RSA Encryption and Decryption Calculator

How RSA works

RSA is a public-key cryptosystem: anyone can encrypt with the public key, but only the holder of the private key can decrypt. Its security rests on the fact that multiplying two large primes is easy while recovering them from their product is believed to be very hard.

Key generation: pick two distinct primes p and q and compute n = pq and φ(n) = (p − 1)(q − 1). Choose a public exponent e with 1 < e < φ(n) and gcd(e, φ(n)) = 1. The private exponent d is the modular inverse of e modulo φ(n). The public key is (n, e) and the private key is d.

Encrypting and decrypting

A message is first encoded as an integer m with 0 ≤ m < n. Encryption computes c = m^e mod n and decryption computes m = c^d mod n. Decryption works because ed = 1 + kφ(n) for some k, and by Euler's theorem m^ed ≡ m (mod n). Both operations are modular exponentiations, done with the square-and-multiply method; the calculator shows the binary table for each.

The standard textbook example is p = 61, q = 53, e = 17. Then n = 3233, φ(n) = 3120 and d = 2753. The message m = 65 encrypts to c = 2790, and decrypting 2790 returns 65.

Why this is only a demonstration

Real RSA uses primes of 1024 bits or more and a padding scheme such as OAEP, because plain ("textbook") RSA is deterministic and has several known weaknesses. With the small primes used in lessons, n can be factored by hand in seconds, which immediately reveals the private key. Use this tool to understand the mathematics and to check homework, never to protect real information.

Choosing valid inputs

The calculator checks that p and q are prime and different, that e is coprime to φ(n), and that your message is smaller than n. If e is not coprime to φ(n), no private exponent exists and you will be asked to pick another e. The common choice in practice is e = 65537, which is prime, so it is coprime to φ(n) unless 65537 divides p − 1 or q − 1.