Rijndael S-box

The Rijndael S-box is a substitution box (lookup table) used in the Rijndael cipher, on which the Advanced Encryption Standard (AES) cryptographic algorithm is based.

Forward S-box

The S-box maps an 8-bit input, c, to an 8-bit output, s = S(c). Both the input and output are interpreted as polynomials over GF(2). First, the input is mapped to its multiplicative inverse in GF(2⁸) = GF(2) [x]/(x⁸ + x⁴ + x³ + x + 1), Rijndael's finite field. Zero, as the identity, is mapped to itself. This transformation is known as the Nyberg S-box after its inventor Kaisa Nyberg. The multiplicative inverse is then transformed using the following affine transformation:

${\displaystyle {\begin{bmatrix}s_{0}\\s_{1}\\s_{2}\\s_{3}\\s_{4}\\s_{5}\\s_{6}\\s_{7}\end{bmatrix}}={\begin{bmatrix}1&0&0&0&1&1&1&1\\1&1&0&0&0&1&1&1\\1&1&1&0&0&0&1&1\\1&1&1&1&0&0&0&1\\1&1&1&1&1&0&0&0\\0&1&1&1&1&1&0&0\\0&0&1&1&1&1&1&0\\0&0&0&1&1&1&1&1\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\\b_{4}\\b_{5}\\b_{6}\\b_{7}\end{bmatrix}}+{\begin{bmatrix}1\\1\\0\\0\\0\\1\\1\\0\end{bmatrix}}}$

where [s₇, ..., s₀] is the S-box output and [b₇, ..., b₀] is the multiplicative inverse as a vector.

This affine transformation is the sum of multiple rotations of the byte as a vector, where addition is the XOR operation:

${\displaystyle s=b\oplus (b\lll 1)\oplus (b\lll 2)\oplus (b\lll 3)\oplus (b\lll 4)\oplus 63_{16}}$

where b represents the multiplicative inverse, ${\displaystyle \oplus }$ is the bitwise XOR operator, ${\displaystyle \lll }$ is a left bitwise circular shift, and the constant 63₁₆ = 01100011₂ is given in hexadecimal.

An equivalent formulation of the affine transformation is

${\displaystyle s_{i}=b_{i}\oplus b_{(i+4)\operatorname {mod} 8}\oplus b_{(i+5)\operatorname {mod} 8}\oplus b_{(i+6)\operatorname {mod} 8}\oplus b_{(i+7)\operatorname {mod} 8}\oplus c_{i}}$

where s, b, and c are 8 bit arrays, c is 01100011₂, and subscripts indicate a reference to the indexed bit.

Another equivalent is:

${\displaystyle s=\left(b\times 31_{10}\mod {257_{10}}\right)\oplus 99_{10}}$

where ${\displaystyle \times }$ is polynomial multiplication of ${\displaystyle b}$ and ${\displaystyle 31_{10}}$ taken as bit arrays.

Inverse S-box

The inverse S-box is simply the S-box run in reverse. For example, the inverse S-box of b8₁₆ is 9a₁₆. It is calculated by first calculating the inverse affine transformation of the input value, followed by the multiplicative inverse. The inverse affine transformation is as follows:

${\displaystyle {\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\\b_{4}\\b_{5}\\b_{6}\\b_{7}\end{bmatrix}}={\begin{bmatrix}0&0&1&0&0&1&0&1\\1&0&0&1&0&0&1&0\\0&1&0&0&1&0&0&1\\1&0&1&0&0&1&0&0\\0&1&0&1&0&0&1&0\\0&0&1&0&1&0&0&1\\1&0&0&1&0&1&0&0\\0&1&0&0&1&0&1&0\end{bmatrix}}{\begin{bmatrix}s_{0}\\s_{1}\\s_{2}\\s_{3}\\s_{4}\\s_{5}\\s_{6}\\s_{7}\end{bmatrix}}+{\begin{bmatrix}1\\0\\1\\0\\0\\0\\0\\0\end{bmatrix}}}$

The inverse affine transformation also represents the sum of multiple rotations of the byte as a vector, where addition is the XOR operation:

${\displaystyle b=(s\lll 1)\oplus (s\lll 3)\oplus (s\lll 6)\oplus 5_{16}}$

where ${\displaystyle \oplus }$ is the bitwise XOR operator, ${\displaystyle \lll }$ is a left bitwise circular shift, and the constant 5₁₆ = 00000101₂ is given in hexadecimal.

Design criteria

The Rijndael S-box was specifically designed to be resistant to linear and differential cryptanalysis. This was done by minimizing the correlation between linear transformations of input/output bits, and at the same time minimizing the difference propagation probability.

The Rijndael S-box can be replaced in the Rijndael cipher, which defeats the suspicion of a backdoor built into the cipher that exploits a static S-box. The authors claim that the Rijndael cipher structure is likely to provide enough resistance against differential and linear cryptanalysis even if an S-box with "average" correlation / difference propagation properties is used (cf. the "optimal" properties of the Rijndael S-box).

Example implementation in C language

The following C code calculates the S-box:

References

1. S-box
2. Advanced Encryption Standard
3. Grøstl
4. SM4 (cipher)
5. AES key schedule
6. ARIA (cipher)
7. Index of cryptography articles
8. Serpent (cipher)
9. Twofish
10. AES implementations
11. Substitution–permutation network
12. SHARK
13. KHAZAD
14. Anubis (cipher)
15. Confusion and diffusion
16. Bricklayer function
17. XSL attack
18. Branch number
19. Advanced Encryption Standard process
20. Data Encryption Standard