arXiv subjects

Mark Zhandry

Publications and source records attributed to Mark Zhandry.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Computational Separation Between Quantum No-cloning and No-telegraphing

Two of the fundamental no-go theorems of quantum information are the no-cloning theorem (that it is impossible to make copies of general quantum states) and the no-teleportation theorem (the prohibition on telegraphing, or sending quantum states over classical channels without pre-shared entanglement). They are known to be equivalent, in the sense that a collection of quantum states is telegraphable if and only if it is clonable. Our main result suggests that this is not the case when computational efficiency is considered. We give a collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable. Given that the opposite scenario is impossible (states that can be telegraphed can always trivially be cloned), this gives the most complete quantum oracle separation possible between these two important no-go properties. We additionally study the complexity class clonableQMA, a subset of QMA whose witnesses are efficiently clonable. As a consequence of our main result, we give a quantum oracle separation between clonableQMA and the class QCMA, whose witnesses are restricted to classical strings. We also propose a candidate oracle-free promise problem separating these classes. We finally demonstrate an application of clonable-but-not-telegraphable states to cryptography, by showing how such states can be used to protect against key exfiltration.

quant-ph↗

arXiv2024

Composability in Watermarking Schemes

Software watermarking allows for embedding a mark into a piece of code, such that any attempt to remove the mark will render the code useless. Provably secure watermarking schemes currently seems limited to programs computing various cryptographic operations, such as evaluating pseudorandom functions (PRFs), signing messages, or decrypting ciphertexts (the latter often going by the name ``traitor tracing''). Moreover, each of these watermarking schemes has an ad-hoc construction of its own. We observe, however, that many cryptographic objects are used as building blocks in larger protocols. We ask: just as we can compose building blocks to obtain larger protocols, can we compose watermarking schemes for the building blocks to obtain watermarking schemes for the larger protocols? We give an affirmative answer to this question, by precisely formulating a set of requirements that allow for composing watermarking schemes. We use our formulation to derive a number of applications.

cs.CR↗

arXiv2024

Quantum State Group Actions

Cryptographic group actions are a leading contender for post-quantum cryptography, and have also been used in the development of quantum cryptographic protocols. In this work, we explore quantum state group actions, which consist of a group acting on a set of quantum states. We show the following results: 1. In certain settings, statistical (even query bounded) security is impossible, analogously to post-quantum classical group actions. 2. We construct quantum state group actions and prove that many computational problems that have been proposed by cryptographers hold it. Depending on the construction, our proofs are either unconditional, rely on LWE, or rely on the quantum random oracle model. While our analysis does not directly apply to classical group actions, we argue it gives at least a sanity check that there are no obvious flaws in the post-quantum assumptions made by cryptographers. 3. Our quantum state group action allows for unifying two existing quantum money schemes: those based on group actions, and those based on non-collapsing hashes. We also explain how they can unify classical and quantum key distribution.

quant-ph↗

arXiv2024

The Space-Time Cost of Purifying Quantum Computations

General quantum computation consists of unitary operations and also measurements. It is well known that intermediate quantum measurements can be deferred to the end of the computation, resulting in an equivalent purely unitary computation. While time efficient, this transformation blows up the space to linear in the running time, which could be super-polynomial for low-space algorithms. Fefferman and Remscrim (STOC'21) and Girish, Raz and Zhan (ICALP'21) show different transformations which are space efficient, but blow up the running time by a factor that is exponential in the space. This leaves the case of algorithms with small-but-super-logarithmic space as incurring a large blowup in either time or space complexity. We show that such a blowup is likely inherent, demonstrating that any "black-box" transformation which removes intermediate measurements must significantly blow up either space or time.

quant-ph↗

arXiv2022

Another Round of Breaking and Making Quantum Money: How to Not Build It from Lattices, and More

Public verification of quantum money has been one of the central objects in quantum cryptography ever since Wiesner's pioneering idea of using quantum mechanics to construct banknotes against counterfeiting. So far, we do not know any publicly-verifiable quantum money scheme that is provably secure from standard assumptions. In this work, we provide both negative and positive results for publicly verifiable quantum money. **In the first part, we give a general theorem, showing that a certain natural class of quantum money schemes from lattices cannot be secure. We use this theorem to break the recent quantum money scheme of Khesin, Lu, and Shor. **In the second part, we propose a framework for building quantum money and quantum lightning we call invariant money which abstracts some of the ideas of quantum money from knots by Farhi et al.(ITCS'12). In addition to formalizing this framework, we provide concrete hard computational problems loosely inspired by classical knowledge-of-exponent assumptions, whose hardness would imply the security of quantum lightning, a strengthening of quantum money where not even the bank can duplicate banknotes. **We discuss potential instantiations of our framework, including an oracle construction using cryptographic group actions and instantiations from rerandomizable functional encryption, isogenies over elliptic curves, and knots.

cs.CR↗

arXiv2022

Commitments to Quantum States

What does it mean to commit to a quantum state? In this work, we propose a simple answer: a commitment to quantum messages is binding if, after the commit phase, the committed state is hidden from the sender's view. We accompany this new definition with several instantiations. We build the first non-interactive succinct quantum state commitments, which can be seen as an analogue of collision-resistant hashing for quantum messages. We also show that hiding quantum state commitments (QSCs) are implied by any commitment scheme for classical messages. All of our constructions can be based on quantum-cryptographic assumptions that are implied by but are potentially weaker than one-way functions. Commitments to quantum states open the door to many new cryptographic possibilities. Our flagship application of a succinct QSC is a quantum-communication version of Kilian's succinct arguments for any language that has quantum PCPs with constant error and polylogarithmic locality. Plugging in the PCP theorem, this yields succinct arguments for NP under significantly weaker assumptions than required classically; moreover, if the quantum PCP conjecture holds, this extends to QMA. At the heart of our security proof is a new rewinding technique for extracting quantum information.

quant-ph↗

arXiv2022

Hidden Cosets and Applications to Unclonable Cryptography

In this work, we study a generalization of hidden subspace states to hidden coset states (first introduced by Aaronson and Christiano [STOC '12]). This notion was considered independently by Vidick and Zhang [Eurocrypt '21], in the context of proofs of quantum knowledge from quantum money schemes. We explore unclonable properties of coset states and several applications: - We show that assuming indistinguishability obfuscation (iO), hidden coset states possess a certain direct product hardness property, which immediately implies a tokenized signature scheme in the plain model. Previously, it was known only relative to an oracle, from a work of Ben-David and Sattath [QCrypt '17]. - Combining a tokenized signature scheme with extractable witness encryption, we give a construction of an unclonable decryption scheme in the plain model. The latter primitive was recently proposed by Georgiou and Zhandry [ePrint '20], who gave a construction relative to a classical oracle. - We conjecture that coset states satisfy a certain natural (information-theoretic) monogamy-of-entanglement property. Assuming this conjecture is true, we remove the requirement for extractable witness encryption in our unclonable decryption construction, by relying instead on compute-and-compare obfuscation for the class of unpredictable distributions. This conjecture was later proved by Culf and Vidick in a follow-up work. - Finally, we give a construction of a copy-protection scheme for pseudorandom functions (PRFs) in the plain model. Our scheme is secure either assuming iO, OWF, and extractable witness encryption, or assuming iO, OWF, compute-and-compare obfuscation for the class of unpredictable distributions, and the conjectured monogamy property mentioned above. This is the first example of a copy-protection scheme with provable security in the plain model for a class of functions that is not evasive.

cs.CR↗

arXiv2022

On the Feasibility of Unclonable Encryption, and More

Unclonable encryption, first introduced by Broadbent and Lord (TQC'20), is a one-time encryption scheme with the following security guarantee: any non-local adversary (A, B, C) cannot simultaneously distinguish encryptions of two equal length messages. This notion is termed as unclonable indistinguishability. Prior works focused on achieving a weaker notion of unclonable encryption, where we required that any non-local adversary (A, B, C) cannot simultaneously recover the entire message m. Seemingly innocuous, understanding the feasibility of encryption schemes satisfying unclonable indistinguishability (even for 1-bit messages) has remained elusive. We make progress towards establishing the feasibility of unclonable encryption. - We show that encryption schemes satisfying unclonable indistinguishability exist unconditionally in the quantum random oracle model. - Towards understanding the necessity of oracles, we present a negative result stipulating that a large class of encryption schemes cannot satisfy unclonable indistinguishability. - Finally, we also establish the feasibility of another closely related primitive: copy-protection for single-bit output point functions. Prior works only established the feasibility of copy-protection for multi-bit output point functions or they achieved constant security error for single-bit output point functions.

cs.CR↗

arXiv2021

Franchised Quantum Money

The construction of public key quantum money based on standard cryptographic assumptions is a longstanding open question. Here we introduce franchised quantum money, an alternative form of quantum money that is easier to construct. Franchised quantum money retains the features of a useful quantum money scheme, namely unforgeability and local verification: anyone can verify banknotes without communicating with the bank. In franchised quantum money, every user gets a unique secret verification key, and the scheme is secure against counterfeiting and sabotage, a new security notion that appears in the franchised model. Finally, we construct franchised quantum money and prove security assuming one-way functions.

cs.CR↗

arXiv2021

Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering

We show polynomial-time quantum algorithms for the following problems: (*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a constant. (*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions, including bounded uniform distributions and Laplace distributions. (*) Extrapolated dihedral coset problem (EDCP) with certain parameters. The SIS, LWE, and EDCP problems in their standard forms are as hard as solving lattice problems in the worst case. However, the variants that we can solve are not in the parameter regimes known to be as hard as solving worst-case lattice problems. Still, no classical or quantum polynomial-time algorithms were known for the variants of SIS and LWE we consider. For EDCP, our quantum algorithm slightly extends the result of Ivanyos et al. (2018). Our algorithms for variants of SIS and EDCP use the existing quantum reductions from those problems to LWE, or more precisely, to the problem of solving LWE given LWE-like quantum states. Our main contribution is solving LWE given LWE-like quantum states with interesting parameters using a filtering technique.

quant-ph↗

arXiv2021

Post-Quantum Succinct Arguments: Breaking the Quantum Rewinding Barrier

We prove that Kilian's four-message succinct argument system is post-quantum secure in the standard model when instantiated with any probabilistically checkable proof and any collapsing hash function (which in turn exist based on the post-quantum hardness of Learning with Errors). This yields the first post-quantum succinct argument system from any falsifiable assumption. At the heart of our proof is a new quantum rewinding procedure that enables a reduction to repeatedly query a quantum adversary for accepting transcripts as many times as desired. Prior techniques were limited to a constant number of accepting transcripts.

cs.CR↗

arXiv2020

New Approaches for Quantum Copy-Protection

Quantum copy protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. Copy protection would be immensely useful, but unfortunately little is known about how to achieve it in general. In this work, we make progress on this goal, by giving the following results: - We show how to copy protect any program that cannot be learned from its input/output behavior, relative to a classical oracle. This improves on Aaronson [CCC'09], which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy protection. -We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public key quantum money. Our construction is general, applicable to many recent watermarking schemes.

cs.CR↗

arXiv2019

On Finding Quantum Multi-collisions

A $k$-collision for a compressing hash function $H$ is a set of $k$ distinct inputs that all map to the same output. In this work, we show that for any constant $k$, $Θ\left(N^{\frac{1}{2}(1-\frac{1}{2^k-1})}\right)$ quantum queries are both necessary and sufficient to achieve a $k$-collision with constant probability. This improves on both the best prior upper bound (Hosoyamada et al., ASIACRYPT 2017) and provides the first non-trivial lower bound, completely resolving the problem.

cs.CR↗

arXiv2018

Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves

We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n >= 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety. Our framework builds a cryptographic invariant map, which is a new primitive closely related to a cryptographic multilinear map, but whose range does not necessarily have a group structure. Nevertheless, we show that a cryptographic invariant map can be used to build several cryptographic primitives, including NIKE, that were previously constructed from multilinear maps and indistinguishability obfuscation.

cs.CR↗

arXiv2017

Quantum Lightning Never Strikes the Same State Twice

Public key quantum money can be seen as a version of the quantum no-cloning theorem that holds even when the quantum states can be verified by the adversary. In this work, investigate quantum lightning, a formalization of "collision-free quantum money" defined by Lutomirski et al. [ICS'10], where no-cloning holds even when the adversary herself generates the quantum state to be cloned. We then study quantum money and quantum lightning, showing the following results: - We demonstrate the usefulness of quantum lightning by showing several potential applications, such as generating random strings with a proof of entropy, to completely decentralized cryptocurrency without a block-chain, where transactions is instant and local. - We give win-win results for quantum money/lightning, showing that either signatures/hash functions/commitment schemes meet very strong recently proposed notions of security, or they yield quantum money or lightning. - We construct quantum lightning under the assumed multi-collision resistance of random degree-2 systems of polynomials. - We show that instantiating the quantum money scheme of Aaronson and Christiano [STOC'12] with indistinguishability obfuscation that is secure against quantum computers yields a secure quantum money scheme

cs.CR↗

arXiv2016

New security notions and feasibility results for authentication of quantum data

We give a new class of security definitions for authentication in the quantum setting. These definitions capture and strengthen existing definitions of security against quantum adversaries for both classical message authentication codes (MACs) and well as full quantum state authentication schemes. The main feature of our definitions is that they precisely characterize the effective behavior of any adversary when the authentication protocol accepts, including correlations with the key. Our definitions readily yield a host of desirable properties and interesting consequences; for example, our security definition for full quantum state authentication implies that the entire secret key can be re-used if the authentication protocol succeeds. Next, we present several protocols satisfying our security definitions. We show that the classical Wegman-Carter authentication scheme with 3-universal hashing is secure against superposition attacks, as well as adversaries with quantum side information. We then present conceptually simple constructions of full quantum state authentication. Finally, we prove a lifting theorem which shows that, as long as a protocol can securely authenticate the maximally entangled state, it can securely authenticate any state, even those that are entangled with the adversary. Thus, this shows that protocols satisfying a fairly weak form of authentication security automatically satisfy a stronger notion of security (in particular, the definition of Dupuis, et al (2012)).

cs.CR↗

arXiv2016

Strong Hardness of Privacy from Weak Traitor Tracing

Despite much study, the computational complexity of differential privacy remains poorly understood. In this paper we consider the computational complexity of accurately answering a family $Q$ of statistical queries over a data universe $X$ under differential privacy. A statistical query on a dataset $D \in X^n$ asks "what fraction of the elements of $D$ satisfy a given predicate $p$ on $X$?" Dwork et al. (STOC'09) and Boneh and Zhandry (CRYPTO'14) showed that if both $Q$ and $X$ are of polynomial size, then there is an efficient differentially private algorithm that accurately answers all the queries, and if both $Q$ and $X$ are exponential size, then under a plausible assumption, no efficient algorithm exists. We show that, under the same assumption, if either the number of queries or the data universe is of exponential size, and the other has size at least $\tilde{O}(n^7)$, then there is no differentially private algorithm that answers all the queries. In both cases, the result is nearly quantitatively tight, since there is an efficient differentially private algorithm that answers $\tildeΩ(n^2)$ queries on an exponential size data universe, and one that answers exponentially many queries on a data universe of size $\tildeΩ(n^2)$. Our proofs build on the connection between hardness results in differential privacy and traitor-tracing schemes (Dwork et al., STOC'09; Ullman, STOC'13). We prove our hardness result for a polynomial size query set (resp., data universe) by showing that they follow from the existence of a special type of traitor-tracing scheme with very short ciphertexts (resp., secret keys), but very weak security guarantees, and then constructing such a scheme.

cs.CR↗

arXiv2015

Quantum Oracle Classification - The Case of Group Structure

The Quantum Oracle Classification (QOC) problem is to classify a function, given only quantum black box access, into one of several classes without necessarily determining the entire function. Generally, QOC captures a very wide range of problems in quantum query complexity. However, relatively little is known about many of these problems. In this work, we analyze the a subclass of the QOC problems where there is a group structure. That is, suppose the range of the unknown function A is a commutative group G, which induces a commutative group law over the entire function space. Then we consider the case where A is drawn uniformly at random from some subgroup A of the function space. Moreover, there is a homomorpism f on A, and the goal is to determine f(A). This class of problems is very general, and covers several interesting cases, such as oracle evaluation; polynomial interpolation, evaluation, and extrapolation; and parity. These problems are important in the study of message authentication codes in the quantum setting, and may have other applications. We exactly characterize the quantum query complexity of every instance of QOC with group structure in terms of a particular counting problem. That is, we provide an algorithm for this general class of problems whose success probability is determined by the solution to the counting problem, and prove its exact optimality. Unfortunately, solving this counting problem in general is a non-trivial task, and we resort to analyzing special cases. Our bounds unify some existing results, such as the existing oracle evaluation and parity bounds. In the case of polynomial interpolation and evaluation, our bounds give new results for secret sharing and information theoretic message authentication codes in the quantum setting.

cs.CC↗