TY - GEN
T1 - Efficient secure linear algebra in the presence of covert or computationally unbounded adversaries
AU - Mohassel, Payman
AU - Weinreb, Enav
PY - 2008/9/24
Y1 - 2008/9/24
N2 - In this work we study the design of secure protocols for linear algebra problems. All current solutions to the problem are either inefficient in terms of communication complexity or assume that the adversary is honest but curious. We design protocols for two different adversarial settings: First, we achieve security in the presence of a covert adversary, a notion recently introduced by [Aumann and Lindell, TCC 2007]. Roughly speaking, this guarantees that if the adversary deviates from the protocol in a way that allows him to cheat, then he will be caught with good probability. Second, we achieve security against arbitrary malicious behaviour in the presence of a computationally unbounded adversary that controls less than a third of the parties. Our main result is a new upper bound of O(n 2∈+∈1/t ) communication for testing singularity of a shared n ×n matrix in constant round, for any constant t in both of these adversarial environments. We use this construction to design secure protocols for computing the rank of a shared matrix and solving a shared linear system of equations with similar efficiency. We use different techniques from computer algebra, together with recent ideas from [Cramer, Kiltz, and Padró, CRYPTO 2007], to reduce the problem of securely deciding singularity to the problem of securely computing matrix product. We then design new and efficient protocols for secure matrix product in both adversarial settings. In the two-party setting, we combine cut-and-choose techniques on random additive decomposition of the input, with a careful use of the random strings of a homomorphic encryption scheme to achieve simulation-based security. Thus, our protocol avoids general zero-knowledge proofs and only makes a black-box use of a homomorphic encryption scheme.
AB - In this work we study the design of secure protocols for linear algebra problems. All current solutions to the problem are either inefficient in terms of communication complexity or assume that the adversary is honest but curious. We design protocols for two different adversarial settings: First, we achieve security in the presence of a covert adversary, a notion recently introduced by [Aumann and Lindell, TCC 2007]. Roughly speaking, this guarantees that if the adversary deviates from the protocol in a way that allows him to cheat, then he will be caught with good probability. Second, we achieve security against arbitrary malicious behaviour in the presence of a computationally unbounded adversary that controls less than a third of the parties. Our main result is a new upper bound of O(n 2∈+∈1/t ) communication for testing singularity of a shared n ×n matrix in constant round, for any constant t in both of these adversarial environments. We use this construction to design secure protocols for computing the rank of a shared matrix and solving a shared linear system of equations with similar efficiency. We use different techniques from computer algebra, together with recent ideas from [Cramer, Kiltz, and Padró, CRYPTO 2007], to reduce the problem of securely deciding singularity to the problem of securely computing matrix product. We then design new and efficient protocols for secure matrix product in both adversarial settings. In the two-party setting, we combine cut-and-choose techniques on random additive decomposition of the input, with a careful use of the random strings of a homomorphic encryption scheme to achieve simulation-based security. Thus, our protocol avoids general zero-knowledge proofs and only makes a black-box use of a homomorphic encryption scheme.
UR - http://www.scopus.com/inward/record.url?scp=52049092094&partnerID=8YFLogxK
U2 - 10.1007/978-3-540-85174-5_27
DO - 10.1007/978-3-540-85174-5_27
M3 - Conference contribution
AN - SCOPUS:52049092094
SN - 3540851739
SN - 9783540851738
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 481
EP - 496
BT - Advances in Cryptology - CRYPTO 2008 - 28th Annual International Cryptology Conference, Proceedings
T2 - 28th Annual International Cryptology Conference, CRYPTO 2008
Y2 - 17 August 2008 through 21 August 2008
ER -