Find the secret key d in RSA if n=1147 and e=7. Find c if m=64. How to extract m from the output?

Sure, let's break this down step by step.

1. First, we need to factorize n to get p and q. The factors of 1147 are 31 and 37. So, p=31 and q=37.

2. Next, we calculate φ(n) which is (p-1)(q-1). So, φ(1147) = (31-1)(37-1) = 30*36 = 1080.

3. Now, we need to find the multiplicative inverse of e mod φ(n) to get d. The multiplicative inverse of 7 mod 1080 is 463. So, d=463.

4. To find c, we use the formula c ≡ m^e (mod n). So, c ≡ 64^7 (mod 1147). Calculating this gives us c=688.

5. To extract m from the output, we use the formula m ≡ c^d (mod n). So, m ≡ 688^463 (mod 1147). Calculating this gives us m=64.

So, the secret key d is 463, c is 688, and the message m is 64.