  The full-information model was introduced by Ben-Or and Linial in 1985 to
study collective coin-flipping: the problem of generating a common bounded-bias
bit in a network of $n$ players with $t=t(n)$ faults. They showed that the
majority protocol can tolerate $t=O(\sqrt n)$ adaptive corruptions, and
conjectured that this is optimal in the adaptive setting. Lichtenstein, Linial,
and Saks proved that the conjecture holds for protocols in which each player
sends a single bit. Their result has been the main progress on the conjecture
in the last 30 years.
  In this work we revisit this question and ask: what about protocols involving
longer messages? Can increased communication allow for a larger fraction of
faulty players?
  We introduce a model of strong adaptive corruptions, where in each round, the
adversary sees all messages sent by honest parties and, based on the message
content, decides whether to corrupt a party (and intercept his message) or not.
We prove that any one-round coin-flipping protocol, regardless of message
length, is secure against at most $\tilde{O}(\sqrt n)$ strong adaptive
corruptions. Thus, increased message length does not help in this setting.
  We then shed light on the connection between adaptive and strongly adaptive
adversaries, by proving that for any symmetric one-round coin-flipping protocol
secure against $t$ adaptive corruptions, there is a symmetric one-round
coin-flipping protocol secure against $t$ strongly adaptive corruptions.
Returning to the standard adaptive model, we can now prove that any symmetric
one-round protocol with arbitrarily long messages can tolerate at most
$\tilde{O}(\sqrt n)$ adaptive corruptions.
  At the heart of our results lies a novel use of the Minimax Theorem and a new
technique for converting any one-round secure protocol into a protocol with
messages of $polylog(n)$ bits. This technique may be of independent interest.
