Secure arithmetic computation with constant computational overhead

Abstract

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 |𝔽| 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 ∈ 𝔽 is the input of one party and a, b ∈ 𝔽w 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.