multi-user deterministic symmetric encryption

Question

I was wondering whether there exists a symmetric encryption scheme such that

1. there is 1 sender and n receivers. Each receiver has 1 random and independent symmetric key. The sender knows the symmetric keys of all receivers in advance.
2. the sender can encrypt one message using the above n keys and produce a ciphertext. Each receiver can decrypt the ciphertext using his own symmetric key to recover the message.
3. the above n keys are used only for encrypting/decrypting 1 message.
4. the encryption is deterministic. This excludes some trivial constructions such that sampling one shared key that is used for encrypting the message, followed by respectively encrypting the shared key using n symmetric keys.
5. the scheme achieves somewhat one-time-CPA security. Of course, compared to the standard notion, the difference is that we need to consider n keys instead of 1 key.

Motivation: Now I am facing one problem that a large file needs to be sent to multiple users (only once) under the condition that the randomness generator of the sender is broken (or unreliable).

Answer 1

Not sure there's anything out of the box.

A trivial construction might be to encrypt the file with a symmetric key and then encrypt the symmetric key n times with receiver symmetric keys and prepend that to the ciphertext. The downsides are obvious.

A broken randomness is going to be a major downside for any symmetric cipher that uses an IV (all the good ones).