Introduction to Cryptography and RSA

The Basics of Cryptography

Cryptography focuses on securing information through scientific methods. This guide explores a family of cryptographic techniques—specifically RSA—enabling secure and convenient data transfer between parties. We'll cover both theoretical foundations and practical applications.

Symmetric Cryptography

The simplest method is the one-time pad. Imagine Alice and Bob (common names in cryptography) wishing to communicate secretly:

1. Key Generation: They meet beforehand to create a shared random binary key (e.g., 10110).
2. Encryption: Alice XORs her message (e.g., 01101) with the key, producing ciphertext (11011).
3. Decryption: Bob XORs the ciphertext with the same key to retrieve the original message.

Challenges:

• The key must match the message length.
• Pre-sharing keys isn’t always practical.

Workarounds:

• Repeating Keys: Shorter keys reused introduce patterns vulnerable to statistical attacks.
• Pseudorandom Number Generators (PRNGs): Use a seed to generate "random" bits dynamically. However, PRNG security isn’t provably robust.

Asymmetric Cryptography: A New Approach

Unlike symmetric systems, asymmetric cryptography uses separate keys for encryption and decryption.

Physical Analogy

Alice and Bob use unique locks:

1. Alice locks a box with her lock and sends it to Bob.
2. Bob adds his lock and returns the box.
3. Alice removes her lock, and Bob unlocks his to access the message.

In cryptography, "locks" are mathematical functions. Bob’s public key (lock) is shared openly, while his private key (unlock) remains secret.

The RSA Algorithm

Key Concepts

1. Terminology:

   • Public Key: Used to encrypt (shared openly).
   • Private Key: Used to decrypt (kept secret).

2. Key Generation:

   • Bob selects two large primes (p and q).
   • Computes N = p × q (public key).

3. Encryption:

   • Alice computes x³ mod N (ensuring x < N < x³ via padding).

4. Decryption:

   • Bob finds k such that 3k ≡ 1 mod (p−1)(q−1).
   • Decrypts via (x³ mod N)^k ≡ x mod N, leveraging Euler’s Theorem.

Benefits and Drawbacks

Advantages

• No Pre-Shared Secrets: Eliminates the need for initial secure meetings.
• Wide Adoption: Used in HTTPS, secure communications.

Challenges

• Security Reliance: Depends on factoring difficulty (unproven for classical computers; broken by quantum algorithms like Shor’s).
• Performance: Slower than symmetric methods; often used to establish seeds for faster protocols.

Authentication via Public Key Cryptography

RSA naturally enables authentication:

1. Alice signs a message with her private key.
2. Bob verifies it using her public key.
3. Combined with encryption, this ensures both secrecy and sender verification.

FAQ

1. Why is RSA considered secure?

RSA relies on the computational difficulty of factoring large primes—a problem with no known efficient classical solution.

2. Can RSA be used for quick communications?

Due to its slower speed, RSA often negotiates a session key first, then switches to symmetric encryption (e.g., AES).

3. How does padding enhance RSA security?

Padding (e.g., PKCS#1) prevents attacks by ensuring messages don’t map to predictable ciphertext patterns.

👉 Explore advanced cryptographic techniques

👉 Learn about quantum-resistant algorithms