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I tried to work this out but I'm stuck at finding a relation between $c$ and $e(k)$:

An encryption system accepts 8 bit-blocks as input.

The encryption is done by XOR'ing the bits with an 8-bit key.

Prove that:

  1. $(m_1 ⊕ m_2) = e_k(m_1) ⊕ e_k(m_2)$

  2. There does not exist a message $m$, such that $e_k(m) = m$, except when $k$ is all 0's.

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  • $\begingroup$ Well, what have you tried then? $\endgroup$ – fkraiem Dec 28 '16 at 3:54
  • $\begingroup$ I tried to copyedit your question a bit, but I'm not entirely sure what you meant by "a relation between c & e(k)" in the first paragraph. You might want to clarify that (and check that I haven't introduced any mistakes in my edit, while you're at it). $\endgroup$ – Ilmari Karonen Dec 28 '16 at 4:09
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Let $k$ be the 8-bit key. Since encryption is simply done by XORin the plaintext with the key, i.e. $e_k(m) = k \oplus m$, we have:

$$e_k(m_1) \oplus e_k(m_2) = (k \oplus m_1) \oplus (k \oplus m_2).$$

Now, the thing you need to know about XOR is that it obeys the same commutative, and associative laws as normal addition, that is:

$$a \oplus b = b \oplus a$$ $$a \oplus (b \oplus c) = (a \oplus b) \oplus c$$

and it also obeys the cancellative laws:

$$a \oplus a = \bar 0$$ $$a \oplus \bar 0 = a$$

where $\bar 0$ denotes the string of all zero bits.

Using these algebraic laws, you should be able to show that the two $k$'s on the right hand side of the equation above will cancel out, leaving only $m_1 \oplus m_2$.

Similarly, you should be able to show that, if $k \oplus m = m$, then $k = \bar 0$. The easiest way is probably to observe that, if $a = b$, than $a \oplus c = b \oplus c$. Thus, you can XOR both sides of the equation $k \oplus m = m$ with $m$, and then apply the cancellative law.

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