Learning with Errors is easy with quantum samples

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE

Predicate Encryption for Circuits from LWE

In predicate encryption, a ciphertext is associated with descriptive attribute values x in addition to a plaintext μ, and a secret key is associated with a predicate f. Decryption returns plaintext μ if and only if f(x)=1. Moreover, security of predicate encryption guarantees that an adversary learns nothing about the attribute x or the plaintext μ from a ciphertext, given arbitrary many secret keys that are not authorized to decrypt the ciphertext individually. We construct a leveled predicate encryption scheme for all circuits, assuming the hardness of the subexponential learning with errors (LWE) problem. That is, for any polynomial function d=d(λ), we construct a predicate encryption scheme for the class of all circuits with depth bounded by d(λ), where λ is the security parameter.Microsoft Corporation (PhD Fellowship)Northrop Grumman Cybersecurity Research ConsortiumUnited States. Defense Advanced Research Projects Agency (Grant FA8750-11-2-0225)National Science Foundation (U.S.) (Awards CNS-1350619)National Science Foundation (U.S.) (Awards CNS-1413920)Alfred P. Sloan Foundation (Fellowship)Microsoft (Faculty Fellowship

CCA-Secure Deterministic Identity-Based Encryption Scheme

Deterministic public-key encryption, encrypting a plaintext into a unique ciphertext without involving any randomness, was introduced by Bellare, Boldyreva, and O'Neill (CRYPTO 2007) as a realistic alternative to some inherent drawbacks in randomized public-key encryption. Bellare, Kiltz, Peikert and Waters (EUROCRYPT 2012) bring deterministic public-key encryption to the identity-based setting, and propose deterministic identity-based encryption scheme (DIBE). Although the construc- tions of chosen plaintext attack (CPA) secure DIBE scheme have been studied intensively, the construction of chosen ciphertext attack (CCA) secure DIBE scheme is still challenging problems. In this paper, we introduce the notion of identity-based all-but-one trapdoor functions (IB-ABO-TDF), which is an extension version of all-but-one lossy trapdoor function in the public-key setting. We give a instantiation of IB-ABO-TDF under decisional linear assumption. Based on an identity-based lossy trapdoor function and our IB-ABO-TDF, we present a generic construction of CCA-secure DIBE scheme

Equivalence-based Security for Querying Encrypted Databases: Theory and Application to Privacy Policy Audits

Motivated by the problem of simultaneously preserving confidentiality and usability of data outsourced to third-party clouds, we present two different database encryption schemes that largely hide data but reveal enough information to support a wide-range of relational queries. We provide a security definition for database encryption that captures confidentiality based on a notion of equivalence of databases from the adversary's perspective. As a specific application, we adapt an existing algorithm for finding violations of privacy policies to run on logs encrypted under our schemes and observe low to moderate overheads.Comment: CCS 2015 paper technical report, in progres

Random Oracles in a Quantum World

The interest in post-quantum cryptography - classical systems that remain secure in the presence of a quantum adversary - has generated elegant proposals for new cryptosystems. Some of these systems are set in the random oracle model and are proven secure relative to adversaries that have classical access to the random oracle. We argue that to prove post-quantum security one needs to prove security in the quantum-accessible random oracle model where the adversary can query the random oracle with quantum states. We begin by separating the classical and quantum-accessible random oracle models by presenting a scheme that is secure when the adversary is given classical access to the random oracle, but is insecure when the adversary can make quantum oracle queries. We then set out to develop generic conditions under which a classical random oracle proof implies security in the quantum-accessible random oracle model. We introduce the concept of a history-free reduction which is a category of classical random oracle reductions that basically determine oracle answers independently of the history of previous queries, and we prove that such reductions imply security in the quantum model. We then show that certain post-quantum proposals, including ones based on lattices, can be proven secure using history-free reductions and are therefore post-quantum secure. We conclude with a rich set of open problems in this area.Comment: 38 pages, v2: many substantial changes and extensions, merged with a related paper by Boneh and Zhandr

Hiding secrets in public random functions

Constructing advanced cryptographic applications often requires the ability of privately embedding messages or functions in the code of a program. As an example, consider the task of building a searchable encryption scheme, which allows the users to search over the encrypted data and learn nothing other than the search result. Such a task is achievable if it is possible to embed the secret key of an encryption scheme into the code of a program that performs the "decrypt-then-search" functionality, and guarantee that the code hides everything except its functionality. This thesis studies two cryptographic primitives that facilitate the capability of hiding secrets in the program of random functions. 1. We first study the notion of a private constrained pseudorandom function (PCPRF). A PCPRF allows the PRF master secret key holder to derive a public constrained key that changes the functionality of the original key without revealing the constraint description. Such a notion closely captures the goal of privately embedding functions in the code of a random function. Our main contribution is in constructing single-key secure PCPRFs for NC^1 circuit constraints based on the learning with errors assumption. Single-key secure PCPRFs were known to support a wide range of cryptographic applications, such as private-key deniable encryption and watermarking. In addition, we build reusable garbled circuits from PCPRFs. 2. We then study how to construct cryptographic hash functions that satisfy strong random oracle-like properties. In particular, we focus on the notion of correlation intractability, which requires that given the description of a function, it should be hard to find an input-output pair that satisfies any sparse relations. Correlation intractability captures the security properties required for, e.g., the soundness of the Fiat-Shamir heuristic, where the Fiat-Shamir transformation is a practical method of building signature schemes from interactive proof protocols. However, correlation intractability was shown to be impossible to achieve for certain length parameters, and was widely considered to be unobtainable. Our contribution is in building correlation intractable functions from various cryptographic assumptions. The security analyses of the constructions use the techniques of secretly embedding constraints in the code of random functions

New lattice-based protocols for proving correctness of a shuffle

In an electronic voting procedure, mixing networks are used to ensure anonymity of the casted votes. Each node of the network re-encrypts the input and randomly permutes it in a process named shuffle, and must prove that the process was applied honestly. State-of-the-art classical proofs achieve logarithmic communication complexity on N (the number of votes to be shuffled) but they are based on assumptions which are weak against quantum computers. To maintain security in a post-quantum scenario, new proofs are based on different mathematical assumptions, such as lattice-based problems. Nonetheless, the best lattice-based protocols to ensure verifiable shuffling have linear communication complexity on N. In this thesis we propose the first sub-linear post-quantum proof for the correctness of a shuffe, for which we have mainly used two ideas: arithmetic circuit satisfiability and Benes networks to model a permutation of N elements

Quantum homomorphic encryption for circuits of low TT-gate complexity

Fully homomorphic encryption is an encryption method with the property that any computation on the plaintext can be performed by a party having access to the ciphertext only. Here, we formally define and give schemes for quantum homomorphic encryption, which is the encryption of quantum information such that quantum computations can be performed given the ciphertext only. Our schemes allows for arbitrary Clifford group gates, but become inefficient for circuits with large complexity, measured in terms of the non-Clifford portion of the circuit (we use the "$\pi/8$" non-Clifford group gate, which is also known as the $T$-gate). More specifically, two schemes are proposed: the first scheme has a decryption procedure whose complexity scales with the square of the number of $T$-gates (compared with a trivial scheme in which the complexity scales with the total number of gates); the second scheme uses a quantum evaluation key of length given by a polynomial of degree exponential in the circuit's $T$-gate depth, yielding a homomorphic scheme for quantum circuits with constant $T$-depth. Both schemes build on a classical fully homomorphic encryption scheme. A further contribution of ours is to formally define the security of encryption schemes for quantum messages: we define quantum indistinguishability under chosen plaintext attacks in both the public and private-key settings. In this context, we show the equivalence of several definitions. Our schemes are the first of their kind that are secure under modern cryptographic definitions, and can be seen as a quantum analogue of classical results establishing homomorphic encryption for circuits with a limited number of multiplication gates. Historically, such results appeared as precursors to the breakthrough result establishing classical fully homomorphic encryption

Lattice-Based zk-SNARKs from Square Span Programs

Zero-knowledge SNARKs (zk-SNARKs) are non-interactive proof systems with short (i.e., independent of the size of the witness) and efficiently verifiable proofs. They elegantly resolve the juxtaposition of individual privacy and public trust, by providing an efficient way of demonstrating knowledge of secret information without actually revealing it. To this day, zk-SNARKs are widely deployed all over the planet and are used to keep alive a system worth billion of euros, namely the cryptocurrency Zcash. However, all current SNARKs implementations rely on so-called pre-quantum assumptions and, for this reason, are not expected to withstand cryptanalitic efforts over the next few decades. In this work, we introduce a new zk-SNARK that can be instantiated from lattice-based assumptions, and which is thus believed to be post-quantum secure. We provide a generalization in the spirit of Gennaro et al. (Eurocrypt'13) to the SNARK of Danezis et al. (Asiacrypt'14) that is based on Square Span Programs (SSP) and relies on weaker computational assumptions. We focus on designated-verifier proofs and propose a protocol in which a proof consists of just 5 LWE encodings. We provide a concrete choice of parameters, showing that our construction is practically instantiable

Impossibility of Quantum Virtual Black-Box Obfuscation of Classical Circuits

Virtual black-box obfuscation is a strong cryptographic primitive: it encrypts a circuit while maintaining its full input/output functionality. A remarkable result by Barak et al. (Crypto 2001) shows that a general obfuscator that obfuscates classical circuits into classical circuits cannot exist. A promising direction that circumvents this impossibility result is to obfuscate classical circuits into quantum states, which would potentially be better capable of hiding information about the obfuscated circuit. We show that, under the assumption that learning-with-errors (LWE) is hard for quantum computers, this quantum variant of virtual black-box obfuscation of classical circuits is generally impossible. On the way, we show that under the presence of dependent classical auxiliary input, even the small class of classical point functions cannot be quantum virtual black-box obfuscated.Comment: v2: Add the notion of decomposable public keys, which allows our impossibility to hold without assuming circular security for QFHE. We also fix an auxiliary lemma (2.9 in v2) where a square root was missing (this does not influence the main result