Symmetric-Key Cryptography - Codes: The Guide to Secrecy from Ancient to Modern Times - page 121

Thus, for example if x =( x 1 x 2 x 3 x 4 ) = (1101) is our input bitstring of
length 4, then if we wish to employ the first S-Box, we get S 0 (1101) = (11),
since ( x 1 x 4 ) = (11) represents the fourth row, and ( x 2 x 3 ) = (10) represents the
third column, the entry at the intersection of which is 11. Similarly, if we want
to use the S-Box S 1 , then S 1 (1101) = (00).
Perhaps the most complicated part of S-DES is the function that does the
combining of permutation and substitution.

The S-DES Round Function
First, we need to describe a mapping F that takes bitstrings of length 4
using a subkey SK , and outputs bitstrings of length 4.
Let x =( x 1 x 2 x 3 x 4 ) be the input. Then F first uses the expansion EP to
produce EP ( x ), as described on page 119. Then this 8-bit result is added to the
subkey SK , modulo 2. Recall that addition modulo 2 is denoted by

⊕ .Thus,

this result is denoted by EP ( x ) SK =( y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 )= y . For the sake

of convenience, we will denote the left four bits of a given byte, such as y ,by

⊕

L ( y )=( y 1 y 2 y 3 y 4 ) and the right four bits by R ( y )=( y 5 y 6 y 7 y 8 ) .

The next action of F is to feed L ( y )into S 0 to produce S 0 ( L ( y )) = ( z 1 z 2 ), and
feed R ( y )into S 1 to get S 1 ( R ( y ))=( z 3 z 4 ). Thus, under this action y gets
sent to ( S 0 ( L ( y )) S 1 ( R ( y )))=( z 1 z 2 z 3 z 4 )= z . Next, we apply the following
permutation to z .

P 4
j 1234
P 4 ( j ) 2431

Therefore, we get P 4 ( z )=( z 2 z 4 z 3 z 1 )= Z , which is the final outcome for F ,
namely, F ( x,SK )= Z .
Now, the definition of the round function, denoted by f SK , which takes an
8-bit plaintext t and a subkey SK , is given as follows.

f SK ( t )=( L ( t ) ⊕ F ( R ( t ) ,SK ) , R ( t )) .

Thus, the round function only alters L ( t ), the left four bits of t , leaving R ( t )
unaltered. However, there is a reason for f SK being called a round function,
since there are two rounds. The next mechanism, the penultimate one, is a
means of swapping left and right bits.

The Switch/Swap Function
The switch function, denoted by SW , merely exchanges the left and right
four bits of an input m . Hence, if m =( L ( m ) , R ( m )) is an 8-bit input, then

SW ( m )=( R ( m ) , L ( m )) .