Secure linear algebra using linearly recurrent sequences

In this work we present secure two-party protocols for various core problems in linear algebra. Our main result is a protocol to obliviously decide singularity of an encrypted matrix: Bob holds an n × n matrix, encrypted with Alice's secret key, and wants to learn whether or not the matrix is singular (while leaking nothing further). We give an interactive protocol between Alice and Bob that solves the above problem in O(log n) communication rounds and with overall communication complexity of roughly O(n²) (note that the input size is n²). Our techniques exploit certain nice mathematical properties of linearly recurrent sequences and their relation to the minimal and characteristic polynomial of the input matrix, 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 O(n^0.275) to O(log n). At the core of our results we use a protocol that securely computes the minimal polynomial of an encrypted matrix. Based on this protocol we exploit certain algebraic reductions to further extend our results to the problems of securely computing rank and determinant, and to solving systems of linear equations (again with low round and communication complexity).