On the Asymptotic Capacity of XX-Secure TT-Private Information Retrieval with Graph Based Replicated Storage

The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary $T$-privacy and $X$-security constraints, where the privacy of the user must be protected against any set of up to $T$ colluding servers and the security of the stored data must be protected against any set of up to $X$ colluding servers. A general achievable scheme for arbitrary storage patterns is presented that achieves the rate $(\rho_{\min}-X-T)/N$, where $N$ is the total number of servers, and each message is replicated at least $\rho_{\min}$ times. Notably, the scheme makes use of a special structure inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is also presented. The two bounds are shown to match for many settings, including symmetric storage patterns. Finally, the asymptotic capacity is fully characterized for the case without security constraints $(X=0)$ for arbitrary storage patterns provided that each message is replicated no more than $T+2$ times. As an example of this result, consider PIR with arbitrary graph based storage ($T=1, X=0$) where every message is replicated at exactly $3$ servers. For this $3$-replicated storage setting, the asymptotic capacity is equal to $2/\nu_2(G)$ where $\nu_2(G)$ is the maximum size of a $2$-matching in a storage graph $G[V,E]$. In this undirected graph, the vertices $V$ correspond to the set of servers, and there is an edge $uv\in E$ between vertices $u,v$ only if a subset of messages is replicated at both servers $u$ and $v$

The Asymptotic Capacity of XX-Secure TT-Private Linear Computation with Graph Based Replicated Storage

The problem of $X$-secure $T$-private linear computation with graph based replicated storage (GXSTPLC) is to enable the user to retrieve a linear combination of messages privately from a set of $N$ distributed servers where every message is only allowed to store among a subset of servers subject to an $X$-security constraint, i.e., any groups of up to $X$ colluding servers must reveal nothing about the messages. Besides, any groups of up to $T$ servers cannot learn anything about the coefficients of the linear combination retrieved by the user. In this work, we completely characterize the asymptotic capacity of GXSTPLC, i.e., the supremum of average number of desired symbols retrieved per downloaded symbol, in the limit as the number of messages $K$ approaches infinity. Specifically, it is shown that a prior linear programming based upper bound on the asymptotic capacity of GXSTPLC due to Jia and Jafar is tight by constructing achievability schemes. Notably, our achievability scheme also settles the exact capacity (i.e., for finite $K$) of $X$-secure linear combination with graph based replicated storage (GXSLC). Our achievability proof builds upon an achievability scheme for a closely related problem named asymmetric $\mathbf{X}$-secure $\mathbf{T}$-private linear computation with graph based replicated storage (Asymm-GXSTPLC) that guarantees non-uniform security and privacy levels across messages and coefficients. In particular, by carefully designing Asymm-GXSTPLC settings for GXSTPLC problems, the corresponding Asymm-GXSTPLC schemes can be reduced to asymptotic capacity achieving schemes for GXSTPLC. In regard to the achievability scheme for Asymm-GXSTPLC, interesting aspects of our construction include a novel query and answer design which makes use of a Vandermonde decomposition of Cauchy matrices, and a trade-off among message replication, security and privacy thresholds.Comment: 39 pages, 2 figure

Double Blind TT-Private Information Retrieval

Double blind $T$-private information retrieval (DB-TPIR) enables two users, each of whom specifies an index ($\theta_1, \theta_2$, resp.), to efficiently retrieve a message $W(\theta_1,\theta_2)$ labeled by the two indices, from a set of $N$ servers that store all messages $W(k_1,k_2), k_1\in\{1,2,\cdots,K_1\}, k_2\in\{1,2,\cdots,K_2\}$, such that the two users' indices are kept private from any set of up to $T_1,T_2$ colluding servers, respectively, as well as from each other. A DB-TPIR scheme based on cross-subspace alignment is proposed in this paper, and shown to be capacity-achieving in the asymptotic setting of large number of messages and bounded latency. The scheme is then extended to $M$-way blind $X$-secure $T$-private information retrieval (MB-XS-TPIR) with multiple ($M$) indices, each belonging to a different user, arbitrary privacy levels for each index ($T_1, T_2,\cdots, T_M$), and arbitrary level of security ($X$) of data storage, so that the message $W(\theta_1,\theta_2,\cdots, \theta_M)$ can be efficiently retrieved while the stored data is held secure against collusion among up to $X$ colluding servers, the $m^{th}$ user's index is private against collusion among up to $T_m$ servers, and each user's index $\theta_m$ is private from all other users. The general scheme relies on a tensor-product based extension of cross-subspace alignment and retrieves $1-(X+T_1+\cdots+T_M)/N$ bits of desired message per bit of download.Comment: Accepted for publication in IEEE Journal on Selected Areas in Information Theory (JSAIT

LightChain: A DHT-based Blockchain for Resource Constrained Environments

As an append-only distributed database, blockchain is utilized in a vast variety of applications including the cryptocurrency and Internet-of-Things (IoT). The existing blockchain solutions have downsides in communication and storage efficiency, convergence to centralization, and consistency problems. In this paper, we propose LightChain, which is the first blockchain architecture that operates over a Distributed Hash Table (DHT) of participating peers. LightChain is a permissionless blockchain that provides addressable blocks and transactions within the network, which makes them efficiently accessible by all the peers. Each block and transaction is replicated within the DHT of peers and is retrieved in an on-demand manner. Hence, peers in LightChain are not required to retrieve or keep the entire blockchain. LightChain is fair as all of the participating peers have a uniform chance of being involved in the consensus regardless of their influence such as hashing power or stake. LightChain provides a deterministic fork-resolving strategy as well as a blacklisting mechanism, and it is secure against colluding adversarial peers attacking the availability and integrity of the system. We provide mathematical analysis and experimental results on scenarios involving 10K nodes to demonstrate the security and fairness of LightChain. As we experimentally show in this paper, compared to the mainstream blockchains like Bitcoin and Ethereum, LightChain requires around 66 times less per node storage, and is around 380 times faster on bootstrapping a new node to the system, while each LightChain node is rewarded equally likely for participating in the protocol

GCSA Codes with Noise Alignment for Secure Coded Multi-Party Batch Matrix Multiplication

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