1 of 3

# Show that the equal difference property exits for a modified DES encryption system

My cryptology professor gave us this problem on a past homework assignment and I only managed to get one of the 3 parts correct. Needless to say, I want to know how to do the other two parts as well. The question is as follows:

 Consider the following DES-like encryption system that operates on 16-bit strings.
The system takes the input and divides it into two 8-bit string Lzero and Rzero.
One round of encryption starts with L sub i-1 and R sub i-1 and the output is
R sub i-1 and (L sub i-1 exclusive or'd with S(R sub i-1) exclusive or'd with K,
the key.

The function S rotates the bits of R sub i-1 to the right. More precisely, if
R sub i-1= b sub 0, b sub 1, b sub 2, b sub 3, b sub 4, b sub 5, b sub 6, b sub 7,
S(R sub i-1)=b sub 7, b sub 0, b sub 1, b sub 2, b sub 3, b sub 4, b sub 5, b sub 6.

The two parts that I was not able to get are:
1) Explain briefly why if A and B are bit strings, then
S(A xor'd with B)=S(A) xor'd with S(B)

2) If M is the plaintext, let E sub k of M denote the process of encrypting M using
one round of the above process. Show that E sub K has the equal difference
property, namely that if A xor'd with B = C or'd with D, then
E sub k of A xor'd with E sub k of B = E sub k of C xor'd with E sub K of D.


Sorry I had to type out all of the math symbols, I'm still not sure how to make them without just typing it all out. Anyway, if someone could provide a complete explanation (answer included) on how to the 2 above questions, I would greatly appreciate it! Thanks in advance!