Symmetric Private Polynomial Computation From Lagrange Encoding

The problem of <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula>-secure <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula>-colluding symmetric Private Polynomial Computation (PPC) fro...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 68; no. 4; pp. 2704 - 2718
Main Authors: Zhu, Jinbao, Yan, Qifa, Tang, Xiaohu, Li, Songze
Format: Journal Article
Language:English
Published: New York IEEE 01.04.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Codes
Complexity
computation complexity
Computational complexity
Costs
Decoding
Encoding
Evaluation
Fields (mathematics)
Functions (mathematics)
Lagrange encoding
Polynomials
Private computation
Servers
symmetric private polynomial computation
Systematics
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:The problem of <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula>-secure <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula>-colluding symmetric Private Polynomial Computation (PPC) from coded storage system with <inline-formula> <tex-math notation="LaTeX">B </tex-math></inline-formula> Byzantine and <inline-formula> <tex-math notation="LaTeX">U </tex-math></inline-formula> unresponsive servers is studied in this paper. Specifically, a dataset consisting of <inline-formula> <tex-math notation="LaTeX">M </tex-math></inline-formula> files is stored across <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> distributed servers according to <inline-formula> <tex-math notation="LaTeX">(N,K+X) </tex-math></inline-formula> Maximum Distance Separable (MDS) codes such that any group of up to <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">M </tex-math></inline-formula> files from the system, while guaranteeing that any <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> colluding servers can not learn anything about the identity of the desired function and the user can not learn anything about the <inline-formula> <tex-math notation="LaTeX">M </tex-math></inline-formula> data files more than the desired polynomial function evaluations, in the presence of <inline-formula> <tex-math notation="LaTeX">B </tex-math></inline-formula> Byzantine servers that can send arbitrary responses maliciously to confuse the user and <inline-formula> <tex-math notation="LaTeX">U </tex-math></inline-formula> unresponsive servers that will not respond any information at all. A novel symmetric PPC scheme using Lagrange encoding is proposed. This scheme achieves a PPC rate of <inline-formula> <tex-math notation="LaTeX">1-\frac {G(K+X-1)+T+2B}{N-U} </tex-math></inline-formula> with secrecy rate <inline-formula> <tex-math notation="LaTeX">\frac {G(K+X-1)+T}{N-(G(K+X-1)+T+2B+U)} </tex-math></inline-formula> and finite field size <inline-formula> <tex-math notation="LaTeX">N+\max \{K,N-(G(K+X-1)+T+2B+U)\} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">G </tex-math></inline-formula> is the maximum degree over all the candidate polynomial functions. Moreover, to further measure the efficiency of PPC schemes, upload cost, query complexity, server computation complexity and decoding complexity required to implement the scheme are analyzed. Remarkably, the PPC setup studied in this paper generalizes all the previous MDS coded PPC setups and the 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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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3140890