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(n0.275 ) to O(log n). At the core of our results we use a protocol that securely computes the mini...

• Type: Conference
• Year: 2007
• Where: TCC
• Authors: Eike Kiltz, Payman Mohassel, Enav Weinreb, Matthew K. Franklin
• Keywords: Communication Complexity, Cryptography, Linear Algebra, Recurrent Sequences, TCC 2007