Communication efficient perfectly secure VSS and MPC in asynchronous networks with optimal resilience

• Authors: Arpita Patra;Ashish Choudhury;C. Pandu Rangan
• Venue: AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa
• Year: 2010

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 ${\mathbb F}$ concurrently, with a communication cost of ${\cal O}(\ell n^2 \log{|{\mathbb F}|})$ bits, where ${\mathbb F}$ 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 dt. Interpreting in a different way, we may also say that our AVSS shares ℓ(d+1−t) secrets simultaneously with a communication cost of ${\cal O}(\ell n^2 \log{|{\mathbb F}|})$ 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 ${\cal O}(n \log{|{\mathbb F}|})$ bits. This is a clear improvement over the AVSS of [2] whose amortized cost of sharing a single secret is ${\cal O}(n^2 \log{|{\mathbb F}|})$ bits. As an interesting application of our AVSS, we propose a new optimally resilient, perfectly secureAsynchronous Multiparty Computation (AMPC) protocol that communicates ${\cal O}(n^2 \log|{\mathbb F}|)$ bits per multiplication gate. The best known optimally resilient perfectly secure AMPC is due to [2], which communicates ${\cal O}(n^3 \log|{\mathbb F}|)$ bits per multiplication gate. Thus our AMPC improves the communication complexity of the best known AMPC of [2] by a factor of Ω(n).