Why is this not a viable key exchange algorithm? [duplicate]

Question

I was just wondering why this kind of algorithm can't be used instead of, say, Diffie-Hellman to exchange keys:

1. Alice decides on a key she wishes to share with Bob.
2. Alice generates a stream of bytes with the same length as the key (securely, say, with a CSPRNG).
3. Alice sends to Bob:

   C1 = (key ^ alice_random_bytes)
   

4. Bob generates a stream of random bytes in a manner similar to Alice.
5. Bob returns to Alice:

   C2 = (C1 ^ bob_random_bytes)
   

6. Alice XORs C2 with her random byte sequence again, leaving only key ^ bob_random_bytes like so and sends it to Bob:

   C3 = (C2 ^ alice_random_bytes)
      = (C1 ^ bob_random_bytes ^ alice_random_bytes)
      = (key ^ alice_random_bytes ^ bob_random_bytes ^ alice_random_bytes)
      = (key ^ bob_random_bytes)
   

7. Bob XORs C3 with his random bytes and obtains the key:

   K = (C3 ^ bob_random_bytes)
     = (key ^ bob_random_bytes ^ bob_random_bytes) 
     = key
   

This seems a lot simpler than Diffie Hellman, so I was wondering: what is the issue with such an algorithm?

Answer 1

I've simplified the Alice random bytes to ARB and Bob random bytes to BRB. Then the protocol follows as;

Alice knows $key$ and $ARB$ and sends $$C_1 = key \oplus ARB$$

Bob knows $C_1$ and $BRB$ and sends

$$C_2 = C_1 \oplus BRB = key \oplus ARB \oplus BRB$$

Alice calculates $C_2 \oplus key \oplus ARB = key \oplus key \oplus ARB \oplus BRB = BRB$

Alice knows $key, ARB,$ and $BRB$ and sends

$$C_3 = (C_2 \oplus ARB) = key \oplus ARB \oplus BRB \oplus ARB = key \oplus BRB$$

Now, first of all, this requires a three-pass protocol.

Now, an observer sees

\begin{align} C_1 & = key \oplus ARB \oplus {}\\ C_2 & = key \oplus ARB \oplus BRB\\ C_3 & = key \oplus \phantom{ARB}\oplus BRB \\ \end{align}

A passive observer (eavesdropper) simply x-ors all to derive the key $$key = C_1 \oplus C_2 \oplus C_3.$$ Therefore it is insecure against the weak assumption on the attacker; passive!.

So, you rely on the xor, however, did not check what an observer can get and calculate from them.

The Diffie–Hellman key exchange (DHKE), on the other hand, leaks $g^a$ and $g^b$ where Alice selects a random integer $a$ and sends $g^a$ and Bob select a random integer $b$ and sends $g^b$. Finding $a$ or $b$ from them is the discrete logarithm problem. On the other hand, The Computational Diffie–Hellman (CDH) assumption, is asked to find $g^{ab}$ given $g^a$ and $g^b$, and the DHKE is relayed on this. If the discrete logarithm is easy then CDH is easy. We don't know the reverse, in the general case.

Answer 2

Key exchange algorithms attempt to protect against eavesdropping. You have to assume what you send over the wire (C1, C2, and C3) are intercepted. That's a problem with the method because C2 is simply C1 xor Bob's random bytes and C3 is simply the key xor Bob's random bytes.

An attacker with C1, C2, and C3 could take C1 xor C2 to get Bob's random bytes, and then xor that with C3 to get the key just like Bob would.