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Private Multiple Linear Computation: A Flexible Communication-Computation Tradeoff

Jinbao Zhu, Lanping Li, Xiaohu Tang, Ping Deng

TL;DR

This work tackles private multiple linear computation (PMLC) over replicated storage with $T$-colluding privacy and $S$ unresponsive servers. It introduces a flexible scheme that trades off uplink communication and computation against download and decoding costs by configuring three integers $K$, $E$, and $R$ under the constraints $K+R\le N-S-T$, with divisibility requirements $E|L$, $N|ME$, and $K|PE$. The construction partitions files, expands the coefficient matrix to form a decodable product with the file matrix via Lagrange polynomial masking and zero forcing, and relies on polynomial interpolation to recover the desired linear computations from a subset of server responses while preserving privacy. The resulting framework provides explicit expressions for upload, download, and computational costs and reveals how to tune parameters to meet system constraints, offering a practical approach to private computation in distributed environments with non-negligible uplink costs and server workloads.

Abstract

We consider the problem of private multiple linear computation (PMLC) over a replicated storage system with colluding and unresponsive constraints. In this scenario, the user wishes to privately compute $P$ linear combinations of $M$ files from a set of $N$ replicated servers without revealing any information about the coefficients of these linear combinations to any $T$ colluding servers, in the presence of $S$ unresponsive servers that do not provide any information in response to user queries. Our focus is on more general performance metrics where the communication and computational overheads incurred by the user are not neglected. Additionally, the communication and computational overheads for servers are also taken into consideration. Unlike most previous literature that primarily focused on download cost from servers as a performance metric, we propose a novel PMLC scheme to establish a flexible tradeoff between communication costs and computational complexities.

Private Multiple Linear Computation: A Flexible Communication-Computation Tradeoff

TL;DR

This work tackles private multiple linear computation (PMLC) over replicated storage with -colluding privacy and unresponsive servers. It introduces a flexible scheme that trades off uplink communication and computation against download and decoding costs by configuring three integers , , and under the constraints , with divisibility requirements , , and . The construction partitions files, expands the coefficient matrix to form a decodable product with the file matrix via Lagrange polynomial masking and zero forcing, and relies on polynomial interpolation to recover the desired linear computations from a subset of server responses while preserving privacy. The resulting framework provides explicit expressions for upload, download, and computational costs and reveals how to tune parameters to meet system constraints, offering a practical approach to private computation in distributed environments with non-negligible uplink costs and server workloads.

Abstract

We consider the problem of private multiple linear computation (PMLC) over a replicated storage system with colluding and unresponsive constraints. In this scenario, the user wishes to privately compute linear combinations of files from a set of replicated servers without revealing any information about the coefficients of these linear combinations to any colluding servers, in the presence of unresponsive servers that do not provide any information in response to user queries. Our focus is on more general performance metrics where the communication and computational overheads incurred by the user are not neglected. Additionally, the communication and computational overheads for servers are also taken into consideration. Unlike most previous literature that primarily focused on download cost from servers as a performance metric, we propose a novel PMLC scheme to establish a flexible tradeoff between communication costs and computational complexities.
Paper Structure (7 sections, 2 theorems, 46 equations)

This paper contains 7 sections, 2 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Result and Discussions
  4. A Flexible Private Multiple Linear Computation Scheme
  5. Illustrative Example
  6. General Description of PMLC Scheme
  7. Performance Analysis

Key Result

Theorem 1

For the $T$-colluding PMLC problem with $M$ files of each length $L$, $P$ desired linear combinations, and $N$ replicated servers among which $S$ are unresponsive, the PMLC scheme proposed in Section PMLC:scheme achieves the following metrics for any given integer parameters $K>0,E>0,R\geq 0$ such t

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Lemma 1: von2013modern