  This paper gives simple distributed algorithms for the fundamental problem of
computing graph distances in the Congested Clique model. One of the main
components of our algorithms is fast matrix multiplication, for which we show
an $O(n^{1/3})$-round algorithm when the multiplication needs to be performed
over a semi-ring, and an $O(n^{0.157})$-round algorithm when the computation
can be performed over a field. We propose to denote by $\kappa$ the exponent of
matrix multiplication in this model, which gives $\kappa < 0.157$.
  We show how to compute all-pairs-shortest-paths (APSP) in $O(n^{1/3}\log{n})$
rounds in weighted graphs of $n$ nodes, implying also the computation of the
graph diameter $D$. In unweighted graphs, APSP can be computed in
$O(\min\{n^{1/3}\log{D},n^{\kappa} D\})$ rounds, and the diameter can be
computed in $O(n^{\kappa}\log{D})$ rounds. Furthermore, we show how to compute
the girth of a graph in $O(n^{1/3})$ rounds, and provide triangle detection and
4-cycle detection algorithms that complete in $O(n^{\kappa})$ rounds.
  All our algorithms are deterministic. Our triangle detection and 4-cycle
detection algorithms improve upon the previously best known algorithms in this
model, and refute a conjecture that $\tilde \Omega (n^{1/3})$ rounds are
required for detecting triangles by any deterministic oblivious algorithm. Our
distance computation algorithms are exact, and improve upon the previously best
known $\tilde O(n^{1/2})$ algorithm of Nanongkai [STOC 2014] for computing a
$(2+o(1))$-approximation of APSP.
  Finally, we give lower bounds that match the above for natural families of
algorithms. For the Congested Clique Broadcast model, we derive unconditioned
lower bounds for matrix multiplication and APSP. The matrix multiplication
algorithms and lower bounds are adapted from parallel computations, which is a
connection of independent interest.
