🔐 AES Subkey Generator

AES (Advanced Encryption Standard) uses a key expansion algorithm to derive multiple subkeys from the original key. These subkeys are used in different rounds of the encryption process. The number of subkeys depends on the key size: AES-128 uses 11 subkeys (10 rounds + initial), AES-192 uses 13 subkeys (12 rounds + initial), and AES-256 uses 15 subkeys (14 rounds + initial).

AES is a symmetric block cipher standardized by NIST. It operates on 128-bit blocks and uses key sizes of 128, 192, or 256 bits. The most common version is AES-128.

• Block size: 128 bits = 16 bytes
• Key size: 128/192/256 bits
• Rounds:
  • 10 for AES-128
  • 12 for AES-192
  • 14 for AES-256

Each block is represented as a 4×4 byte matrix (called the state).

The cipher key is expanded into multiple round keys, each 16 bytes. For AES-128, 11 keys (10 rounds + initial) are generated.

Theoretical Foundation:

The key schedule is designed to provide:

• Diffusion: Each bit of the original key affects many bits of the expanded key
• Non-linearity: Through the use of the S-box in the expansion process
• Resistance to related-key attacks: By using round constants (Rcon)

The expansion algorithm uses operations in the Galois Field GF(2⁸), which provides mathematical properties that strengthen the cipher against algebraic attacks.

Each byte in the state is replaced using a Substitution Box (S-box).

Theoretical Foundation:

The S-box is constructed by:

1. Taking the multiplicative inverse of each byte in GF(2⁸)
2. Applying an affine transformation over GF(2)

This creates a non-linear substitution that provides:

• Confusion: Obscures the relationship between the key and ciphertext
• Non-linearity: Resistance to differential and linear cryptanalysis
• Algebraic complexity: Makes algebraic attacks difficult

Bytes in each row of the state matrix are shifted:

Theoretical Foundation:

ShiftRows provides:

• Diffusion: Ensures that changes in one byte affect multiple bytes after several rounds
• Structural design: Helps prevent columns from being encrypted independently
• Resistance to truncated differential attacks: By ensuring complete diffusion after multiple rounds

Each column of the state is multiplied by a fixed matrix in GF(2⁸).

Fixed Matrix:

[02 03 01 01]
[01 02 03 01]
[01 01 02 03]
[03 01 01 02]

Theoretical Foundation:

MixColumns provides:

• Maximum Distance Separable (MDS) property: Ensures optimal diffusion
• Branch number of 5: Changing one byte affects all 4 bytes in the output column
• Mathematical foundation in GF(2⁸): Operations use finite field arithmetic with the irreducible polynomial x⁸ + x⁴ + x³ + x + 1

The coefficients (02, 03, 01) were carefully chosen to:

• Be efficiently implementable in hardware and software
• Provide strong cryptographic properties
• Create a matrix that is its own inverse when used with different coefficients (for decryption)

Each byte of the state is XORed with the corresponding byte of the round key.

Theoretical Foundation:

AddRoundKey is the only step that directly incorporates the secret key, providing:

• Key dependency: Makes the encryption dependent on the secret key
• Simplicity: XOR is efficient and reversible
• Cryptographic strength: When combined with the other operations, creates a secure cipher

The security of this operation relies on the complexity of the key schedule and the diffusion provided by the other operations.

AES operations are based on finite field mathematics, specifically the Galois Field GF(2⁸).

Key Properties:

• Field structure: GF(2⁸) contains 256 elements (all possible byte values)
• Irreducible polynomial: x⁸ + x⁴ + x³ + x + 1 (0x11B in hexadecimal)
• Operations:
  • Addition: Bitwise XOR
  • Multiplication: Polynomial multiplication modulo the irreducible polynomial

Security Implications:

• Algebraic structure: Provides mathematical properties that strengthen the cipher
• Non-linearity: The S-box introduces non-linearity that prevents linear cryptanalysis
• Diffusion: MixColumns and ShiftRows ensure changes propagate throughout the state
• Confusion: SubBytes and the key schedule hide relationships between plaintext, key, and ciphertext

These mathematical foundations make AES resistant to known cryptanalytic attacks and provide a solid theoretical basis for its security.