In the aspect of theory, Encryption and signatrue scheme both are on the basis of one-way function, which can prevent ciphertext and signature from revealing and forgery respectively. Therefore, at this point, it seems that converting between the two could be possible, such as RSA, elgmal in textbook, etc.
Howerver, in practice, duo to the security requirements of the two are totally different, such that for a encryption scheme, it should meet IND-CCA security; but for a signing scheme, it should meet UF-CMA security.
So, generally speaking, the type of mathematical assumption the two based on is also different:
- The encryption scheme always based on decisional assumptions, such as DDH, DLP, etc., to confuse the adversary that whether the ciphertext c is the encryption result of $m_1$ or $m_0$. Thus, we can conclude that the encryption algorithm should be probabilistic.
- The signing scheme always based on computational assumptions, such as CDH, SDH, etc., to prevent the adversary from forging a valid signature. we also can conclude that the signing algorithm should be deterministic.
This is my opinion about your question, not detailed but should be able to answer your question.