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The Affine Cipher is a type of substitution cipher where each letter in the plaintext is transformed to its corresponding letter in the ciphertext using a mathematical formula. The general form of the encryption function for the Affine Cipher is: \(c = (ap + b) \mod m\), where \(a\) and \(b\) are constants, \(p\) is the plaintext letter, \(c\) is the ciphertext letter, and \(m\) is the size of the alphabet. For example, if we are using the English alphabet, then \(m = 26\).
This transformation ensures that each letter of the plaintext maps to a distinct letter in the ciphertext, making it a simple yet effective encryption method. To decrypt the ciphertext back into the plaintext, specific steps involving modular arithmetic and inverse calculations are required.
It's important to note that for the cipher to work, \(a\) must be chosen such that it has a multiplicative inverse modulo \(m\), meaning \(a\) and \(m\) are coprime (i.e., they have no common divisors other than 1). This ensures that the transformation can be reversed efficiently.

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' upon reaching a certain value—the modulus. In the context of the Affine Cipher, modular arithmetic is critical because it helps to keep letter indices within the range of the alphabet.
When performing operations like addition, subtraction, multiplication, and finding the modular inverse, the results are taken modulo \(m\). For instance, in our example, we use modulo 26 because there are 26 letters in the English alphabet.
Key operations include:

• Addition: \((a + b) \mod m\)
• Subtraction: \((a - b) \mod m\)
• Multiplication: \((a \cdot b) \mod m\)
• Inverse: Finding a number \(a^{-1}\) such that \((a \cdot a^{-1}) \equiv 1 \mod m\)

This wrapping around behavior ensures that our results always stay within the bounds of our alphabet, simplifying calculations.

The Extended Euclidean Algorithm is an extension of the Euclidean Algorithm that not only finds the greatest common divisor (GCD) of two numbers but also finds coefficients that express the GCD as a linear combination of the two numbers.
This is particularly useful for finding the modular inverse, a critical step in the decryption process of the Affine Cipher. In our problem, we need to find \(15^{-1} \mod 26\). This involves solving the equation:

• \((15a^{-1}) \equiv 1 \mod 26\)

By using the Extended Euclidean Algorithm, we determine that \(15^{-1} \equiv 7 \mod 26\). This means 7 is the modular inverse of 15 modulo 26.
The steps involved typically include using the Euclidean Algorithm to express the GCD as a combination of the two numbers and working backward to find the coefficients needed to reach the desired inverse.

Cryptanalysis is the study of analyzing information systems to understand the hidden aspects of the systems. It is usually used to break the encryption and decrypt the information without knowing the encryption key. For the Affine Cipher, one common cryptanalysis attack is the brute-force attack, where all possible keys are tried. Given the need for \(a\) and \(m\) to be coprime, fewer valid keys are possible compared to other types of ciphers.
Cryptanalysts might also look for patterns or frequencies in the ciphertext. Since the Affine Cipher preserves the frequency of letters, analyzing the frequency of each letter in the ciphertext can provide clues.
In practical scenarios, Affine Ciphers are considered relatively weak due to the small key space and vulnerability to frequency analysis. Modern cryptography often employs more complex and secure methods.