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Register a new userOn the Capacity of Private Monomial Computation
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Added by
Yauhen Yakimenka
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this work, we consider private monomial computation (PMC) for replicated noncolluding databases. In PMC, a user wishes to privately retrieve an arbitrary multivariate monomial from a candidate set of monomials in $f$ messages over a finite field $mathbb F_q$, where $q=p^k$ is a power of a prime $p$ and $k ge 1$, replicated over $n$ databases. We derive the PMC capacity under a technical condition on $p$ and for asymptotically large $q$. The condition on $p$ is satisfied, e.g., for large enough $p$. Also, we present a novel PMC scheme for arbitrary $q$ that is capacity-achieving in the asymptotic case above. Moreover, we present formulas for the entropy of a multivariate monomial and for a set of monomials in uniformly distributed random variables over a finite field, which are used in the derivation of the capacity expression.
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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.
We consider the fundamental tradeoff between the storage cost and the download cost in private information retrieval systems, without any explicit structural restrictions on the storage codes, such as maximum distance separable codes or uncoded storage. Two novel outer bounds are provided, which have the following implications. When the messages are stored without any redundancy across the databases, the optimal PIR strategy is to download all the messages; on the other hand, for PIR capacity-achieving codes, each database can reduce the storage cost, from storing all the messages, by no more than one message on average. We then focus on the two-message two-database case, and show that a stronger outer bound can be derived through a novel pseudo-message technique. This stronger outer bound suggests that a precise characterization of the storage-download tradeoff may require non-Shannon type inequalities, or at least more sophisticated bounding techniques.
We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from $N$ non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any $T$ databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when $K>T/textbf{gcd}(N,T)$ where $K$ is the total number of messages in the system and $textbf{gcd}(cdot,cdot)$ means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as $textbf{lcm}(N-T,T)$, where $textbf{lcm}(cdot,cdot)$ means the least common multiple. On the other hand, when $K$ is small, we show that it is in fact possible to design codes with a message size even smaller than $textbf{lcm}(N-T,T)$.
This paper studies the capacity of a general multiple-input multiple-output (MIMO) free-space optical intensity channel under a per-input-antenna peak-power constraint and a total average-power constraint over all input antennas. The focus is on the scenario with more transmit than receive antennas. In this scenario, different input vectors can yield identical distributions at the output, when they result in the same image vector under multiplication by the channel matrix. We first determine the most energy-efficient input vectors that attain each of these image vectors. Based on this, we derive an equivalent capacity expression in terms of the image vector, and establish new lower and upper bounds on the capacity of this channel. The bounds match when the signal-to-noise ratio (SNR) tends to infinity, establishing the high-SNR asymptotic capacity. We also characterize the low-SNR slope of the capacity of this channel.
The interactive capacity of a noisy channel is the highest possible rate at which arbitrary interactive protocols can be simulated reliably over the channel. Determining the interactive capacity is notoriously difficult, and the best known lower bounds are far below the associated Shannon capacity, which serves as a trivial (and also generally the best known) upper bound. This paper considers the more restricted setup of simulating finite-state protocols. It is shown that all two-state protocols, as well as rich families of arbitrary finite-state protocols, can be simulated at the Shannon capacity, establishing the interactive capacity for those families of protocols.
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