I'm wondering if a binary elliptic curve (such as sect571r1 aka B-571) supports pairs of asymmetric operations (for example, either sign/verify or encrypt/decrypt) on a fixed bit or byte input size in a way that this operation becomes fully bijective on those bytes/bits (where the inverse operation requires the opposite key).
If the alignment of such an operation doesn't fall on a byte (8-bit) boundary (but instead falls on just a bit boundary) then this is an alternate route to finding something similar in spirit, since it would then be possible to perform two, four, or eight similar operations until the extra bits are consolidated in a series of aggregate operations. This is workable, but something like a prime field I believe cannot support bijective operation on bits.
This question is also asking for some clarification on the operations of elliptic curves for these pairs of operations (like sign/verify or encrypt/decrypt) that's not the typical Diffie-Hellman key agreement (though maybe there's a simple way to relate that to encrypt/decrypt). Basically, suppose I have an input of "n" bytes, and some key pair (public, private). I'm wondering if it's possible to encrypt that arbitrary input with a public key, and then subsequently decrypt with the private key to fully recover the original "n" bytes - without any additional encoding or exceptional cases/constraints needed on the input/output.
The relative cryptographic strength of binary curves versus prime curves isn't part of the question. Although the other aspect I'm wondering about is the most straightforward, succinct, and self-contained example in code that performs these primitive operations without unnecessary elements like padding, wrapping, hashing, and so on. It can be assumed I can find an appropriate fixed size input to match the binary elliptic curve. And then, I presume if the encrypt-then-decrypt with the respective public-then-private key is doable, then so would sign-then-verify with the private-then-public key (similar to RSA).