The RSA algorithm involves key generation, encryption, and decryption. Given the values:

• $P = 11$
• $Q = 13$
• $e = 11$
• $M = 7$ (the plaintext message)

We will go through each step of the RSA process: key generation, encryption, and decryption.

1. Key Generation:

Step 1: Calculate $N$

$N = P \times Q = 11 \times 13 = 143$

Step 2: Calculate $\phi(N)$ (Euler's Totient Function)

$\phi(N) = (P - 1) \times (Q - 1) = (11 - 1) \times (13 - 1) = 10 \times 12 = 120$

Step 3: Public Key $(N, e)$

We are given $e = 11$. Ensure that $e$ is coprime to $\phi(N)$, i.e., $\gcd(11, 120) = 1$. Since $e = 11$ is coprime with $120$, we can proceed.

• The public key is $(N = 143, e = 11)$.

Step 4: Calculate the Private Key $d$

We need to find $d$, the modular inverse of $e$ modulo $\phi(N)$. This means solving for $d$ in the equation:

$d \times e \equiv 1 \, (\text{mod} \, \phi(N))$

This can be solved using the Extended Euclidean Algorithm to find $d$ such that:

$11d \equiv 1 \, (\text{mod} \, 120)$

Using the Extended Euclidean Algorithm:

• $120 = 10 \times 11 + 10$
• $11 = 1 \times 10 + 1$
• $10 = 10 \times 1 + 0$

Reversing the steps:

• $1 = 11 - 1 \times 10$
• $1 = 11 - 1 \times (120 - 10 \times 11)$
• $1 = 11 \times 11 - 1 \times 120$

Thus, $d = 11$.

• The private key is $(N = 143, d = 11)$.

2. Encryption:

The encryption formula in RSA is:

$C = M^e \, (\text{mod} \, N)$

Given $M = 7$, $e = 11$, and $N = 143$:

$C = 7^{11} \, (\text{mod} \, 143)$

Let's calculate $7^{11} \, (\text{mod} \, 143)$:

$7^2 = 49$ $7^4 = 49^2 = 2401 \quad \Rightarrow \quad 2401 \, (\text{mod} \, 143) = 115$ $7^8 = 115^2 = 13225 \quad \Rightarrow \quad 13225 \, (\text{mod} \, 143) = 4$ $7^{11} = 7^8 \times 7^2 \times 7 = 4 \times 49 \times 7 = 1372 \quad \Rightarrow \quad 1372 \, (\text{mod} \, 143) = 84$

Thus, the ciphertext $C = 84$.

3. Decryption:

The decryption formula in RSA is:

$M = C^d \, (\text{mod} \, N)$

Given $C = 84$, $d = 11$, and $N = 143$:

$M = 84^{11} \, (\text{mod} \, 143)$

Let's calculate $84^{11} \, (\text{mod} \, 143)$:

$84^2 = 7056 \quad \Rightarrow \quad 7056 \, (\text{mod} \, 143) = 61$ $84^4 = 61^2 = 3721 \quad \Rightarrow \quad 3721 \, (\text{mod} \, 143) = 7$ $84^8 = 7^2 = 49$ $84^{11} = 84^8 \times 84^2 \times 84 = 49 \times 61 \times 84 = 250284 \quad \Rightarrow \quad 250284 \, (\text{mod} \, 143) = 7$

Thus, the decrypted message $M = 7$, which matches the original plaintext.

Summary:

• Public Key: $(N = 143, e = 11)$
• Private Key: $(N = 143, d = 11)$
• Encrypted Message (Ciphertext): $C = 84$
• Decrypted Message: $M = 7$

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Would you like any further details or clarification? Here are five related questions to consider:

1. How is the modular inverse calculated in more complex cases?
2. Why is it important for $e$ and $\phi(N)$ to be coprime?
3. What is the significance of the totient function $\phi(N)$?
4. How is RSA security impacted by large prime numbers?
5. What optimizations can be made to RSA for faster encryption?

Tip: In real RSA applications, much larger prime numbers are used to ensure security, typically over 1024 or 2048 bits in size.