It always depends on what you mean by secure, but note that, as SAI Peregrinus said in the other answer, not every cryptosystem is provably secure.
For example showing that it is easy to produce ciphertext from
plaintext but is difficult/impossible to get plain text from
The property you describe here is called one-wayness (OW), and it is expected that any encryption scheme fulfills this property; otherwise, it is a rather crappy scheme, isn't it?
However, it is just a very minimal requirement for an encryption scheme, and it is often considered not enough. There is a stronger security goal called indistinguishability (IND), which dictates that the encryption scheme should not permit an adversary to distinguish which message (from two possible options, $m_0$ and $m_1$) is encrypted by a given ciphertext.
In a real setting, this basically means that even if the adversary cannot extract the original message from the ciphertext, if he suspects that the ciphertext is the encryption of two possible messages, he should not be able to guess correctly which one it is.
Of course, these are security notions from the provable security field. As SAI Peregrinus mentioned, there are many other aspects that can be exploited, such as side-channel attacks, implementation bugs, etc.
Would these things be different from what I need to prove to show it is a good digital
Yes, because the security objectives are different. With digital signatures you are not trying to achieve confidentiality, as in an encryption scheme, but authentication and integrity. Basically, what you want with a digital signature is that the adversary cannot forge signatures, which is called unforgeability.