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The Restricted Congruences Toolbox
Published in Khodakhast Bibak, Restricted Congruences in Computing, 2020
Motivated by applications to cryptography, a notion of Δ-universality was introduced [109, 162, 188]. Suppose that R is an Abelian group. We say that H is a Δ-universal family of hash functions if the probability, over a random h ∈ H, that two distinct keys in D hash to values that are distance b apart for any b in R is 1|R|. Note that the case b = 0 corresponds to universality. Furthermore, we say that H is ɛ-almost-Δ-universal (ɛ-AΔU) if this probability is at most ɛ, 1|R|≤ε<1. We remark that ɛ-AΔU families have applications to message authentication. Informally, it is possible to design a message authentication scheme using ɛ-AΔU families such that two parties can exchange signed messages over an unreliable channel and the probability that an adversary can forge a valid signed message to be sent across the channel is at most ɛ [78]. Also, the well-known leftover hash lemma states that (almost) universal hash functions are good randomness extractors.
Efficient oblivious transfer construction via multiple bits dual-mode cryptosystem for secure selection in the cloud
Published in Journal of the Chinese Institute of Engineers, 2019
Zengpeng Li, Chunguang Ma, Minghao Zhao, Chang Choi
The truncated discrete Gaussian distribution over is denoted by with parameter .If the norm exceeds the bound then we use to replace the output, where is -bounded. Below, we show a variant of the leftover hash lemma (LHL) first proposed by Impagliazzo, Levin, and Luby (1989). Lemma 2.1 ([Brakerski and Vaikuntanathan 2014] Lemma 2.1)Consider the parameters, , , and, and sample a random matrix. Then the statistical distance between the distributionsandis as follows:Lemma 2.2 (Trapdoor Generation [Gentry, Peikert, and Vaikuntanathan 2008; Micciancio and Peikert 2012)) There exists an efficient trapdoor generation algorithmthat takes,,andas input and outputs,wherefor all. Moreover, the statistical distance betweenand uniform distribution is withinand.Definition 2.3 (Decision-) Given independent samples , each one is distributed according to either: (1) the distribution over , .i.e. for the uniform public vector , secret vector , and error element or (2) the uniform distribution. Also, there exists an adversary who can distinguish the above two cases with non-negligible advantage.