How to solve any protocol problem - An efficiency improvement

Abstract

Consider n parties having local inputs x 1,x 2,..,x n respectively. and wishing to compute the value f(x 1,..,x n). where f is a predetermined function. Loosely speaking. an n-party protocol for this purpose has maximum privacy if whatever a subset of the users can efficiently compute when participating in the protocol, they can also compute from their local inputs and the value f(x 1,.., x n). Recently, Goldreich, Micali and Wigderson have presented a polynomial-time algorithm that, given a Turing machine for computing the function f. outputs an n-party protocol with maximum privacy for distributively Computing f(x 1,..,x n). The maximum privacy protocol output uses as a subprotocol a maximum privacy two-party protocol for computing a particular simple function p 1(·,·). More recently, Haber and Micali have improved the efficiency of the above n-party protocols, using a maximum privacy two-party protocol for computing another particular function p 2(·,·). Both works use a general result of Yao in order to implement protocols for the particular functions p 1,and p 2. In this paper, we present direct solutions to the above two particular protocol problems, avoiding the use of Yao’s general result. In fact. we present two alternative approaches for solving both problems. The first approach consists of a simple reduction of these two problems to a variant of Oblivious Transfer. The second approach consists of designing direct solutions to these two problems, assuming the intractability or the Quadratic Residuosity problem. Both approaches yield simpler and more efficient solutions than the ones obtained by Yao’s result.