Syed A. Jafar

Sennur Ulukus, Salman Avestimehr, Michael Gastpar et al.

Most of our lives are conducted in the cyberspace. The human notion of privacy translates into a cyber notion of privacy on many functions that take place in the cyberspace. This article focuses on three such functions: how to privately retrieve information from cyberspace (privacy in information retrieval), how to privately leverage large-scale distributed/parallel processing (privacy in distributed computing), and how to learn/train machine learning models from private data spread across multiple users (privacy in distributed (federated) learning). The article motivates each privacy setting, describes the problem formulation, summarizes breakthrough results in the history of each problem, and gives recent results and discusses some of the major ideas that emerged in each field. In addition, the cross-cutting techniques and interconnections between the three topics are discussed along with a set of open problems and challenges.

Zhen Chen, Zhuqing Jia, Zhiying Wang et al.

A secure multi-party batch matrix multiplication problem (SMBMM) is considered, where the goal is to allow a master to efficiently compute the pairwise products of two batches of massive matrices, by distributing the computation across S servers. Any X colluding servers gain no information about the input, and the master gains no additional information about the input beyond the product. A solution called Generalized Cross Subspace Alignment codes with Noise Alignment (GCSA-NA) is proposed in this work, based on cross-subspace alignment codes. The state of art solution to SMBMM is a coding scheme called polynomial sharing (PS) that was proposed by Nodehi and Maddah-Ali. GCSA-NA outperforms PS codes in several key aspects - more efficient and secure inter-server communication, lower latency, flexible inter-server network topology, efficient batch processing, and tolerance to stragglers. The idea of noise alignment can also be combined with N-source Cross Subspace Alignment (N-CSA) codes and fast matrix multiplication algorithms like Strassen's construction. Moreover, noise alignment can be applied to symmetric secure private information retrieval to achieve the asymptotic capacity.

Zhen Chen, Zhiying Wang, Syed Jafar

The private search problem is introduced, where a dataset comprised of $L$ i.i.d. records is replicated across $N$ non-colluding servers, each record takes values uniformly from an alphabet of size $K$, and a user wishes to search for all records that match a privately chosen value, without revealing any information about the chosen value to any individual server. The capacity of private search is the maximum number of bits of desired information that can be retrieved per bit of download. The asymptotic (large $K$) capacity of private search is shown to be $1-1/N$, even as the scope of private search is further generalized to allow approximate (OR) search over a number of realizations that grows with $K$. The results are based on the asymptotic behavior of a new converse bound for private information retrieval with arbitrarily dependent messages.

Zhen Chen, Zhiying Wang, Syed Jafar

We consider the problem of $T$-Private Information Retrieval with private side information (TPIR-PSI). In this problem, $N$ replicated databases store $K$ independent messages, and a user, equipped with a local cache that holds $M$ messages as side information, wishes to retrieve one of the other $K-M$ messages. The desired message index and the side information must remain jointly private even if any $T$ of the $N$ databases collude. We show that the capacity of TPIR-PSI is $\left(1+\frac{T}{N}+\cdots+\left(\frac{T}{N}\right)^{K-M-1}\right)^{-1}$. As a special case obtained by setting $T=1$, this result settles the capacity of PIR-PSI, an open problem previously noted by Kadhe et al. We also consider the problem of symmetric-TPIR with private side information (STPIR-PSI), where the answers from all $N$ databases reveal no information about any other message besides the desired message. We show that the capacity of STPIR-PSI is $1-\frac{T}{N}$ if the databases have access to common randomness (not available to the user) that is independent of the messages, in an amount that is at least $\frac{T}{N-T}$ bits per desired message bit. Otherwise, the capacity of STPIR-PSI is zero.

Hua Sun, Syed A. Jafar

A $(K, N, T, K_c)$ instance of the MDS-TPIR problem is comprised of $K$ messages and $N$ distributed servers. Each message is separately encoded through a $(K_c, N)$ MDS storage code. A user wishes to retrieve one message, as efficiently as possible, while revealing no information about the desired message index to any colluding set of up to $T$ servers. The fundamental limit on the efficiency of retrieval, i.e., the capacity of MDS-TPIR is known only at the extremes where either $T$ or $K_c$ belongs to $\{1,N\}$. The focus of this work is a recent conjecture by Freij-Hollanti, Gnilke, Hollanti and Karpuk which offers a general capacity expression for MDS-TPIR. We prove that the conjecture is false by presenting as a counterexample a PIR scheme for the setting $(K, N, T, K_c) = (2,4,2,2)$, which achieves the rate $3/5$, exceeding the conjectured capacity, $4/7$. Insights from the counterexample lead us to capacity characterizations for various instances of MDS-TPIR including all cases with $(K, N, T, K_c) = (2,N,T,N-1)$, where $N$ and $T$ can be arbitrary.

Hua Sun, Syed A. Jafar

The capacity has recently been characterized for the private information retrieval (PIR) problem as well as several of its variants. In every case it is assumed that all the queries are generated by the user simultaneously. Here we consider multiround PIR, where the queries in each round are allowed to depend on the answers received in previous rounds. We show that the capacity of multiround PIR is the same as the capacity of single-round PIR (the result is generalized to also include $T$-privacy constraints). Combined with previous results, this shows that there is no capacity advantage from multiround over single-round schemes, non-linear over linear schemes or from $ε$-error over zero-error schemes. However, we show through an example that there is an advantage in terms of storage overhead. We provide an example of a multiround, non-linear, $ε$-error PIR scheme that requires a strictly smaller storage overhead than the best possible with single-round, linear, zero-error PIR schemes.

Hua Sun, Syed A. Jafar

A private information retrieval scheme is a mechanism that allows a user to retrieve any one out of $K$ messages from $N$ non-communicating replicated databases, each of which stores all $K$ messages, without revealing anything about the identity of the desired message index to any individual database. If the size of each message is $L$ bits and the total download required by a PIR scheme from all $N$ databases is $D$ bits, then $D$ is called the download cost and the ratio $L/D$ is called an achievable rate. For fixed $K,N\in\mathbb{N}$, the capacity of PIR, denoted by $C$, is the supremum of achievable rates over all PIR schemes and over all message sizes, and was recently shown to be $C=(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. In this work, for arbitrary $K, N$, we explore the minimum download cost $D_L$ across all PIR schemes (not restricted to linear schemes) for arbitrary message lengths $L$ under arbitrary choices of alphabet (not restricted to finite fields) for the message and download symbols. If the same $M$-ary alphabet is used for the message and download symbols, then we show that the optimal download cost in $M$-ary symbols is $D_L=\lceil\frac{L}{C}\rceil$. If the message symbols are in $M$-ary alphabet and the downloaded symbols are in $M'$-ary alphabet, then we show that the optimal download cost in $M'$-ary symbols, $D_L\in\left\{\left\lceil \frac{L'}{C}\right\rceil,\left\lceil \frac{L'}{C}\right\rceil-1,\left\lceil \frac{L'}{C}\right\rceil-2\right\}$, where $L'= \lceil L \log_{M'} M\rceil$.

Hua Sun, Syed A. Jafar

Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other $K-1$ messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is $1-1/N$ regardless of the number of messages $K$, if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least $1/(N-1)$ bits per desired message bit, and zero otherwise. Extensions to the capacity region of SPIR and the capacity of finite length SPIR are provided.

Hua Sun, Syed A. Jafar

Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ messages) while keeping the identity of the desired message index a secret from each individual database. The information theoretic capacity of PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. $T$-private PIR is a generalization of PIR to include the requirement that even if any $T$ of the $N$ databases collude, the identity of the retrieved message remains completely unknown to them. Robust PIR is another generalization that refers to the scenario where we have $M \geq N$ databases, out of which any $M - N$ may fail to respond. For $K$ messages and $M\geq N$ databases out of which at least some $N$ must respond, we show that the capacity of $T$-private and Robust PIR is $\left(1+T/N+T^2/N^2+\cdots+T^{K-1}/N^{K-1}\right)^{-1}$. The result includes as special cases the capacity of PIR without robustness ($M=N$) or $T$-privacy constraints ($T=1$).

Hua Sun, Syed A. Jafar

In the private information retrieval (PIR) problem a user wishes to retrieve, as efficiently as possible, one out of $K$ messages from $N$ non-communicating databases (each holds all $K$ messages) while revealing nothing about the identity of the desired message index to any individual database. The information theoretic capacity of PIR is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. For $K$ messages and $N$ databases, we show that the PIR capacity is $(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. A remarkable feature of the capacity achieving scheme is that if we eliminate any subset of messages (by setting the message symbols to zero), the resulting scheme also achieves the PIR capacity for the remaining subset of messages.

Hua Sun, Syed A. Jafar

Blind interference alignment (BIA) refers to interference alignment schemes that are designed only based on channel coherence pattern knowledge at the transmitters (the "blind" transmitters do not know the exact channel values). Private information retrieval (PIR) refers to the problem where a user retrieves one out of K messages from N non-communicating databases (each holds all K messages) without revealing anything about the identity of the desired message index to any individual database. In this paper, we identify an intriguing connection between PIR and BIA. Inspired by this connection, we characterize the information theoretic optimal download cost of PIR, when we have K = 2 messages and the number of databases, N, is arbitrary.