Verifiable Secret Sharing (VSS) is a fundamental primitive used in many distributed cryptographic tasks, such as Multiparty Computation (MPC) and Byzantine Agreement (BA). It is a two phase (sharing, reconstruction) protocol. The VSS and MPC protocols are carried out among n parties, where t out of n parties can be under the influence of a Byzantine (active) adversary, having unbounded computing power. It is well known that protocols for perfectly secure VSS and perfectly secure MPC exist in an asynchronous network iff n ≥ 4t + 1. Hence, we call any perfectly secure VSS (MPC) protocol designed over an asynchronous network with n = 4t + 1 as optimally resilient VSS (MPC) protocol. A secret is d-shared among the parties if there exists a random degree-d polynomial whose constant term is the secret and each honest party possesses a distinct point on the degree-d polynomial. Typically VSS is used as a primary tool to generate t-sharing of secret(s). In this paper, we present an optimally resilient, perfectly secure Asynchronous VSS (AVSS) protocol that can generate d-sharing of a secret for any d, where t ≤ d ≤ 2t. This is the first optimally resilient, perfectly secure AVSS of its kind in the literature. Specifically, our AVSS can generate d-sharing of ℓ ≥ 1 secrets from double-struck F concurrently, with a communication cost of script O(ℓn 2 log|double-struck F|) bits, where double-struck F is a finite field. Communication complexity wise, the best known optimally resilient, perfectly secure AVSS is reported in [2]. The protocol of [2] can generate t-sharing of ℓ secrets concurrently, with the same communication complexity as our AVSS. However, the AVSS of [2] and [4] (the only known optimally resilient perfectly secure AVSS, other than [2]) does not generate d-sharing, for any d > t. Interpreting in a different way, we may also say that our AVSS shares ℓ(d + 1 - t) secrets simultaneously with a communication cost of script O(ℓn2 log|double-struck F|) bits. Putting d = 2t (the maximum value of d), we notice that the amortized cost of sharing a single secret using our AVSS is only script O(n log |double-struck F|) bits. This is a clear improvement over the AVSS of [2] whose amortized cost of sharing a single secret is script O(n2 log|double-struck F|) bits. As an interesting application of our AVSS, we propose a new optimally resilient, perfectly secure Asynchronous Multiparty Computation (AMPC) protocol that communicates script O(n2 log|double-struck F|) bits per multiplication gate. The best known optimally resilient perfectly secure AMPC is due to [2], which communicates script O(n3 log|double-struck F|) bits per multiplication gate. Thus our AMPC improves the communication complexity of the best known AMPC of [2] by a factor of Ω(n). © 2010 Springer-Verlag.