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The condition is that: $$g(0, x_2, x_3, ..., x_n) \ne g(1, x_2, x_3, ..., x_n)$$ for all $x_2, x_3, ..., x_n$ This can easily be derived from the condition that implies bijectivity of $f$; that is, $f(x_1, x_2, ..., x_n) = f(y_1, y_2, ..., y_n)$ implies that $x_1 = y_1$, $x_2 = y_2$, ..., $x_n = y_n$


For any fixed $x_2, \ldots, x_n$, $g$ must be a surjective function of $x_1$ (i.e. onto).


(posting as an "answer" because it's a little to long to fit in the "comment" section) It sounds like you are trying to build a 256 bit block cipher, given a 128 bit cipher and 2 keys for that cipher. I'm gonna clarify one more time ... The way I intended this to work: the decryptor uses the second key to decrypt every even block of the full ...


There are two major problems with this method. The first problem is that Susan is likely to be able to recover significant amount of data from a series of such blocks. For example, if Susan knows $subkey_1$, then she could recover the value $subblock_1 \oplus subblock_2 \oplus subkey_2$; if a single block is encrypted with this key, she can't deduce ...

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