Storage-Rate Trade-off in A-XPIR

We consider the storage problem in an asymmetric $X$-secure private information retrieval (A-XPIR) setting. The A-XPIR setting considers the $X$-secure PIR problem (XPIR) when a given arbitrary set of servers is communicating. We focus on the trade-off region between the average storage at the servers and the average download cost. In the case of $N=4$ servers and two non-overlapping sets of communicating servers with $K=2$ messages, we characterize the achievable region and show that the three main inequalities compared to the no-security case collapse to two inequalities in the asymmetric security case. In the general case, we derive bounds that need to be satisfied for the general achievable region for an arbitrary number of servers and messages. In addition, we provide the storage and retrieval scheme for the case of $N=4$ servers with $K=2$ messages and two non-overlapping sets of communicating servers, such that the messages are not replicated (in the sense of a coded version of each symbol) and at the same time achieve the optimal achievable rate for the case of replication. Finally, we derive the exact capacity for the case of asymmetric security and asymmetric collusion for $N=4$ servers, with the communication links $\{1,2\}$ and $\{3,4\}$, which splits the servers into two groups, i.e., $g=2$, and with the collusion links $\{1,3\}$, $\{2,4\}$, as $C=\frac{1}{3}$. More generally, we derive a capacity result for a certain family of asymmetric collusion and asymmetric security cases.

Convergence Properties of Good Quantum Codes for Classical Communication

An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.

Breaking the Storage-Bandwidth Tradeoff in Distributed Storage with Quantum Entanglement

This work investigates the use of quantum resources in distributed storage systems. Consider an $(n,k,d)$ distributed storage system in which a file is stored across $n$ nodes such that any $k$ nodes suffice to reconstruct the file. When a node fails, any $d$ helper nodes transmit information to a newcomer to rebuild the system. In contrast to the classical repair, where helper nodes transmit classical bits, we allow them to send classical information over quantum channels to the newcomer. The newcomer then generates its storage by performing appropriate measurements on the received quantum states. In this setting, we fully characterize the fundamental tradeoff between storage and repair bandwidth (total communication cost). Compared to classical systems, the optimal storage--bandwidth tradeoff can be significantly improved with the enhancement of quantum entanglement shared only among the surviving nodes, particularly at the minimum-storage regenerating point. Remarkably, we show that when $d \geq 2k-2$, there exists an operating point at which \textit{both storage and repair bandwidth are simultaneously minimized}. This phenomenon breaks the tradeoff in the classical setting and reveals a fundamentally new regime enabled by quantum communication.

The Asymptotic Capacity of Byzantine Symmetric Private Information Retrieval and Its Consequences

We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) with $B$ Byzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In \cite{byzantine_tpir}, where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In \cite{unresponsive_byzantine_1}, Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when \emph{symmetric} privacy is required. To that end, we define Byzantine servers, inspired by \cite{byzantine_tpir}, as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.

Entanglement-Assisted Coding for Arbitrary Linear Computations Over a Quantum MAC

We study a linear computation problem over a quantum multiple access channel (LC-QMAC), where $S$ servers share an entangled state and separately store classical data streams $W_1,\cdots, W_S$ over a finite field $\mathbb{F}_d$. A user aims to compute $K$ linear combinations of these data streams, represented as $Y = \mathbf{V}_1 W_1 + \mathbf{V}_2 W_2 + \cdots + \mathbf{V}_S W_S \in \mathbb{F}_d^{K \times 1}$. To this end, each server encodes its classical information into its local quantum subsystem and transmits it to the user, who retrieves the desired computations via quantum measurements. In this work, we propose an achievable scheme for LC-QMAC based on the stabilizer formalism and the ideas from entanglement-assisted quantum error-correcting codes (EAQECC). Specifically, given any linear computation matrix, we construct a self-orthogonal matrix that can be implemented using the stabilizer formalism. Also, we apply precoding matrices to minimize the number of auxiliary qudits required. Our scheme achieves more computations per qudit, i.e., a higher computation rate, compared to the best-known methods in the literature, and attains the capacity in certain cases.

Byzantine-Eavesdropper Alliance: How to Achieve Symmetric Privacy in Quantum $X$-Secure $B$-Byzantine $E$-Eavesdropped $U$-Unresponsive $T$-Colluding PIR?

We consider the quantum \emph{symmetric} private information retrieval (QSPIR) problem in a system with $N$ databases and $K$ messages, with $U$ unresponsive servers, $T$-colluding servers, and $X$-security parameter, under several fundamental threat models. In the first model, there are $\mathcal{E}_1$ eavesdropped links in the uplink direction (the direction from the user to the $N$ servers), $\mathcal{E}_2$ eavesdropped links in the downlink direction (the direction from the servers to the user), where $|\mathcal{E}_1|, |\mathcal{E}_2| \leq E$; we coin this eavesdropper setting as \emph{dynamic} eavesdroppers. We show that super-dense coding gain can be achieved for some regimes. In the second model, we consider the case with Byzantine servers, i.e., servers that can coordinate to devise a plan to harm the privacy and security of the system together with static eavesdroppers, by listening to the same links in both uplink and downlink directions. It is important to note the considerable difference between the two threat models, since the eavesdroppers can take huge advantage of the presence of the Byzantine servers. Unlike the previous works in SPIR with Byzantine servers, that assume that the Byzantine servers can send only random symbols independent of the stored messages, we follow the definition of Byzantine servers in \cite{byzantine_tpir}, where the Byzantine servers can send symbols that can be functions of the storage, queries, as well as the random symbols in a way that can produce worse harm to the system. In the third and the most novel threat model, we consider the presence of Byzantine servers and dynamic eavesdroppers together. We show that having dynamic eavesdroppers along with Byzantine servers in the same system model creates more threats to the system than having static eavesdroppers with Byzantine servers.

Private Counterfactual Retrieval With Immutable Features

In a classification task, counterfactual explanations provide the minimum change needed for an input to be classified into a favorable class. We consider the problem of privately retrieving the exact closest counterfactual from a database of accepted samples while enforcing that certain features of the input sample cannot be changed, i.e., they are \emph{immutable}. An applicant (user) whose feature vector is rejected by a machine learning model wants to retrieve the sample closest to them in the database without altering a private subset of their features, which constitutes the immutable set. While doing this, the user should keep their feature vector, immutable set and the resulting counterfactual index information-theoretically private from the institution. We refer to this as immutable private counterfactual retrieval (I-PCR) problem which generalizes PCR to a more practical setting. In this paper, we propose two I-PCR schemes by leveraging techniques from private information retrieval (PIR) and characterize their communication costs. Further, we quantify the information that the user learns about the database and compare it for the proposed schemes.

Private Counterfactual Retrieval

Transparency and explainability are two extremely important aspects to be considered when employing black-box machine learning models in high-stake applications. Providing counterfactual explanations is one way of catering this requirement. However, this also poses a threat to the privacy of both the institution that is providing the explanation as well as the user who is requesting it. In this work, we propose multiple schemes inspired by private information retrieval (PIR) techniques which ensure the \emph{user's privacy} when retrieving counterfactual explanations. We present a scheme which retrieves the \emph{exact} nearest neighbor counterfactual explanation from a database of accepted points while achieving perfect (information-theoretic) privacy for the user. While the scheme achieves perfect privacy for the user, some leakage on the database is inevitable which we quantify using a mutual information based metric. Furthermore, we propose strategies to reduce this leakage to achieve an advanced degree of database privacy. We extend these schemes to incorporate user's preference on transforming their attributes, so that a more actionable explanation can be received. Since our schemes rely on finite field arithmetic, we empirically validate our schemes on real datasets to understand the trade-off between the accuracy and the finite field sizes.

Quantum $X$-Secure $B$-Byzantine $T$-Colluding Private Information Retrieval

We consider the problems arising from the presence of Byzantine servers in a quantum private information retrieval (QPIR) setting. This is the first work to precisely define what the capabilities of Byzantine servers could be in a QPIR context. We show that quantum Byzantine servers have more capabilities than their classical counterparts due to the possibilities created by the quantum encoding procedure. We focus on quantum Byzantine servers that can apply any reversible operations on their individual qudits. In this case, the Byzantine servers can generate any error, i.e., this covers \emph{all} possible single qudit operations that can be done by the Byzantine servers on their qudits. We design a scheme that is resilient to these kinds of manipulations. We show that the scheme designed achieves superdense coding gain in all cases, i.e., $R_Q= \max \left\{0,\min\left\{1,2\left(1-\frac{X+T+2B}{N}\right)\right\}\right\}$.

Quantum Private Membership Aggregation

We consider the problem of private set membership aggregation of $N$ parties by using an entangled quantum state. In this setting, the $N$ parties, which share an entangled state, aim to \emph{privately} know the number of times each element (message) is repeated among the $N$ parties, with respect to a universal set $\mathcal{K}$. This problem has applications in private comparison, ranking, voting, etc. We propose an encoding algorithm that maps the classical information into distinguishable quantum states, along with a decoding algorithm that exploits the distinguishability of the mapped states. The proposed scheme can also be used to calculate the $N$ party private summation modulo $P$.