A Tight Lower Bound on Adaptively Secure Full-Information Coin Flip

In a distributed coin-flipping protocol, Blum [ACM Transactions on Computer Systems '83], the parties try to output a common (close to) uniform bit, even when some adversarially chosen parties try to bias the common output. In an adaptively secure full-information coin flip, Ben-Or and Linial [FOCS '85], the parties communicate over a broadcast channel and a computationally unbounded adversary can choose which parties to corrupt along the protocol execution. Ben-Or and Linial proved that the n-party majority protocol is resilient to O(√n) corruptions (ignoring poly-logarithmic factors), and conjectured this is a tight upper bound for any n-party protocol (of any round complexity). Their conjecture was proved to be correct for single-turn (each party sends a single message) single-bit (a message is one bit) protocols Lichtenstein, Linial, and Saks [Combinatorica '89], symmetric protocols Goldwasser, Tauman Kalai, and Park [ICALP '15], and recently for (arbitrary message length) single-turn protocols Tauman Kalai, Komargodski, and Raz [DISC '18]. Yet, the question for many-turn protocols was left completely open. In this work we close the above gap, proving that no n-party protocol (of any round complexity) is resilient to ω(√n) (adaptive) corruptions.