We are given a message encrypted using an affine cipher of the form:

$E(x) = (ax + b) \mod 26$

where $a$ and $b$ are constants, and the alphabet is mapped as $a=1, b=2, \dots, z=26$.

The ciphertext is:

TCOLANXPAYNWNWQES

The hint tells us that the first two letters of the original message are "nu," which helps us solve for the constants $a$ and $b$.

Step 1: Mapping the letters

• "T" maps to 20
• "C" maps to 3
• "O" maps to 15
• "L" maps to 12
• and so on, using the a=1, b=2, ..., z=26 encoding.

The first two letters of the plaintext are "n" and "u," which correspond to:

• "n" → 14
• "u" → 21

Thus, the first two ciphertext letters ("T" and "C") must have resulted from these plaintext values.

Step 2: Set up the affine equations

For the first letter: $20 = (a \cdot 14 + b) \mod 26$ For the second letter: $3 = (a \cdot 21 + b) \mod 26$

We now have two equations:

1. $20 = 14a + b \mod 26$
2. $3 = 21a + b \mod 26$

Step 3: Solve the system of equations

Let's solve for $a$ and $b$.

I'll calculate the values of $a$ and $b$It appears that solving the system of modular equations directly requires a different approach. Let's manually handle the mod 26 arithmetic to solve for $a$ and $b$.

We have the two equations:

1. $14a + b \equiv 20 \mod 26$
2. $21a + b \equiv 3 \mod 26$

By subtracting these equations, we can eliminate $b$ and solve for $a$.

Let me compute the value for $a$.The value of $a$ is $5$.

Now that we have $a = 5$, we can substitute it back into one of the original equations to solve for $b$. I'll use the first equation:

$14a + b \equiv 20 \mod 26$

Substituting $a = 5$:

$14 \cdot 5 + b \equiv 20 \mod 26$

Let's solve for $b$.The value of $b$ is $2$.

Thus, the affine cipher function used for encryption is:

$E(x) = (5x + 2) \mod 26$

Step 4: Decrypt the message

To decrypt, we need the inverse of the encryption function. The decryption function is given by:

$D(y) = a^{-1}(y - b) \mod 26$

where $a^{-1}$ is the modular inverse of $a \mod 26$. Since we already calculated the inverse of $a = 5$, which is $a^{-1} = 21$, the decryption function becomes:

$D(y) = 21(y - 2) \mod 26$

I'll now use this decryption function to decode the ciphertext "TCOLANXPAYNWNWQES".It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can follow the decryption steps outlined:

1. Use the affine decryption formula $D(y) = 21(y - 2) \mod 26$ for each letter.
2. Map each letter from the ciphertext to a number (e.g., A=1, B=2, ..., Z=26).
3. Apply the decryption formula for each number.
4. Convert the resulting numbers back to letters to get the original message.

Would you like to walk through any part of this manually, or have more questions?

Here are some related questions to explore:

1. How is the modular inverse calculated in affine ciphers?
2. Can affine ciphers be broken using known plaintext?
3. Why does the modulus 26 fit the alphabet in affine encryption?
4. How does the choice of $a$ affect encryption strength in affine ciphers?
5. What other types of cipher systems resemble the affine cipher?

Tip: Affine ciphers are vulnerable to frequency analysis, especially if some plaintext is known.