Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Matrix multiplication via arithmetic progressions
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Multiparty unconditionally secure protocols
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Non-cryptographic fault-tolerant computing in constant number of rounds of interaction
Proceedings of the eighth annual ACM Symposium on Principles of distributed computing
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
An introduction to parallel algorithms
An introduction to parallel algorithms
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Parallel Coin-Tossing and Constant-Round Secure Two-Party Computation
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Secure Distributed Linear Algebra in a Constant Number of Rounds
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Information-theoretically secure protocols and security under composition
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Protocols for secure computations
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
How to generate and exchange secrets
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Security against covert adversaries: efficient protocols for realistic adversaries
TCC'07 Proceedings of the 4th conference on Theory of cryptography
Secure linear algebra using linearly recurrent sequences
TCC'07 Proceedings of the 4th conference on Theory of cryptography
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Efficient polynomial operations in the shared-coefficients setting
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Communication efficient secure linear algebra
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Secure Arithmetic Computation with No Honest Majority
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
Efficient and secure evaluation of multivariate polynomials and applications
ACNS'10 Proceedings of the 8th international conference on Applied cryptography and network security
Efficient set operations in the presence of malicious adversaries
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
SCN'12 Proceedings of the 8th international conference on Security and Cryptography for Networks
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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(n2 + 1/t) communication for testing singularity of a shared n×nmatrix in constant round, for any constant tin 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.