Can bitcoins security algorithms become outdated due to Moore's law in any reasonable period of time? Is there a possible threat of algorithm compromise by cryptanalysis?
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.)
As for mining, there are enough bits in SHA256 to allow billion-fold increase in difficulty. And if there is a danger of breaking it much quicker, all people holding bitcoins have huge incentive to change the protocol to fix the problem.
As for ECDSA, it can only be broken by quantum computers, but still most balances are protected by sha256*ripemd160 hash, so they are safe in the face of sudden QC attack or a weakness in ECC. People will want to protect their wealth and thus will change the protocol for other algorithms.
When everyone's money is at stake, it'll be very quick. This March, when the network forked, miners with newer version 0.8 agreed to forego mining rewards and abandon their own branch (incompatible with v0.7) just to resolve the issue quickly to not shake the confidence in Bitcoin.
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...