Computational Indistinguishability between Quantum States and Its Cryptographic Application

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem, QSCDff, is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational cryptography. As our major contribution, we show that QSCDff has three properties of cryptographic interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff coincides with its worst-case hardness; and (iii) QSCDff is computationally at least as hard as the graph automorphism problem in the worst case. These cryptographic properties enable us to construct a quantum public-key cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomial-time quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail proofs and follow-up of recent wor

Average-Case Complexity

We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question whether the existence of hard-on-average problems in NP can be based on the P$\neq$NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different ``degrees'' of average-case complexity. We discuss some of these ``hardness amplification'' results

Using Simon's Algorithm to Attack Symmetric-Key Cryptographic Primitives

We present new connections between quantum information and the field of classical cryptography. In particular, we provide examples where Simon's algorithm can be used to show insecurity of commonly used cryptographic symmetric-key primitives. Specifically, these examples consist of a quantum distinguisher for the 3-round Feistel network and a forgery attack on CBC-MAC which forges a tag for a chosen-prefix message querying only other messages (of the same length). We assume that an adversary has quantum-oracle access to the respective classical primitives. Similar results have been achieved recently in independent work by Kaplan et al. Our findings shed new light on the post-quantum security of cryptographic schemes and underline that classical security proofs of cryptographic constructions need to be revisited in light of quantum attackers.Comment: 14 pages, 2 figures. v3: final polished version, more formal definitions adde

Cryptography from Information Loss

© Marshall Ball, Elette Boyle, Akshay Degwekar, Apoorvaa Deshpande, Alon Rosen, Vinod. Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving the former.1 The subject of this work is “lossy” reductions, where the reduction loses some information about the input instance. We show that such reductions, when they exist, have interesting and powerful consequences for lifting hardness into “useful” hardness, namely cryptography. Our first, conceptual, contribution is a definition of lossy reductions in the language of mutual information. Roughly speaking, our definition says that a reduction C is t-lossy if, for any distribution X over its inputs, the mutual information I(X; C(X)) ≤ t. Our treatment generalizes a variety of seemingly related but distinct notions such as worst-case to average-case reductions, randomized encodings (Ishai and Kushilevitz, FOCS 2000), homomorphic computations (Gentry, STOC 2009), and instance compression (Harnik and Naor, FOCS 2006). We then proceed to show several consequences of lossy reductions: 1. We say that a language L has an f-reduction to a language L0 for a Boolean function f if there is a (randomized) polynomial-time algorithm C that takes an m-tuple of strings X = (x1, . . ., xm), with each xi ∈ {0, 1}n, and outputs a string z such that with high probability, L0(z) = f(L(x1), L(x2), . . ., L(xm)) Suppose a language L has an f-reduction C to L0 that is t-lossy. Our first result is that one-way functions exist if L is worst-case hard and one of the following conditions holds: f is the OR function, t ≤ m/100, and L0 is the same as L f is the Majority function, and t ≤ m/100 f is the OR function, t ≤ O(m log n), and the reduction has no error This improves on the implications that follow from combining (Drucker, FOCS 2012) with (Ostrovsky and Wigderson, ISTCS 1993) that result in auxiliary-input one-way functions. 2. Our second result is about the stronger notion of t-compressing f-reductions – reductions that only output t bits. We show that if there is an average-case hard language L that has a t-compressing Majority reduction to some language for t = m/100, then there exist collision-resistant hash functions. This improves on the result of (Harnik and Naor, STOC 2006), whose starting point is a cryptographic primitive (namely, one-way functions) rather than average-case hardness, and whose assumption is a compressing OR-reduction of SAT (which is now known to be false unless the polynomial hierarchy collapses). Along the way, we define a non-standard one-sided notion of average-case hardness, which is the notion of hardness used in the second result above, that may be of independent interest

Unsolvability Cores in Classification Problems

Classification problems have been introduced by M. Ziegler as a generalization of promise problems. In this paper we are concerned with solvability and unsolvability questions with respect to a given set or language family, especially with cores of unsolvability. We generalize the results about unsolvability cores in promise problems to classification problems. Our main results are a characterization of unsolvability cores via cohesiveness and existence theorems for such cores in unsolvable classification problems. In contrast to promise problems we have to strengthen the conditions to assert the existence of such cores. In general unsolvable classification problems with more than two components exist, which possess no cores, even if the set family under consideration satisfies the assumptions which are necessary to prove the existence of cores in unsolvable promise problems. But, if one of the components is fixed we can use the results on unsolvability cores in promise problems, to assert the existence of such cores in general. In this case we speak of conditional classification problems and conditional cores. The existence of conditional cores can be related to complexity cores. Using this connection we can prove for language families, that conditional cores with recursive components exist, provided that this family admits an uniform solution for the word problem

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder