Use the **RSA Encryption Calculator** to perform modular arithmetic for key generation, encryption, and decryption based on the fundamental RSA algorithm. Ensure you use integer representations for your message and choose relatively small primes for demonstration purposes.
RSA Encryption Calculator
RSA Encryption Calculator Formula
The RSA algorithm relies on three core modular arithmetic formulas:
1. Modulus: $$N = P \times Q$$
2. Euler’s Totient Function: $$\Phi(N) = (P-1) \times (Q-1)$$
3. Private Key Derivation: $$D \equiv E^{-1} \pmod \Phi$$
4. Encryption: $$C \equiv M^E \pmod N$$
5. Decryption: $$M \equiv C^D \pmod N$$
Formula Sources: Wikipedia (RSA), NIST FIPS 186-5
Variables
The following variables are used in the calculator:
- P & Q: Two distinct large prime numbers used to generate the key pair.
- N (Modulus): The public modulus, calculated as $N = P \times Q$. It is the core part of both the public and private keys.
- E (Public Exponent): The public key component, often $65537$. It must be coprime to $\Phi(N)$.
- D (Private Exponent): The private key component, calculated as the modular multiplicative inverse of $E$ modulo $\Phi(N)$.
- M (Message): The plaintext integer to be encrypted. Must be less than $N$.
- C (Ciphertext): The resulting encrypted integer.
- $\Phi(N)$ (Phi): Euler’s Totient Function, representing the number of positive integers less than $N$ that are coprime to $N$.
Related Calculators
- Modular Arithmetic Calculator
- Extended Euclidean Algorithm Solver
- Prime Number Checker Tool
- Greatest Common Divisor Finder
What is RSA Encryption?
RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. It is an asymmetric algorithm, meaning it uses two different keys: a public key for encryption and a private key for decryption. Its security relies on the practical difficulty of factoring the product of two large prime numbers.
The process begins by selecting two large, random prime numbers, $P$ and $Q$. The product, $N$, forms the modulus. The private key, $D$, is mathematically derived from $P$, $Q$, and the public exponent, $E$. The entire cryptographic process revolves around modular exponentiation, making it computationally intensive but incredibly secure against all but the most advanced factorization attacks.
How to Calculate RSA Encryption (Example)
Let’s encrypt the message $M=123$ with primes $P=61$, $Q=53$, and public exponent $E=17$.
- Calculate Modulus (N): $N = P \times Q = 61 \times 53 = 3233$.
- Calculate Totient ($\Phi$): $\Phi = (P-1) \times (Q-1) = 60 \times 52 = 3120$.
- Find Private Key (D): Find the modular inverse of $E=17$ modulo $\Phi=3120$. $D \equiv 17^{-1} \pmod{3120}$. The resulting private key is $D=2753$. (Since $17 \times 2753 = 46801$, and $46801 \div 3120 = 15$ remainder $1$).
- Encrypt Message (C): Calculate $C = M^E \pmod N$. $C = 123^{17} \pmod{3233}$.
- Final Ciphertext: $C = 855$.
To decrypt $C=855$, one would calculate $M = C^D \pmod N = 855^{2753} \pmod{3233}$, which yields $123$.
Frequently Asked Questions (FAQ)
What is the typical size of the prime numbers P and Q in real-world RSA?
In modern cryptographic systems, the modulus $N$ is typically $2048$ bits or $4096$ bits long. This means that $P$ and $Q$ are approximately half that size, around $1024$ or $2048$ bits each, resulting in extremely large numbers that are computationally infeasible to factor.
Why must the public exponent E be coprime to $\Phi(N)$?
The public exponent $E$ must be coprime to $\Phi(N)$ (meaning $\text{gcd}(E, \Phi(N)) = 1$) to ensure that the private key $D$ (the modular multiplicative inverse of $E$) exists. Without this condition, the decryption process would be mathematically impossible.
Is RSA still secure?
Yes, when used with sufficiently large key sizes (e.g., 2048-bit or higher), RSA is still considered secure against classical attacks. However, it is being slowly superseded by Elliptic Curve Cryptography (ECC) in many applications due to ECC’s smaller key sizes and faster performance for equivalent security levels.
What happens if I try to encrypt a message $M$ that is larger than the modulus $N$?
If $M \ge N$, the resulting ciphertext $C$ will be the same as encrypting $M \pmod N$. To correctly encrypt a message larger than $N$, the message must first be split into smaller blocks, each of which is less than $N$.