Ok, so I have this question and I honestly have no idea how to prove it. I know

Question

Ok, so I have this question and I honestly have no idea how to prove it. I know that the function is linear and I know it works for every possible combination, I just don't know how to prove it in an arbitrary manner. Here is the question:

Suppose that instead of using those S-boxes, DES would just substitute a 6-bit string with its first 4 bits, that is T(b1 b2 b3 b4 b5 b6 ) = b1 b2 b3 b4. We have called this transformation T. Show that T is a linear transformation (Note: you need to prove that T(x xor y) = T(x) xor T(y), for all 6-bit strings x and y, not just for a particular pair x and y; so write down a proof that works for arbitrary x and y).

I have no idea where to start since this class isn't about writing proofs.

Explanation / Answer

We define T as:

T(B)=(b0,b1,b2,b3)

We use D to represent the difference of X and Y:

D=X?Y

Compute T(D):

T(D)=(d0,d1,d2,d3)

=(x0?y0,x1?y1,x2?y2,x3?y3)

=(x0,x1,x2,x3)?(y0,y1,y2,y3)

which is by definition of T:

T(X)?T(Y)

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