Here, since the key is used more than one time, an attack called “crib dragging” can be used to attack the cipher-text.
The blog post Many Time Pad Attack - Crib Drag could give you a greater understanding on the implementation part:
Many Time Pad Attack – Crib Drag
The one time pad (OTP) is a type of stream cipher that is a perfectly secure method of encryption. It’s very simple to implement and is perfectly secure as long as the length of the key is greater than or equal to the length of the message. That’s its major downfall. However, it also requires that the key never be used more than once. This tutorial shows what happens when you re-use a key to encrypt more than one message. I also show how to uncover the plain-text of two messages that have been encrypted with the same key, without even knowing the key. I use a method called crib dragging.
Let’s begin with a brief description of OTP and how it works. Let’s take the following message and key:
message = "Hello World"
key = "supersecret"
If we convert both the message and key to hex strings, we get the following:
message = "48656c6c6f20576f726c64"
key = "7375706572736563726574"
If we do a simple XOR of the two hex strings we get the following cipher-text:
cipher-text = "3b101c091d53320c000910"
If we XOR the cipher-text with the key, we can recover the plain-text. That’s how OTP works. Without the key, you have no way of uncovering the plain-text.
Let’s consider what happens when you have two messages encrypted with the same key. Take the following two messages and key:
message1 = "Hello World"
message2 = "the program"
key = "supersecret"
If we convert each message and the key to hex strings, and then encrypt each message using a simple XOR with the key, we’ll get the following cipher-texts:
Let’s say that all we have is the two cipher-texts and the knowledge that they were encrypted with a supposed OTP; however, they were both encrypted with the same key. To attack this encryption and uncover the plain-text, follow the steps below.
- Guess a word that might appear in one of the messages
- Encode the word from step 1 to a hex string
- XOR the two cipher-text messages
- XOR the hex string from step 2 at each position of the XOR of the two cipher-texts (from step 3)
- When the result from step 4 is readable text, we guess the English word and expand our crib search.
- If the result is not readable text, we try an XOR of the crib word at the next position.
Step 1 seems difficult (guessing a word that might appear in one of the messages), but when you think about it, the word “the” is the most commonly used English word. So, we’ll start with assuming “the” is in one of the messages. After encoding “the” as a hex string, we’ll get “746865”. That takes care of steps 1 and 2. If we XOR the two cipher-texts, we’ll get the following result:
cipher-text1 XOR cipher-text2 = "3c0d094c1f523808000d09"
The next step is to XOR our crib word “746865” at each position of the XOR of the cipher-texts. What we’ll do is slide “746865” along each position of “3c0d094c1f523808000d09” and analyze the result. After the first XOR, we get the following result:
When we convert the hex string “48656c” to ASCII, we get the following text, “Hel”. This takes us to step 5 from above. Because this looks like readable text, we can assume that the word “the” is in the first position of one message. If we didn’t get readable text, we would slide 48656c one position to the right and try again (and keep repeating until the end of 3c0d094c1f523808000d09).
Note that we don’t know which message contains the word “the”. It could be in either
message2. Next, we need to guess what the word “Hel” is when fully expanded. It could be “Help”, “Hello”, etc. If we guess “Hello”, we can convert “Hello” to a hex string, we get “48656c6c6f”. We then XOR it with the XOR of the two cipher-texts (just like we did with “the”). Here’s the result:
“7468652070”, when converted to ASCII, is “the p”. We then repeat the process, guessing what “the p” might be when expanded and then XOR that result with the XOR of the cipher-texts. Granted, guessing what “the p” might expand to is not super easy, but you get the idea. If we were to guess “the program”, convert it to a hex string, and XOR it with the XOR of the cipher-texts, we’ll get “Hello World”.
This is called crib dragging. My suggestion is to first try “ the ” (note the spaces before and after). Most cipher-texts that you’ll try cracking will contain that word somewhere in the text. If the result of your crib drag yields gibberish, then you can be sure “ the ” isn’t in either of the plain-text messages. So, try another commonly used English word or phrase and keep trying until the result yields something that looks like readable text. Then you can just expand your guess and keep XORing until you uncover the plain-text messages.