Can I build a secure tweakable block cipher from a normal one by adding key and tweak? Let (E,D) be a secure block cipher.

Consider the following tweakable block cipher:

E'(k,t,x) = E( (k+t) mod |K|, x)
D'(k,t,c) = D( (k+t) mod |K|, c)


Is (E',D') secure?
Well, whether it is a secure tweakable block cipher depends on how resistant (E,D) are to related key attacks; that's not a standard assumption for block ciphers.

For example, this would not be a secure tweakable block cipher with 3DES; because every 8th bit is ignored, the attacker can effectively test the value of the 7 adjacent bits (except for the 7 MSBits) by a choosen tweak attack.  For example, if he can find a value t with (for an arbitrary block x):

E'(k,t,x) = E'(k,t+256,x)
E'(k,t+2,x) != E'(k,t+258,x)


then he can deduce that the addition of k and t+258 was just enough to propagate a carry past bit 8 (which doesn't affect the operation of 3DES) into bit 9 (which does).  This allows him to immediate deduce the lowest 7 used bits of the key.

Repeating this observation for the rest of the key bits allows us to recover all but 7 bits of the key with about 400 adaptively chosen tweaks; the last 7 is easy to brute force.