Suppose that we have an EU-CMA or sEU-CMA signature scheme without random oracles. My question is if messages to be signed are in the form H(m), where H is a hash function (doesn't belong to the signature parameters). Does this affect the security of the scheme?
EU-CMA: existentially unforgeable under chosen message attack.
sEU-CMA: Strongly EU-CMA.
See digital signature forgery for more info.