Key Concepts in Cryptography Methods to Know for Discrete Mathematics

Cryptography methods are essential for secure communication, relying on mathematical principles from discrete mathematics. These methods include symmetric and public key cryptography, each with unique approaches to encryption, key exchange, and data integrity, ensuring confidentiality and authenticity.

1. Symmetric Key Cryptography

   • Uses the same key for both encryption and decryption.
   • Key distribution is a major challenge; both parties must securely share the key.
   • Examples include AES (Advanced Encryption Standard) and DES (Data Encryption Standard).

2. Public Key Cryptography

   • Utilizes a pair of keys: a public key for encryption and a private key for decryption.
   • Solves the key distribution problem inherent in symmetric key cryptography.
   • Commonly used in secure communications, such as SSL/TLS for web security.

3. RSA Algorithm

   • A widely used public key cryptosystem based on the difficulty of factoring large integers.
   • Involves generating two large prime numbers to create the public and private keys.
   • Provides both encryption and digital signature capabilities.

4. Diffie-Hellman Key Exchange

   • A method for two parties to securely share a secret key over a public channel.
   • Based on the mathematical properties of modular exponentiation and discrete logarithms.
   • Does not provide authentication; vulnerable to man-in-the-middle attacks without additional measures.

5. Caesar Cipher

   • A simple substitution cipher that shifts letters in the alphabet by a fixed number.
   • Easy to implement but vulnerable to frequency analysis and brute-force attacks.
   • Named after Julius Caesar, who reportedly used it for military communications.

6. Vigenère Cipher

   • A method of encrypting alphabetic text using a simple form of polyalphabetic substitution.
   • Uses a keyword to determine the shift for each letter, making it more secure than the Caesar cipher.
   • Vulnerable to Kasiski examination and other cryptanalysis techniques.

7. One-Time Pad

   • A theoretically unbreakable cipher that uses a random key that is as long as the message.
   • Each key is used only once, ensuring perfect secrecy if the key is truly random and kept secret.
   • Practical challenges include key generation, distribution, and management.

8. Block Ciphers

   • Encrypts data in fixed-size blocks (e.g., 128 bits) rather than as a continuous stream.
   • Common examples include AES and Blowfish.
   • Often used in conjunction with modes of operation (e.g., CBC, ECB) to enhance security.

9. Stream Ciphers

   • Encrypts data one bit or byte at a time, making it suitable for real-time applications.
   • Typically faster than block ciphers and can handle data of arbitrary length.
   • Examples include RC4 and Salsa20.

10. Hash Functions

    • Converts input data of any size into a fixed-size string of characters, typically a digest.
    • Used for data integrity verification, password storage, and digital signatures.
    • Properties include pre-image resistance, collision resistance, and avalanche effect.

11. Digital Signatures

    • A cryptographic technique that verifies the authenticity and integrity of a message.
    • Involves creating a hash of the message and encrypting it with the sender's private key.
    • Provides non-repudiation, ensuring that the sender cannot deny sending the message.

12. Elliptic Curve Cryptography

    • A form of public key cryptography based on the algebraic structure of elliptic curves over finite fields.
    • Offers equivalent security to RSA with smaller key sizes, making it more efficient.
    • Widely used in mobile devices and constrained environments due to its low computational requirements.