Bitcoin uses ECDSA signatures to verify transactions. According to Wikipedia [1], the current record for solving the discrete logarithm problem (which is how you recover a private key from the corresponding public key) was done on a 112-bit key in 6 months. Bitcoin uses 256-bit keys, and the algorithm in question has a complexity of O(2^(N/2)), so breaking its keys is 2^72 times harder.
So even under the ridiculously unlikely assumption that computational power halves in cost every 18 months for the indefinite future: 50 years from now, the entire current global GDP still won't be enough to buy a processor that can crack a Bitcoin key in less than a decade.
(Of course, this is assuming there are no other unknown flaws in the protocol, and that no better algorithms are found to find discrete logarithms over elliptic curves. But if a vulnerability is found, it won't be thanks to Moore's law.)
So even under the ridiculously unlikely assumption that computational power halves in cost every 18 months for the indefinite future: 50 years from now, the entire current global GDP still won't be enough to buy a processor that can crack a Bitcoin key in less than a decade.
(Of course, this is assuming there are no other unknown flaws in the protocol, and that no better algorithms are found to find discrete logarithms over elliptic curves. But if a vulnerability is found, it won't be thanks to Moore's law.)
[1] http://en.wikipedia.org/wiki/Discrete_logarithm_records#Elli...