Communication Efficient Secure Linear Algebra

• Published in: Lecture notes in computer science pp. 522 - 541
• Main Authors: Nissim, Kobbi, Weinreb, Enav
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2006; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Communication Complexity; Computer science; control theory; systems; Constant Probability; Cryptography; Exact sciences and technology; Gaussian Elimination; Information, signal and communications theory; Memory and file management (including protection and security); Memory organisation. Data processing; Secure Protocol; Signal and communications theory; Software; Subspace Versus; Telecommunications and information theory; Triangular matrix; Lower bound; Probabilistic approach; Transmission protocol; Gauss method; Encryption; Modeling; Vector space; Diagonal matrix; Linear algebra; Communication complexity; Cryptography; Computer security
• ISBN: 3540327312; 9783540327318
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11681878_27

We present communication efficient secure protocols for a variety of linear algebra problems. Our main building block is a protocol for computing Gaussian Elimination on encrypted data. As input for this protocol, Bob holds a k × k matrix M, encrypted with Alice’s key. At the end of the protocol run, Bob holds an encryption of an upper-triangular matrix M ′ such that the number of nonzero elements on the diagonal equals the rank of M. The communication complexity of our protocol is roughly O(k2). Building on Oblivious Gaussian elimination, we present secure protocols for several problems: deciding the intersection of linear and affine subspaces, picking a random vector from the intersection, and obliviously solving a set of linear equations. Our protocols match known (insecure) communication complexity lower bounds, and improve the communication complexity of both Yao’s garbled circuits and that of specific previously published protocols.