  We study the problem of reaching agreement in a synchronous distributed
system by $n$ autonomous parties, when the communication links from/to faulty
parties can omit messages. The faulty parties are selected and controlled by an
adaptive, full-information, computationally unbounded adversary. We design a
randomized algorithm that works in $O(\sqrt{n}\log^2 n)$ rounds and sends
$O(n^2\log^3 n)$ communication bits, where the number of faulty parties is
$\Theta(n)$. Our result is simultaneously tight for both these measures within
polylogarithmic factors: due to the $\Omega(n^2)$ lower bound on communication
by Abraham et al. (PODC'19) and $\Omega(\sqrt{n/\log n})$ lower bound on the
number of rounds by Bar-Joseph and Ben-Or (PODC'98). We also quantify how much
randomness is necessary and sufficient to reduce time complexity to a certain
value, while keeping the communication complexity (nearly) optimal. We prove
that no MC algorithm can work in less than $\Omega(\frac{n^2}{\max\{R,n\}\log
n})$ rounds if it uses less than $O(R)$ calls to a random source, assuming a
constant fraction of faulty parties. This can be contrasted with a long line of
work on consensus against an {\em adversary limited to polynomial computation
time}, thus unable to break cryptographic primitives, culminating in a work by
Ghinea et al. (EUROCRYPT'22), where an optimal $O(r)$-round solution with
probability $1-(cr)^{-r}$ is given. Our lower bound strictly separates these
two regimes, by excluding such results if the adversary is computationally
unbounded. On the upper bound side, we show that for $R\in\tilde{O}(n^{3/2})$
there exists an algorithm solving consensus in $\tilde{O}(\frac{n^2}{R})$
rounds with high probability, where tilde notation hides a polylogarithmic
factor. The communication complexity of the algorithm does not depend on the
amount of randomness $R$ and stays optimal within polylogarithmic factor.
