What is the principle of RSA encryption algorithm?

The principle of the RSA encryption algorithm is based on the mathematical difficulty of factoring large prime numbers. It uses a pair of keys: a public key for encryption and a private key for decryption. The security of RSA relies on the fact that it is computationally infeasible to factorize the product of two large prime numbers into its constituent primes.

Here's a simplified explanation with an example:

1. Key Generation:

   • Choose two large prime numbers, $p$ and $q$.
   • Calculate $n = p \times q$.
   • Compute $\phi(n) = (p-1) \times (q-1)$, where $\phi$ is Euler's totient function.
   • Choose an integer $e$ such that $1 < e < \phi(n)$ and $e$ is coprime with $\phi(n)$.
   • Calculate $d$ such that d \times e \equiv 1 \mod \phi(n).

2. Encryption:

   • Suppose Alice wants to send a message $M$ to Bob.
   • She encrypts the message using Bob's public key $(n, e)$: C = M^e \mod n.

3. Decryption:

   • Bob receives the ciphertext $C$ and decrypts it using his private key $(n, d)$: M = C^d \mod n.

Example:

• Let's say $p = 61$ and $q = 53$.
• Then $n = 61 \times 53 = 3233$.
• $\phi(n) = (61-1) \times (53-1) = 3120$.
• Choose $e = 17$ (which is coprime with 3120).
• Calculate $d$ such that 17 \times d \equiv 1 \mod 3120, which gives $d = 2753$.

If Alice wants to send the message $M = 65$:

• Encryption: C = 65^{17} \mod 3233 = 2790.
• Decryption: M = 2790^{2753} \mod 3233 = 65.

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