Some simple questions about tweakable ciphers I have read a few papers on tweakable ciphers (didn't understand them well, though) and looked at many of the questions and answers on this exchange: What is a tweakable block cipher, Tweakable Block ciphers, Tweaking Even-Mansour Ciphers [video]. However, there are a few things I'd like cleared up.


Are the tweaks always just a string of bits? And are they usually shorter than the key?
It seems DES-X is a tweakable cipher where the tweaks are the pre- and post-whitening that use secret extra keys. Does this mean some tweaks are necessarily secret? The reason I am asking is that I got the impression from the papers I read that tweaks are always public.
In view of Q2, what other tweakable ciphers (if any) use secret tweaks or ones that may be secret if desired?
If a tweak is secret does this add strength to the cipher as well as variability?
If a tweak is appended to a key, does this mean to the session key or to each subkey? I am a little confused because I thought part of the idea of tweaks is that they are easier to change than producing new keys through a complex key schedule (for instance).
Perhaps this should be a separate question, but: the term permutation seems to mean two different things. 1) a bit-wise (say) permutation, i.e. excluding XORs, S-boxes, etc. and 2) a complete block cipher encryption that produces an apparently random permutation of the input, i.e. typically including XORs and substitutions, etc. In relevance to tweakable ciphers, when I look at some models I see the cipher is denoted by $E_n$ but when I look at models based on the Even-Mansour ciphers I see $P_n$. I assume the $E$ refers to an cipher that may include several operations (perms, s-boxes, etc. as for $DES$, $AES$, etc.) but the $P$ refers only to bit-wise permutations (or perhaps byte-wise). Is this assumption correct?


Much obliged to anyone who can help.
Suppose the tweakable cipher uses a secret key $k$ and a secret tweak $t$, then syntactically you can regard $K=(k, t)$ as the total secret key.  So, in this case, the 'tweakable cipher' is just an ordinary block cipher, with a longer key.  In order to get a notion different from the usual block cipher, you'll need the tweak to be non-secret.