Many results in cryptography depend on impossibility results/conjectures in complexity theory. For example, public-key cryptography using RSA is believed to be possible because of the conjecture about infeasibility of the factoring (and the modular root finding problems).
My question is :
do we have similar results in computability theory? Are there interesting positive constructions using negative impossibility results?
E.g., does the undecidability of the halting problem allow us to perform tasks that we would not be able to do if the halting problem was decidable?