Efficient Statistical Asynchronous Verifiable Secret Sharing with Optimal Resilience

We present a new statistical asynchronous verifiable secret sharing (AVSS) protocol with optimal resilience; i.e. with n = 3t + 1, where n is the total number of participating parties and t is the maximum number of parties that can be under the control of a computationally unbounded active adversary A(t). Our protocol privately communicates O((ln(3) + n(4)kappa)kappa) bits and A-casts O(n(3) log(n)) bits to simultaneously share l >= 1 elements from a finite field F, where kappa is the error parameter. There are only two known statistical AVSS protocols with n = 3t + 1, reported in [11] and [26]. The AVSS protocol of [11] requires a private communication of O(n(9)kappa(4)) bits and A-cast of O(n(9)kappa(2) log(n)) bits to share a single element from F. Thus our AVSS protocol shows a significant improvement in communication complexity over the AVSS of [11]. The AVSS protocol of [26] requires a private communication of O((ln(3) + n(4))kappa) bits and A-cast of O((ln(3) + n(4))kappa) bits to share l >= 1 elements. However, the shared element(s) may be NULL is not an element of F. Thus our AVSS is better than the AVSS of [26] due to two reasons: (a) The A-cast communication of our AVSS is independent of the number of secrets i.e. l; (b) Our AVSS makes sure that the shared value(s) always belong to F. Using our AVSS, we design a new primitive called Asynchronous Complete Secret Sharing (ACSS) which is an essential building block of asynchronous multiparty computation (AMPC). Using our ACSS scheme, we can design a statistical AMPC with optimal resilience; i.e., with n = 3t + 1, that privately communicates O(n(5) kappa) bits per multiplication gate. This will significantly improve the only known statistical AMPC of [8] with n = 3t + 1, which privately communicates Omega(n(11) kappa(4)) bits and A-cast Omega(n(11) kappa(2) log(n)) bits per multiplication gate.