Let's say I need to verify a digital signature and the public key is stored in untrusted storage which I need to fetch before I verify the digital signature. However, I can store the cryptographic hash of the public key in trusted storage and I decide to use the hash to verify the integrity of the public key. Now say I am using RSA PKCS #1 1.5 signature scheme, the public key contains (n, e), i.e., modulus and public exponent.
Here is the questions: Can I compute the only hash of the modulus for integrity? Do I get any security advantage if I compute the hash of the modulus concatenate with the public exponent? Specifically, I wonder what could go wrong if the attacker can change the my public exponent? (Assume the signature is created with public exponent 65537)