DES decryption of the homomorphic encryption ciphertext

I implemented an application using partial homomorphic encryption for outsourced computations. To get an efficient bandwidth, I am thinking to apply (DES) symmetric algorithm to encrypt the message instead of HE before uploading it to the cloud.

Can I perform DES decryption operations (permutation and S-boxes) on homomorphic encryption ciphertext?

\begin{align} &C = \operatorname{DES}(m)\\ &C'= \operatorname{FHE}(C) = \operatorname{FHE}(\operatorname{DES}(m)) \\ &\operatorname{DES}^{-1} (C' )= \operatorname{DES}^{-1} (\operatorname{FHE}(C))= \operatorname{DES}^{-1} (\operatorname{FHE}(\operatorname{DES}(m)))= \operatorname{FHE}(m) \end{align}

• Keep in mind that Kreyvium is designed from Trivium with increased security for decreasing the data sent to the cloud with FHE. Also, partial will not be enough to achieve that. – kelalaka Jul 17 '20 at 18:51
• Note that there also exist (asymptotically) rate-1 FHE schemes. The relevant papers are here and here. – Mark Jul 19 '20 at 3:47
• This seems dangerous. If anybody can skip the homomorphic encryption then performing $\text{DES}^{-1}$ would directly give an adversary the plaintext. Because I presume you only store $C$ in the cloud. I don't think many FHE schemes allow for efficient bitops, and DES (decryption) is a lot of rounds with a lot of bitops. – Maarten Bodewes Jul 19 '20 at 15:33
• Brainstorm: XOR with a key stream might be easier to perform (storing the nonce next to the ciphertext). – Maarten Bodewes Jul 19 '20 at 15:36
• Besides this interesting thought experiment, I really wonder how you got to something as archaic as DES starting from FHE, by the way. It's like putting a steam locomotive in front of a TGV. – Maarten Bodewes Jul 19 '20 at 15:47