We propose a natural relaxation of differential privacy based on the Renyi divergence. Closely related notions have appeared in several recent papers that analyzed composition of differentially private mechanisms. We argue that the useful analytical tool can be used as a privacy definition, compactly and accurately representing guarantees on the tails of the privacy loss.We demonstrate that the new definition shares many important properties with the standard definition of differential privacy, while additionally allowing tighter analysis of composite heterogeneous mechanisms.

Rényi Differential Privacy

Quantum cryptography exploits principles of quantum physics for the secure processing of information. A prominent example is secure communication, i.e., the task of transmitting confidential messages from one location to another. The cryptographic requirement here is that the transmitted messages remain inaccessible to anyone other than the designated recipients, even if the communication channel is untrusted. In classical cryptography, this can usually only be guaranteed under computational hardness assumptions, e.g., that factoring large integers is infeasible. In contrast, the security of quantum cryptography relies entirely on the laws of quantum mechanics. Here we review this physical notion of security, focusing on quantum key distribution and secure communication.

Security in Quantum Cryptography.

We propose the concept of pseudorandom quantum states, which appear random to any quantum polynomial-time adversary. It offers a computational approximation to perfectly random quantum states analogous in spirit to cryptographic pseudorandom generators, as opposed to statistical notions of quantum pseudorandomness that have been studied previously, such as quantum t-designs analogous to t-wise independent distributions.

Pseudorandom Quantum States

The existence of secure indistinguishability obfuscators ($i\mathcal {O}$) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing $i\mathcal {O}$ rely on d-linear maps. While secure bilinear maps are well established in cryptographic literature, the security of candidates for $d>2$ is poorly understood.

Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification

The existence of secure indistinguishability obfuscators (iO) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing iO rely on d-linear maps which allow the encoding of elements from a large domain, evaluating degree d polynomials on them, and testing if the output is zero. While secure bilinear maps are well established in cryptographic literature, the security of candidates for d > 2 is poorly understood. We propose a new approach to constructing iO for general circuits. Unlike all previously known realizations of iO, we avoid the use of d-linear maps of degree d ≥ 3. At the heart of our approach is the assumption that a new weak pseudorandom object exists, that we call a perturbation resilient generator (∆RG). Informally, a ∆RG maps n integers to m integers, and has the property that for any sufficiently short vector a ∈ Z, all efficient adversaries must fail to distinguish the distributions ∆RG(s) and (∆RG(s)+a), with at least some probability that is inverse polynomial in the security parameter. We require that the ∆RG be computable by some family of low degree polynomials over Z. We use techniques building upon the Dense Model Theorem to deal with adversaries that have nontrivial but non-overwhelming distinguishing advantage. As a result, we obtain iO for general circuits assuming: • Subexponentially secure LWE • Bilinear Maps • (1− 1/poly(λ))-secure 3-block-local PRGs • 1/poly(λ)-secure ∆RGs

/pdf/indistinguishability-obfuscation-without-multilinear-maps-io-1dk5rto805.pdf

Indistinguishability Obfuscation Without Multilinear Maps: iO from LWE, Bilinear Maps, and Weak Pseudorandomness.

We construct a simulator for the simulating auxiliary input problem with complexity better than all previous results and prove the optimality up to logarithmic factors by establishing a black-box lower bound. Specifically, let $\ell $ be the length of the auxiliary input and $\epsilon $ be the indistinguishability parameter. Our simulator is $\tilde{O}(2^{\ell }\epsilon ^{-2})$ more complicated than the distinguisher family. For the lower bound, we show the relative complexity to the distinguisher of a simulator is at least $\varOmega (2^{\ell }\epsilon ^{-2})$ assuming the simulator is restricted to use the distinguishers in a black-box way and satisfy a mild restriction.

/pdf/on-the-complexity-of-simulating-auxiliary-input-154ptfg9i5.pdf

On the Complexity of Simulating Auxiliary Input

Differential privacy is a mathematical definition of privacy for statistical data analysis. It guarantees that any possibly adversarial data analyst is unable to learn too much information that is specific to an individual. Mironov et al.i¾?CRYPTO 2009 proposed several computational relaxations of differential privacy CDP, which relax this guarantee to hold only against computationally bounded adversaries. Their work and subsequent work showed that CDP can yield substantial accuracy improvements in various multiparty privacy problems. However, these works left open whether such improvements are possible in the traditional client-server model of data analysis. In fact, Groce, Katz and Yerukhimovichi¾?TCC 2011 showed that, in this setting, it is impossible to take advantage of CDP for many natural statistical tasks.

Our main result shows that, assuming the existence of sub-exponentially secure one-way functions and 2-message witness indistinguishable proofs zaps for $$\mathbf {NP}$$ , that there is in fact a computational task in the client-server model that can be efficiently performed with CDP, but is infeasible to perform with information-theoretic differential privacy.

/pdf/separating-computational-and-statistical-differential-1taery6ob5.pdf

Separating Computational and Statistical Differential Privacy in the Client-Server Model

We initiate the study of computational entropy in the quantum setting. We investigate to what extent the classical notions of computational entropy generalize to the quantum setting, and whether quantum analogues of classical theorems hold. Our main results are as follows. (1) The classical Leakage Chain Rule for pseudoentropy can be extended to the case that the leakage information is quantum (while the source remains classical). Specifically, if the source has pseudoentropy at least $k$, then it has pseudoentropy at least $k-\ell$ conditioned on an $\ell$-qubit leakage. (2) As an application of the Leakage Chain Rule, we construct the first quantum leakage-resilient stream-cipher in the bounded-quantum-storage model, assuming the existence of a quantum-secure pseudorandom generator. (3) We show that the general form of the classical Dense Model Theorem (interpreted as the equivalence between two definitions of pseudo-relative-min-entropy) does not extend to quantum states. Along the way, we develop quantum analogues of some classical techniques (e.g. the Leakage Simulation Lemma, which is proven by a Non-uniform Min-Max Theorem or Boosting). On the other hand, we also identify some classical techniques (e.g. Gap Amplification) that do not work in the quantum setting. Moreover, we introduce a variety of notions that combine quantum information and quantum complexity, and this raises several directions for future work.

Computational Notions of Quantum Min-Entropy.

We introduce hardness in relative entropy, a new notion of hardness for search problems which on the one hand is satisfied by all one-way functions and on the other hand implies both next-block pseudoentropy and inaccessible entropy, two forms of computational entropy used in recent constructions of pseudorandom generators and statistically hiding commitment schemes, respectively. Thus, hardness in relative entropy unifies the latter two notions of computational entropy and sheds light on the apparent “duality” between them. Additionally, it yields a more modular and illuminating proof that one-way functions imply next-block inaccessible entropy, similar in structure to the proof that one-way functions imply next-block pseudoentropy (Vadhan and Zheng, STOC ‘12).

Unifying computational entropies via Kullback-Leibler divergence

We study entropy flattening: Given a circuit CX implicitly describing an n-bit source X (namely, X is the output of CX on a uniform random input), construct another circuit CY describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have CY evaluate CX altogether Θ(n2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit CY for entropy flattening that repeatedly queries CX as an oracle requires ω(n2) queries.Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [12, 22, 13, 6, 11, 10, 7, 24]. It is also used in reductions between problems complete for statistical zero-knowledge [19, 23, 4, 25]. The Θ(n2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

https://drops.dagstuhl.de/opus/volltexte/2018/8866/pdf/LIPIcs-CCC-2018-23.pdf/

A tight lower bound for entropy flattening

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