| Abstract: |
Two of the most sought-after properties of multi-party computation (MPC) protocols are fairness and guaranteed output delivery (GOD), the latter also referred to as robustness. Achieving both, however, brings in the necessary requirement of malicious-minority. In a generalized adversarial setting where the adversary is allowed to corrupt both actively and passively, the necessary bound for a n -party fair or robust protocol turns out to be $$t_a + t_p < n$$ t a + t p < n , where $$t_a,t_p$$ t a , t p denote the threshold for active and passive corruption with the latter subsuming the former. Subsuming the malicious-minority as a boundary special case, this setting, denoted as dynamic corruption, opens up a range of possible corruption scenarios for the adversary. While dynamic corruption includes the entire range of thresholds for $$(t_a,t_p)$$ ( t a , t p ) starting from $$(\lceil \frac{n}{2} \rceil - 1, \lfloor \frac{n}{2} \rfloor )$$ ( ⌈ n 2 ⌉ - 1 , ⌊ n 2 ⌋ ) to $$(0,n-1)$$ ( 0 , n - 1 ) , the boundary corruption restricts the adversary only to the boundary cases of $$(\lceil \frac{n}{2} \rceil - 1, \lfloor \frac{n}{2} \rfloor )$$ ( ⌈ n 2 ⌉ - 1 , ⌊ n 2 ⌋ ) and $$(0,n-1)$$ ( 0 , n - 1 ) . Notably, both corruption settings empower an adversary to control majority of the parties, yet ensuring the count on active corruption never goes beyond $$\lceil \frac{n}{2} \rceil - 1$$ ⌈ n 2 ⌉ - 1 . We target the round complexity of fair and robust MPC tolerating dynamic and boundary adversaries. As it turns out, $$\lceil \frac{n}{2} \rceil + 1$$ ⌈ n 2 ⌉ + 1 rounds are necessary and sufficient for fair as well as robust MPC tolerating dynamic corruption. The non-constant barrier raised by dynamic corruption can be sailed through for a boundary adversary. The round complexity of 3 and 4 is necessary and sufficient for fair and GOD protocols, respectively, with the latter having an exception of allowing 3-round protocols in the presence of a single active corruption. While all our lower bounds assume pairwise-private and broadcast channels and hold in the presence of correlated randomness setup (which subsumes both public (CRS) and private (PKI) setup), our upper bounds are broadcast-only and assume only public setup. The traditional and popular setting of malicious-minority, being restricted compared to both dynamic and boundary setting, requires 3 and 2 rounds in the presence of public and private setup, respectively, for both fair and GOD protocols. |
