{primary_keyword}
Calculate RSA modulus, totient, private exponent and encrypt/decrypt a message instantly.
RSA Calculator Inputs
Enter a prime number (e.g., 61)
Enter a second prime number (e.g., 53)
Common choices: 3, 5, 17, 257, 65537
Integer less than n = p·q
| Variable | Value |
|---|---|
| Modulus n = p·q | |
| Totient φ(n) = (p‑1)(q‑1) | |
| Private exponent d | |
| Ciphertext c = m^e mod n | |
| Decrypted m’ = c^d mod n |
Prime Values Chart
What is {primary_keyword}?
The {primary_keyword} is an online tool that helps you compute the essential components of the RSA cryptosystem. RSA is a widely used public‑key encryption algorithm that relies on the mathematical difficulty of factoring large numbers. This calculator is useful for students, security professionals, and developers who need to understand or demonstrate RSA key generation, encryption, and decryption.
Common misconceptions include believing that any two numbers can be used as primes or that the public exponent can be any value. In reality, p and q must be prime and e must be coprime with φ(n).
{primary_keyword} Formula and Mathematical Explanation
RSA key generation follows a clear set of formulas:
- Choose two distinct prime numbers p and q.
- Compute the modulus n = p·q.
- Calculate the totient φ(n) = (p‑1)(q‑1).
- Select a public exponent e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1.
- Determine the private exponent d as the modular inverse of e modulo φ(n), i.e., d·e ≡ 1 (mod φ(n)).
- Encryption of a message m (0 ≤ m < n) is c = m^e mod n.
- Decryption is m’ = c^d mod n.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| p, q | Prime numbers | – | 2 – 10⁶ (for teaching) or > 10⁹ (real keys) |
| n | Modulus | – | p·q |
| φ(n) | Euler’s totient | – | (p‑1)(q‑1) |
| e | Public exponent | – | 3, 5, 17, 257, 65537 |
| d | Private exponent | – | Modular inverse of e |
| m | Plaintext message | integer | 0 ≤ m < n |
| c | Ciphertext | integer | 0 ≤ c < n |
Practical Examples (Real‑World Use Cases)
Example 1
Input: p = 61, q = 53, e = 17, m = 65.
Calculated values:
- n = 61·53 = 3233
- φ(n) = (61‑1)(53‑1) = 3120
- d = 2753 (since 17·2753 ≡ 1 mod 3120)
- c = 65¹⁷ mod 3233 = 2790
- Decrypted m’ = 2790²⁷⁵³ mod 3233 = 65
The message encrypts to 2790 and decrypts back to the original 65, demonstrating correct RSA operation.
Example 2
Input: p = 101, q = 113, e = 65537, m = 1234.
Results:
- n = 11413
- φ(n) = 11200
- d = 2753 (modular inverse of 65537 mod 11200)
- c = 1234⁶⁵⁵³⁷ mod 11413 = 8552
- Decrypted m’ = 8552²⁷⁵³ mod 11413 = 1234
This example uses a larger public exponent, common in real implementations.
How to Use This {primary_keyword} Calculator
- Enter two prime numbers p and q.
- Provide a public exponent e (default 17 works for most small examples).
- Enter the integer message m you wish to encrypt (must be less than n).
- The calculator instantly shows n, φ(n), private exponent d, ciphertext c, and decrypted message m’.
- Read the highlighted result for the private exponent d – this is the key you keep secret.
- Use the copy button to copy all results for documentation or further analysis.
Key Factors That Affect {primary_keyword} Results
- Prime size (p, q): Larger primes increase n, making factorization harder.
- Public exponent e: Must be coprime with φ(n); common choices balance security and performance.
- Message size m: Must be smaller than n; otherwise encryption fails.
- Totient φ(n): Directly influences the private exponent d; errors here break decryption.
- Modular inverse calculation: Incorrect d leads to wrong decryption.
- Implementation precision: Using big integers is essential for real‑world key sizes.
Frequently Asked Questions (FAQ)
- What if I enter non‑prime numbers for p or q?
- The calculator will still compute values, but the resulting RSA system is insecure because φ(n) will be incorrect.
- Can I use the same prime for both p and q?
- No. Using identical primes makes n a perfect square, drastically reducing security.
- Why does the calculator require e to be coprime with φ(n)?
- Only then does a modular inverse d exist, which is essential for decryption.
- What happens if my message m is larger than n?
- Encryption will produce an incorrect ciphertext; always ensure m < n.
- Is the private exponent d always smaller than n?
- Not necessarily; d can be larger than n depending on the chosen e.
- Can I use this calculator for real‑world key sizes (2048‑bit)?
- JavaScript’s Number type cannot handle such large integers; for production use a big‑integer library.
- How do I verify that my RSA keys are correct?
- Encrypt a test message and then decrypt it; the output should match the original message.
- Is the RSA algorithm still considered secure?
- When used with sufficiently large primes (≥ 2048 bits) and proper padding, RSA remains secure.
Related Tools and Internal Resources
- RSA Key Generator – Quickly generate strong RSA key pairs.
- Modular Inverse Calculator – Compute d = e⁻¹ mod φ(n) for any values.
- Prime Number Checker – Verify if a number is prime.
- BigInt Tutorial – Learn how to handle large integers in JavaScript.
- Encryption Basics – Overview of symmetric and asymmetric encryption.
- Security Best Practices – Guidelines for implementing cryptography safely.