Adiabatic Quantum State Generation and Statistical Zero Knowledge

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'. This approach provides intriguing links between many different areas: quantum computation, adiabatic evolution, analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing Markov chains, the complexity class statistical zero knowledge, quantum random walks, and more. We first show that many natural candidates for quantum algorithms can be cast as a state generation problem. We define a paradigm for state generation, called 'adiabatic state generation' and develop tools for adiabatic state generation which include methods for implementing very general Hamiltonians and ways to guarantee non negligible spectral gaps. We use our tools to prove that adiabatic state generation is equivalent to state generation in the standard quantum computing model, and finally we show how to apply our techniques to generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure

An Improved BKW Algorithm for LWE with Applications to Cryptography and Lattices

In this paper, we study the Learning With Errors problem and its binary variant, where secrets and errors are binary or taken in a small interval. We introduce a new variant of the Blum, Kalai and Wasserman algorithm, relying on a quantization step that generalizes and fine-tunes modulus switching. In general this new technique yields a significant gain in the constant in front of the exponent in the overall complexity. We illustrate this by solving p within half a day a LWE instance with dimension n = 128, modulus $q = n^2$, Gaussian noise $\alpha = 1/(\sqrt{n/\pi} \log^2 n)$ and binary secret, using $2^{28}$ samples, while the previous best result based on BKW claims a time complexity of $2^{74}$ with $2^{60}$ samples for the same parameters. We then introduce variants of BDD, GapSVP and UniqueSVP, where the target point is required to lie in the fundamental parallelepiped, and show how the previous algorithm is able to solve these variants in subexponential time. Moreover, we also show how the previous algorithm can be used to solve the BinaryLWE problem with n samples in subexponential time $2^{(\ln 2/2+o(1))n/\log \log n}$. This analysis does not require any heuristic assumption, contrary to other algebraic approaches; instead, it uses a variant of an idea by Lyubashevsky to generate many samples from a small number of samples. This makes it possible to asymptotically and heuristically break the NTRU cryptosystem in subexponential time (without contradicting its security assumption). We are also able to solve subset sum problems in subexponential time for density $o(1)$, which is of independent interest: for such density, the previous best algorithm requires exponential time. As a direct application, we can solve in subexponential time the parameters of a cryptosystem based on this problem proposed at TCC 2010.Comment: CRYPTO 201

Zero-Knowledge Password Policy Check from Lattices

Passwords are ubiquitous and most commonly used to authenticate users when logging into online services. Using high entropy passwords is critical to prevent unauthorized access and password policies emerged to enforce this requirement on passwords. However, with current methods of password storage, poor practices and server breaches have leaked many passwords to the public. To protect one's sensitive information in case of such events, passwords should be hidden from servers. Verifier-based password authenticated key exchange, proposed by Bellovin and Merrit (IEEE S\&P, 1992), allows authenticated secure channels to be established with a hash of a password (verifier). Unfortunately, this restricts password policies as passwords cannot be checked from their verifier. To address this issue, Kiefer and Manulis (ESORICS 2014) proposed zero-knowledge password policy check (ZKPPC). A ZKPPC protocol allows users to prove in zero knowledge that a hash of the user's password satisfies the password policy required by the server. Unfortunately, their proposal is not quantum resistant with the use of discrete logarithm-based cryptographic tools and there are currently no other viable alternatives. In this work, we construct the first post-quantum ZKPPC using lattice-based tools. To this end, we introduce a new randomised password hashing scheme for ASCII-based passwords and design an accompanying zero-knowledge protocol for policy compliance. Interestingly, our proposal does not follow the framework established by Kiefer and Manulis and offers an alternate construction without homomorphic commitments. Although our protocol is not ready to be used in practice, we think it is an important first step towards a quantum-resistant privacy-preserving password-based authentication and key exchange system

On the hardness of the shortest vector problem

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 77-84).An n-dimensional lattice is the set of all integral linear combinations of n linearly independent vectors in Rm. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest non-zero vector in it. We prove that the shortest vector problem is NP-hard (for randomized reductions) to approximate within some constant factor greater than 1 in any 1, norm (p >\=1). In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2. The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an n-dimensional ball of radius greater than 1 + [square root of] 2 grows exponentially in n, and a new constructive version of Sauer's lemma (a combinatorial result somehow related to the notion of VC-dimension) is presented, considerably simplifying all previously known constructions.by Daniele Micciancio.Ph.D

Lattice-Based Techniques for Accountable Anonymity: Composition of Abstract Stern’s Protocols and Weak PRF with Efficient Protocols from LWR

In an accountable anonymous system, a user is guaranteed anonymity and unlinkability unless some well-defined condition is met. A line of research focus on schemes that do not rely on any trusted third party capable of de-anonymising users. Notable examples include $k$-times anonymous authentication ($k$-TAA), blacklistable anonymous credentials (BLAC) and linkable ring signatures (LRS). All instances of these schemes are based on traditional number theoretic assumptions, which are vulnerable to quantum attacks. One common feature of these schemes is the need to limit the number of times a key can be (mis-)used. Traditionally, it is usually achieved through the use of a pseudorandom function (PRF) which maps a user\u27s key to a pseudonym, along with a proof of correctness. However, existing lattice-based PRFs do not interact well with zero-knowledge proofs. To bridge this gap, we propose and develop the following techniques and primitives: We formalize the notion of weak PRF with efficient protocols, which allows a prover to convince a verifier that the function $\mathsf{F}$ is evaluated correctly. Specifically, we provide an efficient construction based on the learning with rounding problem, which uses abstract Stern\u27s Protocol to prove $y = \mathsf{F}_k(x)$ and $y \neq \mathsf{F}_k(x)$ without revealing $x$, $y$ or $k$. We develop a general framework, which we call extended abstract Stern\u27s protocol, to construct zero-knowledge arguments system for statements formed by conjunction and disjunction of sub-statements, who (or whose variants) are provable using abstract Stern\u27s Protocol. Specifically, our system supports arbitrary monotonic propositions and allows a prover to argue polynomial relationships of the witnesses used in these sub-statements. As many existing lattice-based primitives also admit proofs using abstract Stern\u27s protocol, our techniques can easily glue different primitives together for privacy-enhancing applications in a simple and clean way. Indeed, we propose three new schemes, all of which are the first of its kind, in the lattice setting. They also enjoy additional advantages over instances of the number-theoretic counterpart. Our $k$-TAA and BLAC schemes support concurrent enrollment while our LRS features logarithmic signature size without relying on a trusted setup. Our techniques enrich the arsenal of privacy-enhancing techniques and could be useful in the constructions of other schemes such as e-cash, unique group signatures, public key encryption with verifiable decryption, etc