RSA Calculator

What Is RSA Encryption?

RSA (Rivest–Shamir–Adleman) is an asymmetric cryptographic algorithm widely used for secure data transmission, digital signatures, and key exchange. Unlike symmetric encryption systems where the same secret key is used to lock and unlock messages, RSA uses a mathematically linked key pair:

• Public Key $(e, N)$: Shared openly with anyone who wants to encrypt a message or verify a digital signature.
• Private Key $(d, N)$: Kept secret by the recipient to decrypt incoming ciphertexts or sign digital documents.

The security of RSA rests on the mathematical hardness of integer factorization: multiplying two large prime numbers is computationally trivial, but finding the original prime factors from their product is computationally infeasible for modern supercomputers when sufficiently large keys are used.

How RSA Key Generation Works Step-by-Step

The RSA key creation pipeline involves five core mathematical steps:

1. Select Two Distinct Primes: Choose two prime numbers $p$ and $q$. In real-world cryptographic applications, these primes are hundreds of digits long (e.g. 1024 or 2048 bits).
2. Calculate the Modulus $N$: Multiply the two primes together: $$N = p \times q$$ The value $N$ serves as the modulus for both the public and private keys. Its bit length represents the key size.
3. Compute the Totient: Calculate Euler's totient function $\phi(N)$ or Carmichael's totient function $\lambda(N)$: $$\phi(N) = (p - 1)(q - 1)$$ $$\lambda(N) = \text{lcm}(p - 1, q - 1)$$
4. Choose Public Exponent $e$: Pick an integer $e$ such that $1 < e < \phi(N)$ and $\gcd(e, \phi(N)) = 1$. The exponent $e$ must be coprime with the totient. Common choices in cryptographic standards include 3, 17, and 65537 ($2^{16} + 1$).
5. Calculate Private Exponent $d$: Solve for the modular multiplicative inverse of $e$ modulo $\lambda(N)$ (or modulo $\phi(N)$): $$d \equiv e^{-1} \pmod{\lambda(N)} \implies e \times d \equiv 1 \pmod{\lambda(N)}$$ This is efficiently solved using the Extended Euclidean Algorithm. If you want to compute individual inverses, try our Modular Multiplicative Inverse Calculator.

Encryption and Decryption Formulas

Once the keys $(e, N)$ and $(d, N)$ are generated, numerical messages can be encrypted and decrypted:

• Encryption: Convert the plaintext message into an integer $m$ where $0 \le m < N$. The ciphertext $c$ is: $$c \equiv m^e \pmod{N}$$ You can verify exponentiation modular steps with our Power Modulo Calculator.
• Decryption: The recipient recovers the original plaintext message $m$ using the private exponent $d$: $$m \equiv c^d \pmod{N}$$

Worked Example

Let us trace an example using small primes:

• Choose primes: $p = 61$ and $q = 53$
• Modulus: $N = 61 \times 53 = 3233$
• Totient: $\phi(N) = (61 - 1) \times (53 - 1) = 60 \times 52 = 3120$
• Carmichael totient: $\lambda(N) = \text{lcm}(60, 52) = 780$
• Choose $e = 17$ (since $\gcd(17, 780) = 1$)
• Compute $d$: $17 \times d \equiv 1 \pmod{780} \implies d = 413$

To encrypt the plaintext number $m = 65$: $$c = 65^{17} \pmod{3233} = 2790$$ To decrypt ciphertext $c = 2790$: $$m = 2790^{413} \pmod{3233} = 65$$ The message is recovered accurately! If you need PEM-formatted key certificates for production development, explore our RSA Key Pair Generator or calculate general remainders with our Modulo Calculator.