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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.
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 $
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
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
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 sid