RSA (Rivest–Shamir–Adleman, 1977) secures web traffic, e-mail and digital signatures. Its security rests on the difficulty of factoring N = p·q when p and q have hundreds of digits. This calculator uses small numbers to show the mechanics.
How to use the RSA Calculator
- Prime p
- Prime q
- Public exponent e
- Message M (number smaller than N)
- Received ciphertext C to decrypt
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Enter two different primes, a public exponent coprime to λ(N) (65537 is standard; small values like 17 work for small primes), a message M smaller than N and a ciphertext C to decrypt. Exact big-integer arithmetic is used.
Formula
N = p·q, λ(N) = lcm(p − 1, q − 1), d = e⁻¹ mod λ(N)
C = Mᵉ mod N, M = Cᵈ mod N
Worked example
Inputs — Prime p: 61; Prime q: 53; Public exponent e: 17; Message M (number smaller than N): 65; Received ciphertext C to decrypt: 2,790.
Results — Modulus N = p·q = 3233; λ(N) = lcm(p − 1, q − 1) = 780; Private exponent d = 413; Encrypted message Mᵉ mod N = 2790; Decrypted ciphertext Cᵈ mod N = 65.
Key terms
- Public key
- (N, e), shared openly.
- Private key
- d, kept secret.
Frequently asked questions
Is this secure?
No – real RSA uses primes of 1,024 bits or more, plus padding.
Why 65537?
It is prime, and its binary form 10000000000000001 makes encryption fast.
What is λ(N)?
Carmichael's function; φ(N) = (p − 1)(q − 1) also works.
Why must M be smaller than N?
Arithmetic is mod N, so larger messages would wrap around.