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Register a new userSymmetric Private Polynomial Computation From Lagrange Encoding
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Jinbao Zhu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The problem of $X$-secure $T$-colluding symmetric Private Polynomial Computation (PPC) from coded storage system with $B$ Byzantine and $U$ unresponsive servers is studied in this paper. Specifically, a dataset consisting of $M$ files are stored across $N$ distributed servers according to $(N,K+X)$ Maximum Distance Separable (MDS) codes such that any group of up to $X$ colluding servers can not learn anything about the data files. A user wishes to privately evaluate one out of a set of candidate polynomial functions over the $M$ files from the system, while guaranteeing that any $T$ colluding servers can not learn anything about the identity of the desired function and the user can not learn anything about the $M$ data files more than the desired polynomial function, in the presence of $B$ Byzantine servers that can send arbitrary responses maliciously to confuse the user and $U$ unresponsive servers that will not respond any information at all. Two novel symmetric PPC schemes using Lagrange encoding are proposed. Both the two schemes achieve the same PPC rate $1-frac{G(K+X-1)+T+2B}{N-U}$, secrecy rate $frac{G(K+X-1)+T}{N-(G(K+X-1)+T+2B+U)}$, finite field size and decoding complexity, where $G$ is the maximum degree over all the candidate polynomial functions. Particularly, the first scheme focuses on the general case that the candidate functions are consisted of arbitrary polynomials, and the second scheme restricts the candidate functions to be a finite-dimensional vector space (or sub-space) of polynomials over $mathbb{F}_p$ but requires less upload cost, query complexity and server computation complexity. Remarkably, the PPC setup studied in this paper generalizes all the previous MDS coded PPC setups and the two degraded schemes strictly outperform the best known schemes in terms of (asymptotical) PPC rate, which is the main concern of the PPC schemes.
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We study the problem of private set intersection (PSI). In this problem, there are two entities $E_i$, for $i=1, 2$, each storing a set $mathcal{P}_i$, whose elements are picked from a finite field $mathbb{F}_K$, on $N_i$ replicated and non-colluding databases. It is required to determine the set intersection $mathcal{P}_1 cap mathcal{P}_2$ without leaking any information about the remaining elements to the other entity with the least amount of downloaded bits. We first show that the PSI problem can be recast as a multi-message symmetric private information retrieval (MM-SPIR) problem. Next, as a stand-alone result, we derive the information-theoretic sum capacity of MM-SPIR, $C_{MM-SPIR}$. We show that with $K$ messages, $N$ databases, and the size of the desired message set $P$, the exact capacity of MM-SPIR is $C_{MM-SPIR} = 1 - frac{1}{N}$ when $P leq K-1$, provided that the entropy of the common randomness $S$ satisfies $H(S) geq frac{P}{N-1}$ per desired symbol. This result implies that there is no gain for MM-SPIR over successive single-message SPIR (SM-SPIR). For the MM-SPIR problem, we present a novel capacity-achieving scheme that builds on the near-optimal scheme of Banawan-Ulukus originally proposed for the multi-message PIR (MM-PIR) problem without database privacy constraints. Surprisingly, our scheme here is exactly optimal for the MM-SPIR problem for any $P$, in contrast to the scheme for the MM-PIR problem, which was proved only to be near-optimal. Our scheme is an alternative to the SM-SPIR scheme of Sun-Jafar. Based on this capacity result for MM-SPIR, and after addressing the added requirements in its conversion to the PSI problem, we show that the optimal download cost for the PSI problem is $minleft{leftlceilfrac{P_1 N_2}{N_2-1}rightrceil, leftlceilfrac{P_2 N_1}{N_1-1}rightrceilright}$, where $P_i$ is the cardinality of set $mathcal{P}_i$
The rate region of the task-encoding problem for two correlated sources is characterized using a novel parametric family of dependence measures. The converse uses a new expression for the $rho$-th moment of the list size, which is derived using the relative $alpha$-entropy.
Classical reversible circuits, acting on $w$~bits, are represented by permutation matrices of size $2^w times 2^w$. Those matrices form the group P($2^w$), isomorphic to the symmetric group {bf S}$_{2^w}$. The permutation group P($n$), isomorphic to {bf S}$_n$, contains cycles with length~$p$, ranging from~1 to $L(n)$, where $L(n)$ is the so-called Landau function. By Lagrange interpolation between the $p$~matrices of the cycle, we step from a finite cyclic group of order~$p$ to a 1-dimensional Lie group, subgroup of the unitary group U($n$). As U($2^w$) is the group of all possible quantum circuits, acting on $w$~qubits, such interpolation is a natural way to step from classical computation to quantum computation.
We study the fundamental problem of index coding under an additional privacy constraint that requires each receiver to learn nothing more about the collection of messages beyond its demanded messages from the server and what is available to it as side information. To enable such private communication, we allow the use of a collection of independent secret keys, each of which is shared amongst a subset of users and is known to the server. The goal is to study properties of the key access structures which make the problem feasible and then design encoding and decoding schemes efficient in the size of the server transmission as well as the sizes of the secret keys. We call this the private index coding problem. We begin by characterizing the key access structures that make private index coding feasible. We also give conditions to check if a given linear scheme is a valid private index code. For up to three users, we characterize the rate region of feasible server transmission and key rates, and show that all feasible rates can be achieved using scalar linear coding and time sharing; we also show that scalar linear codes are sub-optimal for four receivers. The outer bounds used in the case of three users are extended to arbitrary number of users and seen as a generalized version of the well-known polymatroidal bounds for the standard non-private index coding. We also show that the presence of common randomness and private randomness does not change the rate region. Furthermore, we study the case where no keys are shared among the users and provide some necessary and sufficient conditions for feasibility in this setting under a weaker notion of privacy. If the server has the ability to multicast to any subset of users, we demonstrate how this flexibility can be used to provide privacy and characterize the minimum number of server multicasts required.
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