Secure Linear Algebra Using Linearly Recurrent Sequences

Abstract

In this work we present secure two-party protocols for
various core problems in linear algebra.
Our main building block is a protocol to obliviously decide singularity
of an encrypted matrix:
Bob holds an $n 	imes n$ matrix $M$, encrypted with Alice's secret
key, and wants to learn whether
the matrix is singular or not (and nothing beyond that).
We give an interactive protocol between Alice and Bob that solves the
above problem
with optimal communication complexity while at the same time achieving
low round complexity.
More precisely, the number of communication rounds in our protocol
is $polylog(n)$ and
the overall communication is roughly $O(n^2)$ (note that the input size is $n^2$).
At the core of our protocol we exploit some nice mathematical
properties of linearly recurrent sequences and their
relation to the characteristic polynomial of the matrix $M$, following [Wiedemann, 1986].
With our new techniques we are able to improve the round complexity of
the communication efficient solution of [Nissim and Weinreb, 2006] from $n^{0.275}$ to $polylog(n)$.

Based on our singularity protocol we further
extend our result to the problems of securely computing the rank of an
encrypted matrix and solving systems of linear equations.