a. In Section 3.1, under the subsection on the motivation for the Feistel cipher structure, it was stated that, for a block of \(n\) bits, the number of different reversible mappings for the ideal block cipher is \(2^{n}\) !. Justify.

b. In that same discussion, it was stated that for the ideal block cipher, which allows all possible reversible mappings, the size of the key is \(n \times 2^{n}\) bits. But, if there are \(2^{n}\) !

possible mappings, it should take \(\log _{2} 2^{n}\) ! bits to discriminate among the different mappings, and so the key length should be \(\log _{2} 2^{n}\) !. However, \(\log _{2} 2^{n} !Explain the discrepancy.