In SKE, can we assume without loss that the ciphers of a fixed plaintext distribute identically?
Is this possible without using any cryptographic assumptions which don't already follow from the existence of an IND-CCA1 secure SKE?

Yes, symmetric-key encryption implies the existence of &quot;uniform&quot; symmetric-key encryption. But I don't know of a particularly direct way to show it. Below is an extremely convoluted, theoretical, impractical way to do such a transformation.

If CPA-secure symmetric-key encryption schemes exist, then so do one-way functions. This is shown in Impagliazzo-Luby.

If one-way functions exist, then PRGs exist; this is the classic &quot;HILL&quot; result.

If PRGs exist then PRFs exist; this is the classic GGM result.

Once you have a PRF you can construct an authenticated encryption scheme based on counter mode with a MAC like Carter-Wegman. This scheme will have the property that you need, since the ciphertext is a deterministic function of the (uniform) initialization vector + key + plaintext, and no two choices of initialization vector give the same output (since the IV is included in the output). Hence the output is uniform over the support.