How to know the difference between two ciphertexts without key stream in stream ciphers [duplicate]

Question

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:

$$C_1: \texttt{96 C6 A1 08 E7 F2 33 3B 3F 5C AB}$$

$$C_2: \texttt{90 C6 A1 1E E6 F3 31 2B 37 4A B6}$$

$C_1$ is encrypted as ($P_1 \oplus \text{Keystream}$) and $C_2$ by ($P_2 \oplus \text{Keystream}$) 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.

Answer 1

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.

Answer 2

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 = 0).

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.