Common Principles of Cryptography

Author

Affiliation

Andrei Biziuk

VSTU

Introduction

We’ll delve into various crucial aspects, including the classification of encryption algorithms, the classification of threats, the concept of algorithm robustness, necessary knowledge in Boolean algebra and number theory, basic encryption algorithms, principles of block cipher construction, private key cryptosystems, the Feistel scheme, and public key cryptosystems along with practical examples.

1. Classification of Encryption Algorithms

Cryptographic algorithms are divided into two main categories:

1. Symmetric Key Algorithms: Both encryption and decryption use the same key.
   • Block Ciphers: Encrypts data in fixed-size blocks.
     • Examples: AES (Advanced Encryption Standard), DES (Data Encryption Standard).
   • Stream Ciphers: Encrypts data as a stream of bits or bytes.
     • Examples: RC4, Salsa20.

---

2. Asymmetric Key Algorithms: Uses a pair of keys – one public for encryption and one private for decryption.
   • Examples: RSA (Rivest-Shamir-Adleman), ECC (Elliptic Curve Cryptography).

2. Classification of Threats

Cryptographic systems encounter various threats, classified as follows:

1. Passive Attacks: Eavesdropping on communications without altering the content.
   • Example: Traffic analysis – monitoring the data flow to deduce information.

---

2. Active Attacks: Involves the modification or creation of false data streams.
   • Examples:
     • Man-in-the-Middle Attack: Intercepting and altering communication between two parties.
     • Replay Attack: Reusing valid data transmission to deceive the receiver.
     • Modification Attack: Altering the transmitted message.

3. Concept of Algorithm Robustness

Algorithm robustness is a measure of a cryptographic algorithm’s ability to resist attacks. Key factors include:

1. Key Length: Longer keys offer stronger security. For example, AES can use 128, 192, or 256-bit keys.
2. Algorithm Complexity: Algorithms using complex mathematical operations are more secure.
3. Nonlinearity: Introducing non-linear transformations increases difficulty for attackers.
4. Diffusion and Confusion: Ensuring that the structure of the ciphertext significantly differs from the plaintext.

4. Necessary Knowledge in Boolean Algebra and Number Theory

Cryptography heavily relies on the following mathematical foundations:

1. Boolean Algebra:
   • Fundamental operations: AND, OR, NOT, XOR.
   • Boolean functions: Key in designing symmetric ciphers like AES.

---

2. Number Theory:
   • Prime Numbers: Basis for public key algorithms (e.g., RSA).
   • Modular Arithmetic: Essential for operations in many algorithms.
   • Greatest Common Divisors (GCD): Used in key generation algorithms.

5. Basic Encryption Algorithms

1. Caesar Cipher:
   • Description: Shifts each letter in the plaintext by a fixed number of positions.
   • Example: Shift by 3, A becomes D, B becomes E, etc.
2. Vigenère Cipher:
   • Description: Uses a keyword to shift letters; the shift depends on the corresponding letter in the keyword.
   • Example: For keyword “KEY”, A shifted by K (10 positions), B by E (4 positions), etc.

6. Principles of Block Cipher Construction

Block ciphers process plaintext in fixed-size blocks and involve multiple rounds of transformation:

1. Substitution-Permutation Network (SPN): Involves substituting (S-boxes) and permuting bits.
   • Example: AES employs multiple rounds of SPN.
2. Feistel Network: Divides the block into two halves and applies a round function iteratively.
   • Example: DES uses a 16-round Feistel structure.

7. Private Key Cryptosystems

Private key (or symmetric key) cryptosystems involve a single key for both encryption and decryption:

1. Key Management: Secure distribution and storage of keys are critical.
2. Performance: Generally faster than public key systems due to simpler mathematical operations.

8. The Feistel Scheme

The Feistel scheme is a symmetric structure used in many block ciphers. Key components include:

1. Round Function: Applies a function to one half of the data block, combined with the other half.
2. Swapping Halves: After each round, the halves are swapped to ensure diffusion.
3. Example: DES (Data Encryption Standard):
   • Structure: 16 rounds of Feistel structure.
   • Process: Each round involves substitution and permutation based on the round key.

9. Examples of Private Key Algorithms

1. DES (Data Encryption Standard):
   • Block Size: 64 bits.
   • Key Size: 56 bits.
   • Rounds: 16 rounds of Feistel structure.
2. AES (Advanced Encryption Standard):
   • Block Size: 128 bits.
   • Key Sizes: 128, 192, or 256 bits.
   • Rounds: 10, 12, or 14 rounds depending on key size.

10. Public Key Cryptosystems

Public key (or asymmetric key) cryptosystems use two different keys – one public and one private:

1. Key Pair Generation: A public and private key pair is generated using complex mathematical problems.
2. Encryption and Decryption: Public key encrypts data, private key decrypts.

11. Examples of Public Key Algorithms

1. RSA (Rivest-Shamir-Adleman):
   • Basis: Difficulty of factoring large integers.
   • Key Generation: Involves large prime numbers.
   • Usage: Secure data transmission, digital signatures.
2. ECC (Elliptic Curve Cryptography):
   • Basis: Mathematical structure of elliptic curves.
   • Key Size: Provides same security with smaller keys compared to RSA.
   • Usage: Mobile devices, IoT due to efficiency.