May an adversary be fooled by a random-looking input that in reality is fixed? I want to prove the EUF-CMA-security of a signature scheme. It is a variation on an established scheme, therefore I would ideally like to reduce the new-scheme-security to the old-scheme-security.
the (very abstract) construction
Let the old scheme be called S and let k denote a number. In S, k is fixed, say k=5. Imagine the new scheme N is identical, only with k becoming random and secret (because it is now related to the key material). So, k=5 is in theory still possible but may just as well be 1,2,3, .. and k=5 is just as likely as k = any other number.
One needs to know k to verify the signature, and k acts similar to a symmetric key. The setting is a group where you want to identify an individual sender but still share symmetric keys.
the reduction attempt
An adversary may query oracles for different values of k and the challenger can simply draw them randomly.
However, in the challenge setting, it may obviously not know k, which is secret.
Now, it would be easy to reduce the security of one scheme to the other if the adversary were to be tricked into outputting a signature for k=5. Then the security of N can be reduced to the security of S.
The adversary can not tell from the key material or the signatures or the verification equation that k=5 (because everything still appears random), except if it can break the DLog assumption.
So the questions are:

An adversary can easily check whether k=5 or k!=5 in the challenge setting by attempting to verify a signature using k=5. Would an adversary realize that it is not interacting with the
scheme but with a &quot;simulation&quot; if it finds out that k=5 in the challenge setting? (In which case, as far as I understand, it would not
output a signature but break off the game.) Or is 5 a good a number as any, from the perspective of the adversary?
I can not play the game differently for different adversaries, depending on whether they compute k or not, because I know nothing about how A works - correct? (See Katz, Lindell, Introduction to Modern Cryptography, pg.59) Or could I play differently for &quot;DLog-Breakers&quot; and &quot;DLog-Ignorants&quot;?
Would an adversary realize that it is not interacting with the scheme but with a &quot;simulation&quot; if it finds out that k=5 in the challenge setting?

This is commonly stated as Kerckhoff's Principal, which I prefer to cite as Shannon's Maxim (they both stated it, and Shannon's Maxim sounds better). The idea is, the adversary should be assumed to know everything about a secure system, except the key, if it is to be considered secure at all. As such, the adversary must be assumed to know that you could lie to them about the value of $k$.
Also, a signature scheme could not be considered secure if it involves an oracle capable of lying about what signatures are valid or invalid. As such, if the system is to be secure at all, it must behave the same for an attacker as if they were a legitimate verifier.