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.