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Lets fix that description of the signature generation. Please read the PKCS#1 v2.2 specifications to get the full detail.
Canonical binary plaintext gets run through a hashing mechanism to make a message digest (IBM's terminology, there appear to be plenty of other terms for the hashed data).
Message digest is used as input for the configured padding ...
2
TLDR: Can't do.
16-bit is simply not enough for digital signature, which requires many times more bits¹, and can't be checked if truncated (becoming useless).
One may want to consider a Message Authentication Code, such as HMAC with SHA-256. At least, that's cryptographically better than an encrypted CRC². But there must be a shared secret between sender ...
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