Symmetric-Key Cryptography - Codes: The Guide to Secrecy from Ancient to Modern Times - page 122

The last aspect of S-DES is the inverse of the initial permutation.
The Inverse of IP
The inverse of IP , naturally denoted by IP − 1 , is given by the following.
IP − 1
j 12345678
IP − 1 ( j ) 41357286
An easy means for finding the inverse of any permutation is as follows. Take
the table for IP on page 118, for instance. To find the inverse, just read off in
numeric order (determined by the second row), the terms in the first row. For
instance, the term in the first row sitting above 1, in the aforementioned table
for IP , is 4, so 4 is the first term in the table for IP − 1 . The term in the first
row sitting above 2 is 1, so 1 is the second entry in the table for IP − 1 , and so
on, to construct the above. Note that the reason for the above to work is that
IP ( IP − 1 ( j )) = j for all j under consideration. So since IP takes 1 to 2, then
IP − 1 must take 2 to 1, and so forth. (See Definition A.5 on page 467.)
Now, we are in a position to describe the totality of the S-DES cipher.
The S-DES Cryptosystem
Given a 10-bit key k and an 8-bit plaintext m , to encipher, we execute the
following.
S-DES Encryption
1. Apply IP to m .
2. Apply f k 1 to the output from step 1. (This is round 1.)
3. Apply SW to the output of step 2.
4. Apply f k 2 to the output of step 3. (This is round 2.)
5. Apply IP − 1 to the output of step 4.
Hence, the plaintext 8-bit message unit m gets sent to the 8-bit ciphertext
message unit c , the output of step 5, under this sequence of steps of the S-DES
cipher. To decrypt, we perform the following.
S-DES Decryption
1. Apply IP to c .
2. Apply f k 2 to the output from step 1. (This is round 1.)
3. Apply SW to the output of step 2.