### New Protocols for Secure Linear Algebra: Pivoting-Free Elimination and Fast Block-Recursive Matrix Decomposition

Niek J. Bouman and Niels de Vreede

##### Abstract

Cramer and Damg\aa{}rd were the first to propose a constant-rounds protocol for securely solving a linear system of unknown rank over a finite field in multiparty computation (MPC). For $m$ linear equations and $n$ unknowns, and for the case $m\leq n$, the computational complexity of their protocol is $O(n^5)$. Follow-up work (by Cramer, Kiltz, and Padró) proposes another constant-rounds protocol for solving this problem, which has complexity $O(m^4+n^2 m)$. For certain applications, such asymptotic complexities might be prohibitive. In this work, we improve the asymptotic computational complexity of solving a linear system over a finite field, thereby sacrificing the constant-rounds property. We propose two protocols: (1) a protocol based on pivoting-free Gaussian elimination with computational complexity $O(n^3)$ and linear round complexity, and (2) a protocol based on block-recursive matrix decomposition, having $O(n^2)$ computational complexity (assuming cheap'' secure inner products as in Shamir's secret-sharing scheme) and $O(n^{1.585})$ (super-linear) round complexity.

Note: Corrected a misleading typo in Protocol 2a

Available format(s)
Category
Cryptographic protocols
Publication info
Preprint. Minor revision.
Keywords
secure linear algebramultiparty computation
Contact author(s)
n j bouman @ tue nl
History
2018-08-22: revised
See all versions
Short URL
https://ia.cr/2018/703

CC BY

BibTeX

@misc{cryptoeprint:2018/703,
author = {Niek J.  Bouman and Niels de Vreede},
title = {New Protocols for Secure Linear Algebra: Pivoting-Free Elimination and Fast Block-Recursive Matrix Decomposition},
howpublished = {Cryptology ePrint Archive, Paper 2018/703},
year = {2018},
note = {\url{https://eprint.iacr.org/2018/703}},
url = {https://eprint.iacr.org/2018/703}
}

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