DES 48-bit round keys replaced with 48-bit block keys

Question

In a variant of DES with a 48*16 = 768-bit key, the sixteen 48-bit round keys scheduled for successive rounds in the encryption process for DES are replaced in this variant by successive 48-bit blocks of the 768-bit key.

1. What are the weak keys for this variant of DES, i.e. the keys for which encryption and decryption give the same transformation on input blocks?
2. What are the semi-weak key pairs, i.e. the pairs $k_1, k_2$ of keys for which encryption with $k_1$ is the same transformation as decryption with $k_2$?

Answer 1

1. Let the DES-variant's key $k=sk_0|sk_1|...|sk_{15}$, where each $sk_i$ denotes a 48-bit subkey and "|" concatenation. Due to the Feistel nature of DES, a sufficient condition for $sk_i$ is that for $i \in [0,15]$, subkeys $sk_i = sk_{15-i}$. They will work as each others' complements and encryption will become identical to decryption. There will be $2^{48*8}=2^{384}$ such sets, or keys $k$. Relatively speaking this is still far less than the number of weak keys in original DES, due to the large number of freely selected key bits.

2. The semi-weak key definition makes sense only in the original DES, where the round keys depend on the secret key: both the definition of weak and semi-weak keys aim at producing identical round keys for encryption and decryption; the weak keys are sufficient to accomplish this by themselves, as semi-weak keys come in sets.

P.S. I have hard time imagining any other purpose for this kind of cryptosystem than purely educational