The problem provided relates to the RSA cryptosystem. Let's break it down step by step to determine the plaintext $M$:

Problem Breakdown

1. RSA Encryption Formula: $C = M^e \mod n$ where:

   • $C$ is the ciphertext (given as 20),
   • $M$ is the plaintext to be found,
   • $e$ is the public key exponent (given as 13),
   • $n$ is the modulus (given as 77).

2. Goal: To find $M$, we need to compute the modular inverse operation: $M = C^d \mod n$ where $d$ is the private key, calculated as: $d = e^{-1} \mod \phi(n)$ Here, $\phi(n)$ (Euler's totient function) is calculated as: $\phi(n) = (p - 1)(q - 1)$ where $n = p \cdot q$, and $p$, $q$ are the prime factors of $n$.

Step-by-Step Solution

Step 1: Factorize $n = 77$

The prime factors of $n = 77$ are $p = 7$ and $q = 11$.

Step 2: Compute $\phi(n)$

$\phi(n) = (p - 1)(q - 1) = (7 - 1)(11 - 1) = 6 \cdot 10 = 60$

Step 3: Find $d$ (the private key)

We need to compute $d$ such that: $d \cdot e \equiv 1 \mod \phi(n)$ or equivalently: $d \cdot 13 \equiv 1 \mod 60$

Using the Extended Euclidean Algorithm, we find: $d = 37$

Step 4: Decrypt to find $M$

The plaintext $M$ is given by: $M = C^d \mod n$ Substituting $C = 20$, $d = 37$, and $n = 77$, we calculate: $M = 20^{37} \mod 77$

Simplify with Modular Arithmetic

To compute $20^{37} \mod 77$, we use repeated squaring:

1. Compute powers of 20 modulo 77:

   • $20^2 \mod 77 = 400 \mod 77 = 15$,
   • $20^4 \mod 77 = 15^2 \mod 77 = 225 \mod 77 = 71$,
   • $20^8 \mod 77 = 71^2 \mod 77 = 5041 \mod 77 = 22$,
   • $20^{16} \mod 77 = 22^2 \mod 77 = 484 \mod 77 = 22$.

2. Use the exponents to calculate $20^{37} \mod 77$: $20^{37} = 20^{32} \cdot 20^4 \cdot 20^1$ Substituting:

   • $20^{32} \mod 77 = (20^{16})^2 \mod 77 = 22^2 \mod 77 = 22$,
   • $20^4 \mod 77 = 71$,
   • $20^1 \mod 77 = 20$.

   Combine: $20^{37} \mod 77 = (22 \cdot 71 \cdot 20) \mod 77 = 15620 \mod 77 = 44$

Thus, the plaintext $M = 44$.

Final Answer:

The plaintext $M$ is: $\boxed{44}$

Would you like further clarification or a deeper explanation of any step?

Relative Questions:

1. How does the Extended Euclidean Algorithm find the modular inverse?
2. Why is modular arithmetic crucial in cryptographic algorithms like RSA?
3. Can $n$ be any number, or must it have specific properties in RSA encryption?
4. What happens if the primes $p$ and $q$ are too small?
5. How is the security of RSA affected by advances in computational power?

Tip:

When performing modular exponentiation, always use the method of repeated squaring to simplify calculations efficiently.