At least get the terminology right in your FUD: ECDSA is not encryption.
yeah ok, my error,
the ECDSA scheme that is used, and if it can be reverse engineered makeing all coins remaining in a address that has spent coins, able to be acquired.
The loss of confidence by itself would be very damaging.
Is there any way to replace this with a more secure mathematical scheme?
gmaxwell makes a fair point that the secp256k1 curve parameters are not especially magical and can be deduced based on logic. however, that does not rule out the curve having some hidden degeneracy that substantially weakens the encryption.
gmaxwell has, in the past, suggested that djb's Ed25519 is an interesting alternative to secp256k1. the trouble with swapping out ecdsa is that it is rather smart about keeping the key size small, meaning that any replacement would have to compete on the key size vs security parameter. most post quantum algorithms have a relatively large key size vs security parameter and there aren't a whole lot of alternatives out there to ecc. the average tx size would go up along with the average key size, leading to blockchain bloat.
even if the USG et al can break secp256k1, i'll take that over banking as it currently stands. unless the weakness in ecdsa, specifically secp256k1, is really acute, you can get an edge by spreading your coins out between many addresses, i.e. keep less than X BTC at each address. this way it makes it more expensive to steal those coins, assuming someone can break ecdsa.
need to dig for it! 
piotr_n you were part of the thread where we reproduced our curve parameters, were you not?