Cryptography can largely be divided into two classes: asymmetric and symmetric.
Like homomorphic encryption now greatly restricted in the asymmetric cryptography, the difference of asymmetric and symmetric, goes beyond "one-key or two-key", but flexibility, funcationality.
In some system setting, the utilization of asymmetric cryptography is not to provide a "public-key", but to offer funcationality like "additive homomorphic". To guarantee the security of such systems, the adversary with the knowledge of public key should be taken into account.
But, but, what if, we turn back to the history. We want to keep public key "secret".
If both public key and private key are not public, are secret key (i.e. secret key 1, secret key 2).
Some asymmetric cryptography could, if security proof can shows, to have less flexibility like "public-key", but contribute greater security to real-world system. The advantage of "additive-homomorphic" may remain but some security problems vanish.
My question is:
Is it generally acceptable to use asymmetric encryption in a symmetric manner?
Is there any existing works on this conception?
Does the security definition lack here for such case?