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In RSA public key cryptosystem suppose $n=p*q$ where $p$ and $q$ are primes. $(e,n)$ and $(d,n)$ are public and private keys respectively. Let $M$ be an integer such that $o< M< n$ and $\phi(n)=(p-1)(q-1)$.

Which of the following equations represent RSA public key cryptosystem?

1. $\begin{array}{} C \equiv M^{e} \text{(mod n)} \\ M \equiv C^{d} \text{(mod n)} \end{array} \\$
2. $ed\equiv1 \text{(mod n)} \\$
3. $ed\equiv1 ( \text{mod } \phi( n)) \\$
4. $\begin{array}{} C\equiv M^{e} ( \text{mod } \phi(n)) \\ M \equiv C^{d}( \text{mod } \phi (n)) \end{array}$
1. I and II
2. I and III
3. II and III
4. I and IV

Comparing it with RSA algorithm , I and III is correct ahere...see the example

• Choose p = 3 and q = 11
• Compute n = p * q = 3 * 11 = 33
• Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20
• Choose e such that 1 < e < φ(n) and e and n are coprime. Let e = 7
• Compute a value for d such that (d * e) % φ(n) = 1. One solution is d = 3 [(3 * 7) % 20 = 1]
• Public key is (e, n) => (7, 33)
• Private key is (d, n) => (3, 33)
• The encryption of m = 2 is c = 27 % 33 = 29
• The decryption of c = 29 is m = 293 % 33 = 2

so B is correct answer here

by

Comparing it with RSA algorithm , I and III is correct ahere...see the example

• Choose p = 3 and q = 11
• Compute n = p * q = 3 * 11 = 33
• Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20
• Choose e such that 1 < e < φ(n) and e and n are coprime. Let e = 7
• Compute a value for d such that (d * e) % φ(n) = 1. One solution is d = 3 [(3 * 7) % 20 = 1]
• Public key is (e, n) => (7, 33)
• Private key is (d, n) => (3, 33)
• The encryption of m = 2 is c = 27 % 33 = 29
• The decryption of c = 29 is m = 293 % 33 = 2

so B is correct answer here

In RSA algorithm

1) Cipher text is obtained as $Cipher Text = (Plain Text)^{public\ key} \ mod n$

2) Plain Text is obtained as $Plain Text = (Cipher Text)^{private\ key} \ mod n$

3) e should be chosen in such a way that it satisfy $e*d = 1(mod)\phi (n)$

Hence option (2) is correct

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