Extraction of uniform randomness from (noisy) non-uniform sources is an important primitive in many security applications, e.g. (pseudo-)random number generators, privacy-preserving biometrics, and key storage based on Physical Unclonable Functions. Generic extraction methods exist, using universal hash functions. There is a trade-off between the length of the extracted bit string and the uniformity of the string. In the literature there are proven lower bounds on this length as a function of the desired uniformity. The best known bound involves a quantity known as smooth min-entropy. Unfortunately, there exist at least three definitions of smooth entropy. In this paper we compare three of these definitions, and we derive improved lower bounds on the extractable randomness.\nWe also investigate the use of almost universal hash functions, which are slightly worse at extracting randomness than universal hash functions, but are preferable in practice because they require far less resources in devices. We show that using them has negligible effect on the extractable randomness.\nKeywords: Leftover Hash Lemma; Universal hash function; Randomness extraction
A fuzzy extractor is a security primitive that allows for reproducible extraction of an almost uniform key from a non-uniform noisy source. We analyze a fuzzy extractor scheme that uses universal hash functions for both information reconciliation and privacy amplification. This is a useful scheme when the number of error patterns likely to occur is limited, regardless of the error probabilities. We derive a sharp bound on the uniformity of the extracted key, making use of the concatenation property of universal hash functions and a recent tight formulation of the leftover hash lemma.
Multi-source extractors are functions that extract uniform randomness from multiple (weak) sources of randomness. Quantum multi-source extractors were considered by Kasher and Kempe (2010) (for the quantum independent adversary and the quantum bounded storage adversary), Chung et al. (2014) (for the general entangled adversary) and Arnon-Friedman et al. (2016) (for the quantum Markov adversary). One of the main objectives of this work is to unify all the existing quantum multi-source adversary models. We propose two new models of adversaries: 1) the quantum measurement adversary (
qma), which generates side information using entanglement and on post-measurement; and 2) the quantum communication adversary (
qca), which generates side information using entanglement and communication between multiple sources. We show that: 1)
qmais the strongest adversary among all the known adversaries, in the sense that the side information of all other adversaries can be generated by
qma; 2) The (generalized) inner-product function (in fact a general class of two-wise independent functions) continues to work as a good extractor with matching parameters as that of Chor and Goldreich (1985) against classical adversaries; 3) A non-malleable extractor proposed by Li (2012) (against classical adversaries) continues to be secure against quantum side information. This result implies a non-malleable extractor result of Aggarwal et al. (2019) with uniform seed. We strengthen their result via a completely different proof to make the non-malleable extractor of Li secure against quantum side information even when the seed is not uniform; 4) A modification (working with weak local randomness instead of uniform local randomness) of the Dodis and Wichs (2009) protocol for privacy-amplification is secure against active quantum adversaries (those who arbitrarily modify the messages exchanged in the protocol). This strengthens on a recent result due to Aggarwal et al. (2019) which uses uniform local randomness; 5) A tight efficiency lower bound for the (generalized) inner-product function (in fact a general class of two-wise independent functions).