RSA Calculator

The RSA algorithm relies on the mathematical asymmetry of prime factorization. First, two distinct prime numbers, p and q, are multiplied together to find the modulus n:

n = p × q

Next, Carmichael's totient function or the least common multiple (LCM) of the decremented primes gives us Lambda (λ), which determines the step structure:

λ(n) = lcm(p - 1, q - 1)

The public exponent e is chosen such that it is coprime to λ(n). Finally, the private exponent d is calculated as the modular multiplicative inverse of e modulo λ(n):

d × e ≡ 1 (mod λ(n))

Messages are then converted to cipher text via c ≡ m^e (mod n) and reverted back using m ≡ c^d (mod n).

Let's walk through a concrete RSA computation using small prime numbers to see how keys and cipher text are generated.

Step 1: Choose prime numbers. Let p = 17 and q = 23.

Step 2: Calculate the modulus (n). Multiply p and q together: 17 × 23 = 391.

Step 3: Calculate Lambda (λ). Find the least common multiple of (p - 1) and (q - 1): lcm(16, 22) = 176.

Step 4: Select public exponent (e). Choose a standard prime like e = 5, which is coprime to 176.

Step 5: Compute the private key (d). Solve for d such that (5 × d) mod 176 = 1. Testing values yields d = 105 (since 5 × 105 = 525, and 525 mod 176 = 1).

Step 6: Encrypt a message. If our message (mess) is 12, the cipher text is calculated as 12⁵ mod 391, which equals 316.

Use Genuine Primes: For educational purposes, small numbers work well, but real-world security requires primes that are hundreds of bits long to prevent factorization attacks.

Verify Coprimality: Ensure your chosen public exponent e shares no common factors other than 1 with your calculated Lambda value, or the modular inverse for d will fail.

Keep Private Keys Confidential: Never input production-grade private keys into online testing tools. Treat computed private components with absolute discretion.