DES and using the S-Box 1

Question

Can someone please check my work and correct me if I am wrong, I am just starting to learn about DES. I am trying to solve the following problem:

Show that S1(x1) XOR S1 (x2) != S1(x1 XOR x2).
x1 = 000000, x2 = 000001

So, we get S1(x1) first: 00 is our row, and 0000 is our column which is (0,0) and in the S1 Box table is the value 14. 14 in binary is 1110.

Now we get S1(x2): 01 is our row, and 0000 is our column which is (1,0) and in the S1 Box table is the value 0. In binary, it's 0000.

Now we XOR the two outputs:

    1110
XOR 0000
  = 0001 

Thus, S1(x1) XOR S1(x2) = 0001.

Now we XOR X1 and X2 first:

     0000000
 XOR 0000001
  =  1111110

Now we lookup our output in the S1 table: Our row is 10, and our column is 1111 which is (2,15) and in the S1 Box Table is the value 08 which is 1000.

Thus, 0001 != 1000. Is that correct? (And in case it isn’t, where did I take a wrong turn?)

Answer 1

Your XOR equations are wrong. You wrote:

    1110
XOR 0000
  = 0001 

But it's not how the XOR works. 1 xor 0 = 1, and in your example it somehow turns into 0 wich is an error. Reference the xor table: https://en.wikipedia.org/wiki/Exclusive_or

So, the correct answer for S1(x1) XOR S1(x2) is:

    1110
XOR 0000
  = 1110

and for x1 xor x2:

    0000000
XOR 0000001
  = 0000001