either uniform or fixed). [ABSTRACT FROM AUTHOR]