  Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$
with $n$ vertices, in every step $i \geq 1$ the graph $G_i$ is formed by taking
an edge chosen uniformly at random among the non-existing ones and adding it to
the graph $G_{i - 1}$. The classical `hitting-time' result of Ajtai,
Koml\'{o}s, and Szemer\'{e}di, and independently Bollob\'{a}s, states that
asymptotically almost surely the graph becomes Hamiltonian as soon as the
minimum degree reaches $2$, that is if $\delta(G_i) \ge 2$ then $G_i$ is
Hamiltonian. We establish a resilience version of this result. In particular,
we show that the random graph process almost surely creates a sequence of
graphs such that for $m \geq (\tfrac{1}{6} + o(1))n\log n$ edges, the $2$-core
of the graph $G_m$ remains Hamiltonian even after an adversary removes
$(\tfrac{1}{2} - o(1))$-fraction of the edges incident to every vertex. A
similar result is obtained for perfect matchings.
