Rijndael S-box
substitution box used in the Rijndael cipher
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:
[
s
0
s
1
s
2
s
3
s
4
s
5
s
6
s
7
]
=
[
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
]
[
b
0
b
1
b
2
b
3
b
4
b
5
b
6
b
7
]
+
[
1
1
0
0
0
1
1
0
]
{\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:
s
=
b
⊕
(
b
⋘
1
)
⊕
(
b
⋘
2
)
⊕
(
b
⋘
3
)
⊕
(
b
⋘
4
)
⊕
63
16
{\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
s
i
=
b
i
⊕
b
(
i
+
4
)
mod
8
⊕
b
(
i
+
5
)
mod
8
⊕
b
(
i
+
6
)
mod
8
⊕
b
(
i
+
7
)
mod
8
⊕
c
i
{\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:
s
=
(
b
×
31
10
mod
257
10
)
⊕
99
10
{\displaystyle s=\left(b\times 31_{10}\mod {257_{10}}\right)\oplus 99_{10}}
where
×
{\displaystyle \times }
is polynomial multiplication of
b
{\displaystyle b}
and
31
10
{\displaystyle 31_{10}}
taken as bit arrays.
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