In Statistical Mechanics (physics), we have something called The Law of *really* Large Numbers. This law basically says that if X is a normal number (like 2 or 44 or 193) and Z is a *really* large number, then we can approximate their product as X*Z = Z.
2^256 is not as big as the really large numbers in stat mech, but the rule still basically applies to bitcoin: the probability of brute forcing *any* private key that actually contains funds is essentially unchanged regardless of how many addresses get created or how many times those addresses sign transactions: nil.
2^256 is not as big as the really large numbers in stat mech, but the rule still basically applies to bitcoin: the probability of brute forcing *any* private key that actually contains funds is essentially unchanged regardless of how many addresses get created or how many times those addresses sign transactions: nil.
Just to clarify, this vulnerability does not involve computing private keys corresponding to a re-used public key, it is purely the synthesizing of valid signatures for an individual public key.