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Consider a network of processors among which elements in a nite eld K can be veriably shared in a constant number of rounds. Assume furthermore constant-round protocols are available for generat- ing random shared values, for secure multiplication and for addition of shared values. These requirements can be met by known techniques in all standard models of communication. In this model we construct protocols allowing the network to securely solve standard computational problems in linear algebra. In particular, we show how the network can securely, eciently and in constant-round compute determinant, characteristic polynomial, rank, and the solution space of linear systems of equations. Constant round solutions follow for all problems which can be solved by direct application of such linear algebraic methods, such as deciding whether a graph contains a perfect match. If the basic protocols (for shared random values, addition and multiplica- tion) we start from are unconditionally secure, then so are our protocols. Our results oer solutions that are signicantly more ecient than pre- vious techniques for secure linear algebra, they work for arbitrary elds and therefore extend the class of functions previously known to be com- putable in constant round and with unconditional security. In particular, we obtain an unconditionally secure protocol for computing a function f in constant round, where the protocol has complexity polynomial in the span program size of f over an arbitrary nite eld.

Secure Distributed Linear Algebra in a Constant Number of Rounds