Suppose you have a public program P(n) which takes message n and generates an encrypted output (utilizing asymmetric cryptography) for some entity which has the private key to decrypt it.
Using a scheme such as Elligator (for elliptic curves) it is possible to have P which has output indistinguishable from random.
But a problem arises when the output is expected to be subject to corruption (bit flips). An application of error correcting codes seems to break the 'indistinguishable from random' requirement.
After considering the problem I arrived at the following consideration. The actual message n is encrypted by a symmetric encryption scheme using some shared secret, the same shared secret (after KDF) can be used to whiten (w/ stream cipher) the result after the application of error correcting codes to it. Would there be a problem with this approach?
This leaves the ephemeral public key (~256 bits) which due to Elligator is indistinguishable from random. But is subject to corruption. How can error correction be used on it such that the result is itself indistinguishable from random?
There appears to be some confusion regarding the encryption part. It's a standard anonymous sender sealed box. https://libsodium.gitbook.io/doc/public-key_cryptography/sealed_boxes
ephemeral_pk ‖ box(m, recipient_pk, ephemeral_sk, nonce=blake2b(ephemeral_pk ‖ recipient_pk))
The 'indistinguishable from random' requirement is mandatory, there is a Markov chain that mixes it with a structured format. Extraction of this embed cannot be allowed to yield something containing detectable error correcting codes, betraying its presence. In its absence, actual noise with no statistical pattern is extracted instead.