Secure Arithmetic Computation with Constant Computational Overhead

We study the complexity of securely evaluating an arithmetic circuit over a finite field F in the setting of secure two-party computation with semi-honest adversaries. In all existing protocols, the number of arithmetic operations per multiplication gate grows either linearly with log vertical bar F vertical bar or polylogarithmically with the security parameter. We present the first protocol that only makes a constant (amortized) number of field operations per gate. The protocol uses the underlying field F as a black box, and its security is based on arithmetic analogues of well-studied cryptographic assumptions. Our protocol is particularly appealing in the special case of securely evaluating a "vector-OLE" function of the form ax + b, where x is an element of F is the input of one party and a, b is an element of F-w are the inputs of the other party. In this case, which is motivated by natural applications, our protocol can achieve an asymptotic rate of 1/3 (i.e., the communication is dominated by sending roughly 3w elements of F). Our implementation of this protocol suggests that it outperforms competing approaches even for relatively small fields F and over fast networks. Our technical approach employs two new ingredients that may be of independent interest. First, we present a general way to combine any linear code that has a fast encoder and a cryptographic ("LPN-style") pseudorandomness property with another linear code that supports fast encoding and erasure-decoding, obtaining a code that inherits both the pseudorandomness feature of the former code and the efficiency features of the latter code. Second, we employ local arithmetic pseudo-random generators, proposing arithmetic generalizations of boolean candidates that resist all known attacks.