Cryptographic Execution

Definition

• An execution of Cryptographic Protocol $\pi$ with environment $ϵ$ and adversary $A$ on initial input $z$ Begins by running $ϵ$ on $z$
• Machines take turns
• When $\mu =(\text{ID},c,\tilde{\mu })$ executes “transmit to ${\text{ID}}^{\prime }$”, $\mu$ is suspended
• Then:
  • If $\mu =ϵ$, message written to ${\mu }^{\prime }$ on the input tape with sender as some external entity, then, ${\mu }^{\prime }$ runs (Environment gives input to Protocol)
  • if $\mu \ne ϵ$, and the destination ${\text{ID}}^{\prime }$ is external, the message is written to $ϵ$‘s output tape with $\text{ID}$ and $\text{ID’}$ and $ϵ$ runs. (Protocol or adversary gives output to environment)
  • Otherwise, the message is written to ${\mu }^{\prime }$ with the given tape that the machine specifies and the $\text{ID}$ then ${\mu }^{\prime }$ runs. (Cryptographic Machine gives message to other machine in the Protocol)
  • If a machine yields without transmitting $ϵ$ runs
• When $ϵ$ halts, it gives some output ${\text{EXEC}}_{\pi ,A,ϵ}(z)$