1. What is RSA Decryption?

RSA decryption is the process of converting ciphertext back to plaintext using a private key in the RSA cryptosystem. It's based on the mathematical properties of modular exponentiation and the difficulty of factoring large numbers.

2. How Does the Calculator Work?

The calculator uses the RSA decryption formula:

Where:

• \( m \) — Decrypted message (plaintext)
• \( c \) — Ciphertext (encrypted message)
• \( d \) — Private exponent (part of private key)
• \( n \) — Modulus (part of both public and private keys)

Explanation: The formula performs modular exponentiation to reverse the encryption process, using the private key components that only the recipient knows.

3. Importance of RSA Decryption

Details: RSA decryption is fundamental to secure communications, allowing only authorized parties with the private key to read encrypted messages. It's widely used in SSL/TLS, digital signatures, and secure email.

4. Using the Calculator

Tips: Enter the ciphertext (integer), private exponent (integer), and modulus (integer). All values must be positive integers (n > 0, d > 0, c ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: Why is RSA secure?
A: RSA's security relies on the practical difficulty of factoring the product of two large prime numbers (the modulus n).

Q2: What key sizes are secure?
A: Currently, 2048-bit or larger keys are recommended for security. This calculator is for demonstration with small numbers.

Q3: Can this calculator handle large numbers?
A: It uses PHP's bcpowmod function which can handle larger numbers than standard operators, but has practical limits.

Q4: What if I get wrong results?
A: Ensure all inputs are correct integers and that you're using the proper private key components for the ciphertext.

Q5: Is this suitable for real encryption?
A: This demonstrates the math, but real-world implementations need proper padding (like OAEP) and key management.