Secret key generation refers to the problem of generating a common secret key without revealing any information about it to an eaves-dropper. All users observe correlated components of a common source and can further use a rate-limited public channel for discussion which is open to eavesdroppers. This paper studies the Turing computability of the secret key capacity with a single rate-limited public forward transmission. Turing computability provides fundamental performance limits for today's digital computers. It is shown that the secret key capacity under rate constraints is not Turing computable, and consequently there is no algorithm that can simulate or compute the secret key capacity, even if there are no limitations on computational complexity and computing power. On the other hand, if there are no rate constraints on the forward transmission, the secret key capacity is Turing computable. This shows that restricting the communication rate over the public channel transforms a Turing computable problem into a non-computable problem. To the best of our knowledge, this is the first time that such a phenomenon has been observed.