Entropy bounds for multiparty device-independent cryptography
Federico Grasselli, Gláucia Murta, Hermann Kampermann, Dagmar Bruß
备注：
Added discussion on the tightness of the entropy bounds, with comparison between analytical bounds and numerical estimations. Added explicit proof of the necessity of genuine multipartite entanglement for the secrecy of a party's outcome
Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks. The security of many multipartite device-independent (DI) protocols, such as DI conference key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality. We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes. From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improve previous results. We obtain the entropy bounds thanks to two general results of independent interest. The first one drastically simplifies the quantum setup of an $N$-partite Bell scenario. The second one provides an upper bound on the violation of the MABK inequality by an arbitrary $N$-qubit state, as a function of the state's parameters.
We propose a scheme for multiparty hierarchical quantum-information splitting (QIS) with a multipartite entangled state, where a boss distributes a secret quantum state to two grades of agents asymmetrically. The agents who belong to different grades have different authorities for recovering boss's secret. Except for boss's Bell-state measurement, no nonlocal operation is involved. The presented scheme is also shown to be secure against eavesdropping. Such a hierarchical QIS is expected to find useful applications in the field of modern multipartite quantum cryptography.
Quantum information science breaks limitations of conventional information transfer, cryptography and computation by using quantum superpositions or entanglement as resources for information processing. Here, we report on the experimental realization of three-party quantum communication protocols using single three-level quantum system (qutrit) communication: secret sharing, detectable Byzantine agreement, and communication complexity reduction for a three-valued function. We have implemented these three schemes using the same optical fiber interferometric setup. Our realization is easily scal…
Methods of quantum mechanics promise information-theoretic security for various protocols in cryptography. However, impossibility of some cryptographic applications such as standard bit commitment, oblivious transfer, multiparty secure computations and ideal coin tossing in quantum regime leaves an obvious question on the completeness of quantum cryptography. Instead of using wide range of rules and techniques for a variety of cryptographic applications, we demonstrate here a unified structure for quantum cryptography based on quantum non-local correlations. The unified framework achieves same…
We derive complementarity relations for arbitrary quantum states of multiparty systems, of arbitrary number of parties and dimensions, between the purity of a part of the system and several correlation quantities, including entanglement and other quantum correlations as well as classical and total correlations, of that part with the remainder of the system. We subsequently use such a complementarity relation, between purity and quantum mutual information in the tripartite scenario, to provide a bound on the secret key rate for individual attacks on a quantum key distribution protocol.
Generalizing Amari's work titled "Information geometry on hierarchy of probability distributions", we define the degrees of irreducible multiparty correlations in a multiparty quantum state based on quantum relative entropy. We prove that these definitions are equivalent to those derived from the maximal von Neaumann entropy principle. Based on these definitions, we find a counterintuitive result on irreducible multiparty correlations: although the degree of the total correlation in a three-party quantum state does not increase under local operations, the irreducible three-party correlation ca…
The oblivious transfer primitive is sufficient to implement secure multiparty computation. However, secure multiparty computation based only on classical cryptography is severely limited by the security and efficiency of the oblivious transfer implementation. We present a method to efficiently and securely generate and distribute oblivious keys by exchanging qubits and by performing commitments using classical hash functions. With the presented hybrid approach, quantum and classical, we obtain a practical and high-speed oblivious transfer protocol, secure even against quantum computer attacks.…
Consider the problem: Alice wishes to send the same key to $n-1$ users (Bob, Carol,. . . , Nathan), while preventing eavesdropper Eve from acquiring information without being detected. The problem has no solution in the classical cryptography but in quantum telecommunication there are some codes to solve the problem. In the paper \cite{zengetall}, Guo-Jyun Zeng, Kuan-Hung Chen, Zhe-Hua Chang, Yu-Shan Yang, and Yao-Hsin Chou from one side and Cabello in \cite{cabello} from other side, used Hadamard gates, Pauli gates in providing the quantum communication code for two-partity telecommunication …
Secret sharing and multiparty computation (also called "secure function evaluation") are fundamental primitives in modern cryptography, allowing a group of mutually distrustful players to perform correct, distributed computations under the sole assumption that some number of them will follow the protocol honestly. This paper investigates how much trust is necessary -- that is, how many players must remain honest -- in order for distributed quantum computations to be possible. We present a verifiable quantum secret sharing (VQSS) protocol, and a general secure multiparty quantum computation (MP…
Here we propose a general relativistic quantum framework for cryptography that exploits the fascinating connection of quantum non-locality and special theory of relativity with cryptography. The underlying principle of unconditional security is causality and two-fold quantum non-local correlations: first entanglement swapping and then teleportation over causally independent entangled systems. We show that the proposed framework has following remarkable and novel features in mistrustful cryptography: (i) It helps in defining a new notion of oblivious transfer where both the data transferred and…
Reliable Entropy Estimation from Observed Statistics for Device-Independent Quantum Cryptography
Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；Institut für Theoretische Physik, Leibniz Universität Hannover（汉诺威莱布尼茨大学理论物理研究所）
We study various distance-like entanglement measures of multipartite states under certain symmetries. Using group averaging techniques we provide conditions under which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equivalent. We consider important classes of multiparty states, and in particular show that these measures are equivalent for all stabilizer states, symmetric basis and antisymmetric basis states. We rigorously prove a conjecture that the closest product state of permutation symmetric states can always be chosen to be …
Digital signatures are widely used in electronic communications to secure important tasks such as financial transactions, software updates, and legal contracts. The signature schemes that are in use today are based on public-key cryptography and derive their security from computational assumptions. However, it is possible to construct unconditionally secure signature protocols. In particular, using quantum communication, it is possible to construct signature schemes with security based on fundamental principles of quantum mechanics. Several quantum signature protocols have been proposed, but n…
Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are thereby relations in terms of the smooth min-entropy for BB84 and six-state encodings. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove a new uncertainty relation in terms of the smooth min-entropy that is only marginally less strong, but has the crucial property that it can be applied to rather small block lengths. This paves the way for a p…
In the classical model of computation, it is well established that one-way functions (OWF) are minimal for computational cryptography: They are essential for almost any cryptographic application that cannot be realized with respect to computationally unbounded adversaries. In the quantum setting, however, OWFs appear not to be essential (Kretschmer 2021; Ananth et al., Morimae and Yamakawa 2022), and the question of whether such a minimal primitive exists remains open. We consider EFI pairs -- efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quant…
Classical and quantum information theory are simply explained. To be more specific it is clarified why Shannon entropy is used as measure of classical information and after a brief review of quantum mechanics it is possible to demonstrate why the density matrix is the main tool of quantum information theory. Then von Neumann entropy is introduced and with its help a great difference between classical and quantum information theory is presented: quantum entanglement. Moreover an information theoretic interpretation of quantum measurement is discussed. Data compression, error correction and nois…
In this article I present a protocol for quantum cryptography which is secure against attacks on individual signals. It is based on the Bennett-Brassard protocol of 1984 (BB84). The security proof is complete as far as the use of single photons as signal states is concerned. Emphasis is given to the practicability of the resulting protocol. For each run of the quantum key distribution the security statement gives the probability of a successful key generation and the probability for an eavesdropper's knowledge, measured as change in Shannon entropy, to be below a specified maximal value.
We give an example of a wide class of problems for which quantum information protocols based on multi-system entanglement can be mapped into much simpler ones involving one system. Secret sharing is a cryptographic primitive which plays a central role in various secure multiparty computation tasks and management of keys in cryptography. In secret sharing protocols, a classical message is divided into shares given to recipient parties in such a way that some number of parties need to collaborate in order to reconstruct the message. Quantum protocols for the task commonly rely on multi-partite G…
Relativistic protocols have been proposed to overcome some impossibility results in classical and quantum cryptography. In such a setting, one takes the location of honest players into account, and uses the fact that information cannot travel faster than the speed of light to limit the abilities of dishonest agents. For example, various relativistic bit commitment protocols have been proposed. Although it has been shown that bit commitment is sufficient to construct oblivious transfer and thus multiparty computation, composing specific relativistic protocols in this way is known to be insecure…
Pseudorandom states, introduced by Ji, Liu and Song (Crypto'18), are efficiently-computable quantum states that are computationally indistinguishable from Haar-random states. One-way functions imply the existence of pseudorandom states, but Kretschmer (TQC'20) recently constructed an oracle relative to which there are no one-way functions but pseudorandom states still exist. Motivated by this, we study the intriguing possibility of basing interesting cryptographic tasks on pseudorandom states. We construct, assuming the existence of pseudorandom state generators that map a $λ$-bit seed to a $ω…
We propose a variational quantum attack algorithm (VQAA) for classical AES-like symmetric cryptography, as exemplified the simplified-data encryption standard (S-DES). In the VQAA, the known ciphertext is encoded as the ground state of a Hamiltonian that is constructed through a regular graph, and the ground state can be found using a variational approach. We designed the ansatz and cost function for the S-DES's variational quantum attack. It is surprising that sometimes the VQAA is even faster than Grove's algorithm as demonstrated by our simulation results. The relationships of the entanglem…
We introduce "binding complexity", a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show t…
Random bit generators (RBGs) are key components of a variety of information processing applications ranging from simulations to cryptography. In particular, cryptographic systems require "strong" RBGs that produce high-entropy bit sequences, but traditional software pseudo-RBGs have very low entropy content and therefore are relatively weak for cryptography. Hardware RBGs yield entropy from chaotic or quantum physical systems and therefore are expected to exhibit high entropy, but in current implementations their exact entropy content is unknown. Here we report a quantum random bit generator (…
In the context of network information theory, one often needs a multiparty probability distribution to be typical in several ways simultaneously. When considering quantum states instead of classical ones, it is in general difficult to prove the existence of a state that is jointly typical. Such a difficulty was recently emphasized and conjectures on the existence of such states were formulated. In this paper, we consider a one-shot multiparty typicality conjecture. The question can then be stated easily: is it possible to smooth the largest eigenvalues of all the marginals of a multipartite st…
The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party to a smaller number $m$ of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with …
To construct a quantum network with many end users, it is critical to have a cost-efficient way to distribute entanglement over different network ends. We demonstrate an entanglement access network, where the expensive resource, the entangled photon source at the telecom wavelength and the core communication channel, is shared by many end users. Using this cost-efficient entanglement access network, we report experimental demonstration of a secure multiparty computation protocol, the privacy-preserving secure sum problem, based on the network quantum cryptography.
Lectures on classical and quantum cryptography. Contents: Private key cryptosystems. Elements of number theory. Public key cryptography and RSA cryptosystem. Shannon`s entropy and mutual information. Entropic uncertainty relations. The no cloning theorem. The BB84 quantum cryptographic protocol. Security proofs. Bell`s theorem. The EPRBE quantum cryptographic protocol.
Using the relative entropy of total correlation, we derive an expression relating the mutual information of $n$-partite pure states to the sum of the mutual informations and entropies of its marginals and analyze some of its implications. Besides, by utilizing the extended strong subadditivity of von Neumann entropy, we obtain generalized monogamy relations for the total correlation in three-partite mixed states. These inequalities lead to a tight lower bound for this correlation in terms of the sum of the bipartite mutual informations. We use this bound to propose a measure for residual three…
We find that a bipartite quantum state is entangled if and only if it is quantum coherent with respect to complete bases of states in the corresponding system that are distinguishable under local quantum operations and classical communication. The corresponding minimal quantum coherence is the entanglement of formation. Connections to the relative entropy of entanglement and quantum coherence, and generalizations to the multiparty case are also considered.
Entropic uncertainty relations are quantitative characterizations of Heisenberg's uncertainty principle, which make use of an entropy measure to quantify uncertainty. In quantum cryptography, they are often used as convenient tools in security proofs. We propose a new entropic uncertainty relation. It is the first such uncertainty relation that lower bounds the uncertainty in the measurement outcome for all but one choice for the measurement from an arbitrarily large (but specifically chosen) set of possible measurements, and, at the same time, uses the min-entropy as entropy measure, rather t…
We present first measure of quantum correlation of an ensemble of multiparty states. It is based on the idea of minimal entropy production in a locally distinguishable basis measurement. It is shown to be a relative entropy distance from a set of ensembles. For bipartite ensembles, which span the whole bipartite Hilbert space, the measure is bounded below by average relative entropy of entanglement. We naturally obtain a monotonicity axiom for any measure of quantum correlation of ensembles. We evaluate this measure for certain cases. Subsequently we use this measure to propose a complementari…
Oblivious transfer is an important primitive in modern cryptography. Applications include secure multiparty computation, oblivious sampling, e-voting, and signatures. Information-theoretically secure perfect 1-out-of 2 oblivious transfer is impossible to achieve. Imperfect variants, where both participants' ability to cheat is still limited, are possible using quantum means while remaining classically impossible. Precisely what security parameters are attainable remains unknown. We introduce a theoretical framework for studying semirandom quantum oblivious transfer, which is shown to be equiva…
Device-independent quantum cryptography allows security even if the devices used to execute the protocol are untrusted - whether this is due to unknown imperfections in the implementation, or because the adversary himself constructed them to subvert the security of the protocol. While device-independence has seen much attention in the domain of quantum key distribution, relatively little is known for general protocols. Here we introduce a new model for device-independence for two-party protocols and position verification in the noisy-storage model. For the first time, we show that such protoco…
Random numbers have significant applications in fundamental science, high-level scientific research, cryptography, and several other areas where there is a pressing need for high-quality random numbers. We present an experimental demonstration of a non-deterministic random bit generator from a quantum entropy source and a deterministic random bit generator mechanism to provide high quality random numbers providing a throughput of 1 Gb. Quantum entropy is realized by a series of quantum chips based on radioactive isotope Americium-241. The extracted raw random numbers are further post-processed…
In multiparty quantum systems, the monogamy inequality proposes an upper bound on the distribution of bipartite quantum correlation between a single party and each of the remaining parties in the system, in terms of the amount of quantum correlation shared by that party with the rest of the system taken as a whole. However, it is well-known that not all quantum correlation measures universally satisfy the monogamy inequality. In this work, we aim at determining the non-trivial value by which the monogamy inequality can be violated by a quantum correlation measure. Using an information-theoreti…
In this Thesis, several results in quantum information theory are collected, most of which use entropy as the main mathematical tool. *While a direct generalization of the Shannon entropy to density matrices, the von Neumann entropy behaves differently. A long-standing open question is, whether there are quantum analogues of unconstrained non-Shannon type inequalities. Here, a new constrained non-von-Neumann type inequality is proven, a step towards a conjectured unconstrained inequality by Linden and Winter. *IID quantum state merging can be optimally achieved using the decoupling technique. …
This thesis initiates the study of cryptographic protocols in the bounded-quantum-storage model. On the practical side, simple protocols for Rabin Oblivious Transfer, 1-2 Oblivious Transfer and Bit Commitment are presented. No quantum memory is required for honest players, whereas the protocols can only be broken by an adversary controlling a large amount of quantum memory. The protocols are efficient, non-interactive and can be implemented with today's technology. On the theoretical side, new entropic uncertainty relations involving min-entropy are established and used to prove the security o…
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entan…
We initiate the study of two-party cryptographic primitives with unconditional security, assuming that the adversary's quantum memory is of bounded size. We show that oblivious transfer and bit commitment can be implemented in this model using protocols where honest parties need no quantum memory, whereas an adversarial player needs quantum memory of size at least n/2 in order to break the protocol, where n is the number of qubits transmitted. This is in sharp contrast to the classical bounded-memory model, where we can only tolerate adversaries with memory of size quadratic in honest players'…
Uncertainty relations lie at the very core of quantum mechanics, and form the cornerstone of essentially all quantum cryptographic applications. In particular, they play an important role in cryptographic protocols in the bounded-quantum-storage model, where proving the security of all existing protocols ultimately reduces to bounding such relations. Yet, very little is known about such uncertainty relations for more than two measurements. Here, we prove optimal entropic uncertainty relations for anti-commuting binary observables for the Shannon entropy, and nearly optimal relations for the co…
A central question for causal inference is to decide whether a set of correlations fit a given causal structure. In general, this decision problem is computationally infeasible and hence several approaches have emerged that look for certificates of compatibility. Here we review several such approaches based on entropy. We bring together the key aspects of these entropic techniques with unified terminology, filling several gaps and establishing new connections regarding their relation, all illustrated with examples. We consider cases where unobserved causes are classical, quantum and post-quant…
From the output produced by a memoryless deletion channel with a uniformly random input of known length $n$, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length $m$. We first conjecture on the basis of experimental data that the entropy of the posterior is minimized by the constant strings $\texttt{000}\ldots$, $\texttt{111}\ldots$ and maximized by the alternating strings $\texttt{0101}\ldots$, …
We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the poss…
Random numbers are a fundamental resource in science and engineering with important applications in simulation and cryptography. The inherent randomness at the core of quantum mechanics makes quantum systems a perfect source of entropy. Quantum random number generation is one of the most mature quantum technologies with many alternative generation methods. We discuss the different technologies in quantum random number generation from the early devices based on radioactive decay to the multiple ways to use the quantum states of light to gather entropy from a quantum origin. We also discuss rand…
