RSA Encryption Calculator

Number Theory & Sequences

Generate RSA keys, encrypt and decrypt messages using the RSA algorithm. Perfect for learning public key cryptography and number theory concepts.

RSA Calculator Examples

Explore different RSA operations with these practical examples

Small Prime Key Generation

keyGeneration

Generate RSA keys using small prime numbers for educational purposes

p: 7

q: 11

e: 3

Medium Prime Key Generation

keyGeneration

Generate RSA keys with larger prime numbers

p: 61

q: 53

e: 17

Message Encryption

encryption

Encrypt a message using RSA public key

n: 77

e: 3

m: 65

Message Decryption

decryption

Decrypt a ciphertext using RSA private key

n: 77

d: 27

m: 31

Other Titles
Understanding RSA Encryption Calculator: A Comprehensive Guide
Master the fundamentals of public key cryptography, RSA algorithm implementation, and secure communication principles

What is RSA Encryption?

  • Mathematical Foundation
  • Public Key Cryptography
  • Security Principles
RSA (Rivest-Shamir-Adleman) is a public-key cryptographic algorithm that enables secure communication over insecure channels. Named after its inventors Ron Rivest, Adi Shamir, and Leonard Adleman, RSA was first published in 1977 and remains one of the most widely used encryption systems today.
Core Mathematical Concepts
RSA security relies on the mathematical difficulty of factoring large composite numbers. The algorithm uses modular arithmetic, prime number theory, and Euler's totient function to create a mathematically secure encryption system. The fundamental principle is that while it's easy to multiply two large prime numbers, it's extremely difficult to factor their product back into the original primes.
Public Key Cryptography Revolution
RSA introduced the revolutionary concept of asymmetric cryptography, where encryption and decryption use different keys. This solved the key distribution problem that plagued symmetric encryption systems, enabling secure communication without prior key exchange.
Security and Trust
The security of RSA depends on the computational infeasibility of factoring large numbers. Modern RSA implementations use keys of 2048 bits or larger, providing security levels that would require centuries to break with current computing technology.

RSA Security Examples

  • RSA-2048 provides approximately 112 bits of security
  • Factoring a 2048-bit number would require ~2^112 operations

Step-by-Step Guide to Using the RSA Calculator

  • Key Generation Process
  • Encryption Procedure
  • Decryption Method
Our RSA calculator supports three main operations: key generation, message encryption, and message decryption. Each operation follows the mathematical principles of the RSA algorithm while providing educational insights into the underlying processes.
RSA Key Generation
1. Select two distinct prime numbers p and q. 2. Calculate the modulus n = p × q. 3. Compute Euler's totient function φ(n) = (p-1)(q-1). 4. Choose a public exponent e that is coprime to φ(n). 5. Calculate the private exponent d as the multiplicative inverse of e modulo φ(n).
Message Encryption
To encrypt a message m using the public key (n, e): 1. Ensure the message m < n. 2. Calculate the ciphertext c = m^e mod n. The result is the encrypted message that can only be decrypted with the corresponding private key.
Message Decryption
To decrypt a ciphertext c using the private key (n, d): 1. Calculate the plaintext m = c^d mod n. 2. The result is the original message. This process demonstrates the mathematical relationship between public and private keys.

RSA Calculation Examples

  • Key generation with p=7, q=11 produces n=77, φ(n)=60
  • Encrypting message 65 with e=3: 65^3 mod 77 = 31

Real-World Applications of RSA Encryption

  • Digital Security
  • Internet Communications
  • Authentication Systems
RSA encryption forms the backbone of modern internet security, enabling secure online transactions, private communications, and digital authentication. Its applications span from everyday web browsing to high-security government communications.
HTTPS and Web Security
Every time you see 'https://' in your browser, RSA encryption is likely protecting your connection. RSA is used in SSL/TLS protocols to establish secure connections between your browser and websites, protecting sensitive data like passwords and credit card information.
Digital Signatures and Authentication
RSA enables digital signatures that verify the authenticity and integrity of digital documents. This technology is crucial for software distribution, legal documents, and any situation where you need to prove that a message came from a specific sender and hasn't been tampered with.
Cryptocurrency and Blockchain
Many cryptocurrency systems use RSA-based cryptography for wallet security and transaction verification. The technology ensures that only the rightful owner of a cryptocurrency wallet can authorize transactions.

RSA Application Examples

  • Online banking uses RSA to protect financial transactions
  • Email encryption with PGP/GPG uses RSA for key exchange

Common Misconceptions and Correct Methods

  • Security Myths
  • Implementation Errors
  • Best Practices
Understanding RSA requires dispelling common misconceptions and learning proper implementation practices. Many security vulnerabilities arise from incorrect RSA usage rather than weaknesses in the algorithm itself.
Key Size Misconceptions
A common misconception is that RSA keys can be arbitrarily small for educational purposes without security implications. While small keys (like 512 bits) are useful for learning, they provide no real security and can be factored in minutes with modern computers.
Random Number Generation
RSA security critically depends on using truly random prime numbers. Weak random number generators have led to real-world security breaches where attackers could predict or duplicate private keys. Always use cryptographically secure random number generators.
Padding and Message Formatting
Raw RSA encryption (as demonstrated in educational examples) is vulnerable to various attacks. Production RSA implementations must use proper padding schemes like OAEP (Optimal Asymmetric Encryption Padding) to prevent chosen-ciphertext attacks and ensure semantic security.

RSA Security Examples

  • RSA-512 can be factored in hours with modern hardware
  • Textbook RSA without padding is vulnerable to chosen-ciphertext attacks

Mathematical Derivation and Advanced Concepts

  • Theoretical Foundation
  • Algorithmic Complexity
  • Modern Developments
The mathematical elegance of RSA lies in its foundation on well-established number theory principles. Understanding these underlying concepts provides insight into why RSA works and how its security properties emerge from mathematical relationships.
Euler's Theorem and Totient Function
RSA's correctness relies on Euler's theorem, which states that for coprime integers a and n: a^φ(n) ≡ 1 (mod n). This theorem ensures that encryption and decryption are inverse operations when e and d satisfy ed ≡ 1 (mod φ(n)).
Computational Complexity
The security of RSA depends on the computational complexity of integer factorization. The best known factoring algorithms have sub-exponential complexity, but remain impractical for large numbers. Shor's algorithm could factor RSA keys efficiently on quantum computers, driving research into post-quantum cryptography.
Extended Euclidean Algorithm
Computing the private key d requires finding the multiplicative inverse of e modulo φ(n). This is efficiently accomplished using the Extended Euclidean Algorithm, which not only finds the greatest common divisor but also the coefficients needed for the inverse calculation.

Mathematical Examples

  • For n=77, φ(n)=60, and e=3: d = 27 because 3×27 ≡ 1 (mod 60)
  • Factoring RSA-2048 requires approximately 2^112 operations