# RSA Calculator

Explore RSA encryption and decryption with small primes. See how public/private keys are generated and how messages are encrypted and decrypted. Educational RSA tool.

## What this calculates

Explore how RSA encryption works using small prime numbers. This educational calculator generates public and private keys, encrypts a message, and decrypts it back to demonstrate the complete RSA process.

## Inputs

- **Prime p** — min 2 — First prime number. Keep small for demonstration (under 1000).
- **Prime q** — min 2 — Second prime number, different from p. Keep small for demonstration.
- **Message (integer m)** — min 0 — The plaintext message as a number. Must be less than n = p x q.

## Outputs

- **n = p x q (modulus)** — The RSA modulus, part of both the public and private keys.
- **phi(n) = (p-1)(q-1)** — Euler's totient of n, used to compute the keys.
- **Public exponent e** — The public encryption exponent. Coprime to phi(n); commonly 65537 or smaller.
- **Private exponent d** — The private decryption exponent. Satisfies e x d = 1 (mod phi(n)).
- **Encrypted message (ciphertext)** — formatted as text — The ciphertext: c = m^e mod n.
- **Decrypted message** — formatted as text — Decryption of the ciphertext: m = c^d mod n.

## Details

RSA (Rivest-Shamir-Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. This calculator demonstrates the algorithm with small numbers so you can see every step.

**How RSA Works:**

1. **Choose two primes** p and q. Compute n = p x q.
2. **Compute Euler's totient:** phi(n) = (p-1)(q-1).
3. **Choose public exponent e** such that 1 < e < phi(n) and gcd(e, phi(n)) = 1. The value 65537 is a common choice.
4. **Compute private exponent d** such that e x d = 1 (mod phi(n)).
5. **Encrypt:** ciphertext c = m^e mod n.
6. **Decrypt:** plaintext m = c^d mod n.

**Why It Works:**

The security relies on the difficulty of factoring n back into p and q. Multiplying two primes is fast, but reversing that multiplication (factoring) is computationally hard for very large numbers. Real RSA uses primes with hundreds of digits.

**Important Note:**

This calculator uses small numbers for educational purposes. Real RSA implementations use 2048-bit or larger keys, which involve primes with over 300 digits each.

## Frequently Asked Questions

**Q: What is RSA encryption?**

A: RSA is a public-key cryptographic algorithm where encryption uses a public key (n, e) and decryption uses a private key (n, d). Anyone can encrypt a message with the public key, but only the holder of the private key can decrypt it. The security depends on the difficulty of factoring the product of two large primes.

**Q: Why do p and q need to be prime?**

A: RSA relies on Euler's totient function, which has a simple formula when n is a product of two distinct primes: phi(n) = (p-1)(q-1). If p or q were not prime, the totient calculation would be different and the key generation process would not work correctly. The primes must also be kept secret.

**Q: Is this calculator secure for real encryption?**

A: No. This calculator is for learning only. Real RSA uses primes with hundreds of digits (2048-bit keys or larger). The small numbers used here can be factored instantly. Never use small primes for actual encryption. Use established libraries like OpenSSL for real cryptographic needs.

**Q: What is the public exponent e = 65537?**

A: 65537 (which is 2^16 + 1) is the most commonly used public exponent in real RSA implementations. It is a Fermat prime, which makes encryption fast due to its binary representation (only two 1-bits). It is large enough to be secure against certain attacks while remaining efficient for computation.

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Source: https://vastcalc.com/calculators/math/rsa
Category: Math
Last updated: 2026-04-08