| """ |
| AES S-box: Algebraic Construction and Verification |
| |
| S(x) = A(x^254) + c |
| where x^254 = x^{-1} in GF(2^8) and A is the affine transformation. |
| """ |
|
|
| from gf256 import gf_inv |
|
|
| AFFINE_MATRIX = [ |
| 0b11110001, |
| 0b11100011, |
| 0b11000111, |
| 0b10001111, |
| 0b00011111, |
| 0b00111110, |
| 0b01111100, |
| 0b11111000, |
| ] |
|
|
| AFFINE_CONSTANT = 0x63 |
|
|
| STANDARD_SBOX = [ |
| 0x63, 0x7c, 0x77, 0x7b, 0xf2, 0x6b, 0x6f, 0xc5, |
| 0x30, 0x01, 0x67, 0x2b, 0xfe, 0xd7, 0xab, 0x76, |
| 0xca, 0x82, 0xc9, 0x7d, 0xfa, 0x59, 0x47, 0xf0, |
| 0xad, 0xd4, 0xa2, 0xaf, 0x9c, 0xa4, 0x72, 0xc0, |
| 0xb7, 0xfd, 0x93, 0x26, 0x36, 0x3f, 0xf7, 0xcc, |
| 0x34, 0xa5, 0xe5, 0xf1, 0x71, 0xd8, 0x31, 0x15, |
| 0x04, 0xc7, 0x23, 0xc3, 0x18, 0x96, 0x05, 0x9a, |
| 0x07, 0x12, 0x80, 0xe2, 0xeb, 0x27, 0xb2, 0x75, |
| 0x09, 0x83, 0x2c, 0x1a, 0x1b, 0x6e, 0x5a, 0xa0, |
| 0x52, 0x3b, 0xd6, 0xb3, 0x29, 0xe3, 0x2f, 0x84, |
| 0x53, 0xd1, 0x00, 0xed, 0x20, 0xfc, 0xb1, 0x5b, |
| 0x6a, 0xcb, 0xbe, 0x39, 0x4a, 0x4c, 0x58, 0xcf, |
| 0xd0, 0xef, 0xaa, 0xfb, 0x43, 0x4d, 0x33, 0x85, |
| 0x45, 0xf9, 0x02, 0x7f, 0x50, 0x3c, 0x9f, 0xa8, |
| 0x51, 0xa3, 0x40, 0x8f, 0x92, 0x9d, 0x38, 0xf5, |
| 0xbc, 0xb6, 0xda, 0x21, 0x10, 0xff, 0xf3, 0xd2, |
| 0xcd, 0x0c, 0x13, 0xec, 0x5f, 0x97, 0x44, 0x17, |
| 0xc4, 0xa7, 0x7e, 0x3d, 0x64, 0x5d, 0x19, 0x73, |
| 0x60, 0x81, 0x4f, 0xdc, 0x22, 0x2a, 0x90, 0x88, |
| 0x46, 0xee, 0xb8, 0x14, 0xde, 0x5e, 0x0b, 0xdb, |
| 0xe0, 0x32, 0x3a, 0x0a, 0x49, 0x06, 0x24, 0x5c, |
| 0xc2, 0xd3, 0xac, 0x62, 0x91, 0x95, 0xe4, 0x79, |
| 0xe7, 0xc8, 0x37, 0x6d, 0x8d, 0xd5, 0x4e, 0xa9, |
| 0x6c, 0x56, 0xf4, 0xea, 0x65, 0x7a, 0xae, 0x08, |
| 0xba, 0x78, 0x25, 0x2e, 0x1c, 0xa6, 0xb4, 0xc6, |
| 0xe8, 0xdd, 0x74, 0x1f, 0x4b, 0xbd, 0x8b, 0x8a, |
| 0x70, 0x3e, 0xb5, 0x66, 0x48, 0x03, 0xf6, 0x0e, |
| 0x61, 0x35, 0x57, 0xb9, 0x86, 0xc1, 0x1d, 0x9e, |
| 0xe1, 0xf8, 0x98, 0x11, 0x69, 0xd9, 0x8e, 0x94, |
| 0x9b, 0x1e, 0x87, 0xe9, 0xce, 0x55, 0x28, 0xdf, |
| 0x8c, 0xa1, 0x89, 0x0d, 0xbf, 0xe6, 0x42, 0x68, |
| 0x41, 0x99, 0x2d, 0x0f, 0xb0, 0x54, 0xbb, 0x16, |
| ] |
|
|
|
|
| def affine_transform(x: int) -> int: |
| """Apply the AES affine transformation over GF(2).""" |
| result = 0 |
| for i in range(8): |
| bit = 0 |
| for j in range(8): |
| bit ^= ((x >> j) & 1) & ((AFFINE_MATRIX[i] >> j) & 1) |
| result |= (bit << i) |
| return result ^ AFFINE_CONSTANT |
|
|
|
|
| def sbox(x: int) -> int: |
| """S(x) = A(x^254) + c = A(x^{-1}) + c""" |
| return affine_transform(gf_inv(x)) |
|
|
|
|
| def verify_sbox(): |
| """Verify algebraic S-box matches the standard AES S-box for all 256 inputs.""" |
| mismatches = [] |
| for x in range(256): |
| computed = sbox(x) |
| expected = STANDARD_SBOX[x] |
| if computed != expected: |
| mismatches.append((x, computed, expected)) |
| return len(mismatches) == 0, mismatches |
|
|
|
|
| def differential_uniformity(): |
| """Compute the differential uniformity of the AES S-box. |
| For each nonzero input difference a, count max solutions to S(x+a)+S(x)=b.""" |
| max_count = 0 |
| for a in range(1, 256): |
| counts = [0] * 256 |
| for x in range(256): |
| diff = sbox(x ^ a) ^ sbox(x) |
| counts[diff] += 1 |
| max_count = max(max_count, max(counts)) |
| return max_count |
|
|
|
|
| if __name__ == "__main__": |
| valid, mismatches = verify_sbox() |
| print(f"S-box verification: {'PASS' if valid else 'FAIL'} (mismatches: {len(mismatches)})") |
| du = differential_uniformity() |
| print(f"Differential uniformity: {du} (expected: 4, giving 2^-6 per S-box)") |
|
|
