Leftover Hash Lemma, Revisited
(2011) CRYPTO 2011 — Location: Santa Barbara (14.August.2011)
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- Abstract
- The famous Leftover Hash Lemma (LHL) states that (almost) universal hash functions are good randomness extractors. Despite its numerous applications, LHL-based extractors suffer from the following two drawbacks: (1) Large Entropy Loss: to extract v bits from distribution X of min-entropy m which are e-close to uniform, one must set v <= m - 2*log(1/e), meaning that the entropy loss L = m-v >= 2*log(1/e). (2) Large Seed Length: the seed length n of (almost) universal hash function required by the LHL must be at least n >= min(u-v, v + 2*log(1/e))-O(1), where u is the length of the source. Quite surprisingly, we show that both limitations of the LHL --- large entropy loss and large seed --- can often be overcome (or, at least, mitigated) in various quite general scenarios. First, we show that entropy loss could be reduced to L=log(1/e) for the setting of deriving secret keys for a wide range of cryptographic applications. Specifically, the security of these schemes gracefully degrades from e to at most e + sqrt(e * 2^{-L}). (Notice that, unlike standard LHL, this bound is meaningful even for negative entropy loss, when we extract more bits than the the min-entropy we have!) Based on these results we build a general *computational extractor* that enjoys low entropy loss and can be used to instantiate a generic key derivation function for *any* cryptographic application. Second, we study the soundness of the natural *expand-then-extract* approach, where one uses a pseudorandom generator (PRG) to expand a short "input seed" S into a longer "output seed" S', and then use the resulting S' as the seed required by the LHL (or, more generally, any randomness extractor). Unfortunately, we show that, in general, expand-then-extract approach is not sound if the Decisional Diffie-Hellman assumption is true. Despite that, we show that it is sound either: (1) when extracting a "small" (logarithmic in the security of the PRG) number of bits; or (2) in *minicrypt*. Implication (2) suggests that the sample-then-extract approach is likely secure when used with "practical" PRGs, despite lacking a reductionist proof of security! Finally, we combine our main results to give a very *simple and efficient* AES-based extractor, which easily supports variable-length messages, and is likely to offer our *improved entropy loss bounds* for any computationally-secure application, despite having a *fixed-length* seed.
- Affiliations
Microsoft Research New England New York University Department of Computer Science IBM Research CWI East China Normal University Department of Computer Science
Citations
Barak, B., Dodis, Y., Krawczyk, H., Pereira, O., Pietrzak, K., Standaert, F.-X., & Yu, Y. (2011). Leftover Hash Lemma, Revisited. In Phillip Rogaway (ed.), Advances in Cryptology - CRYPTO 2011 (p. p. 1-20). https://doi.org/10.1007/978-3-642-22792-9