This is a problem involving the El Gamal cryptosystem. Let's break it down step by step to find the value of $m$.

Given:

• $p = 11$, $g = 2$, $B = 10$
• Alice sends $A = 4$, $X = 6$
• Bob's private key $b = 5$

Steps to Decrypt:

1. Compute $A^b \mod p$: Bob computes the shared secret $s$ using: $s = A^b \mod p$ Substituting $A = 4$, $b = 5$, and $p = 11$: $s = 4^5 \mod 11$ Calculating step-by-step: $4^2 = 16 \equiv 5 \pmod{11}$ $4^4 = (4^2)^2 = 5^2 = 25 \equiv 3 \pmod{11}$ $4^5 = 4^4 \cdot 4 = 3 \cdot 4 = 12 \equiv 1 \pmod{11}$ So, $s = 1$.

2. Compute $s^{-1} \mod p$: The decryption process requires the multiplicative inverse of $s \mod p$. Since $s = 1$, its inverse is $s^{-1} = 1$.

3. Decrypt $X$: Bob decrypts the message $m$ using: $m = X \cdot s^{-1} \mod p$ Substituting $X = 6$, $s^{-1} = 1$, and $p = 11$: $m = 6 \cdot 1 \mod 11 = 6$

Final Answer:

$m = 6$

Would you like further clarification or details? Here are 5 related questions for deeper understanding:

1. How does the El Gamal cryptosystem ensure secure communication?
2. Why is modular arithmetic critical in cryptography?
3. How do you calculate the modular inverse efficiently?
4. What role does the generator $g$ play in the El Gamal cryptosystem?
5. How is the private key $b$ kept secure during communication?

Tip: Always double-check modular arithmetic calculations, as they are crucial for correctness in cryptographic protocols!