Security of Fast Two-Party ECDSA Signing
In real life, $P_1$ of course does not abort randomly; this would make no sense. The idea is as follows. Assume that a cheating $P_2$ can forge a signature with probability $\epsilon$ when interacting to obtain $p(n)$ signatures are signed with the system. Now, if $P_2$ cheats then there is a first place where $P_1$ aborts. An adversary for ECDSA can simulate everything and guess where this first abort happens and will be correct with probability $1/p(n)$. This implies that the adversary for ECDSA can forge a signature with probability $\epsilon/p(n)$. Since ECDSA is assumed to be secure, this means that $\epsilon$ must be negligible.