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If I have two cipher texts lets say $C_1$ and $C_2$ of the same length encrypted through stream cipher technique using the same keystream. Let's say they are:

$$C_1: texttt96 C6 A1 08 E7 F2 33 3B 3F 5C AB$$

$$C_2: texttt90 C6 A1 1E E6 F3 31 2B 37 4A B6$$

$C_1$ is encrypted as ($P_1 oplus textKeystream$) and $C_2$ by ($P_2 oplus textKeystream$) where $P_1$ and $P_2$ are corresponding plaintexts.

• I am asked to tell how can I differentiate between corresponding plain text $P_1$ and plain text $P_2$ from $C_1$ and $C_2$ as an attacker without knowing the keystream?

So, I think the answer would be since both ciphers are encrypted through the same key stream, they would have similarities where the same plain text and keystream value exists. In this way, I can differentiate the other parts of the plain text. Is there anything more to it?
Thanks.

Let's say $C_1 = P_1 oplus K$ and $C_2 = P_2 oplus K$ where $P$ is a plaintext, $K$ is the key stream and $C$ is the ciphertext.

Then if you XOR the two ciphertext together you get:

$$C_1 oplus C_2 =\
P_1 oplus K oplus P2 oplus K =\
P_1 oplus P_2$$

There are all kinds of interesting properties of the XOR of two plaintext together. For instance, one of the most common characters is the space, so you can easily guess many characters by just flipping a bit (space is 0x20 or 0b0010_0000 after all). You can see that a lot of combinations are not possible or unlikely and you can perform frequency analysis.

This becomes even more powerful if you have 3 or more ciphertexts, as you can compare each and every pair, and if there are $n$ ciphertext then there are $n cdot (n - 1) over 2$ combinations to be made.

In the stream-ciphers, same key-stream is not used two times, I mean that when you encrypt P1 with a Keystream (P1⊕Keystream), the same key-stream should never used for encrypting P2 (P2⊕Keystream). if you use same key-stream for two different encryption, then you cipher-texts are susceptible to "two time pad Attack". In this attack, Attacker captures C1 and C2 which they are encrypted in this way:

 C1 = (P1⊕Keystream) 
 C2 = (P2⊕Keystream) 

Then attacker works out C1 ⊕ C2; which leads to P1 ⊕ P2. We know that (Keystream ⊕ Keystream = 1).

Now attacker bases on some characteristics of plaintext (P1,P2) such as redundancy of ASCII codes, we can get the original plaintext.

But we should remember that we use IV beside the Key for preventing of producing the same keystream.