Rediscovered Affine Equivalence Between CLEFIA and Rijndael S-boxes
An interesting Link shows that the CLEFIA S-box and the Rijndael (AES) S-box are affine equivalent—meaning one can be transformed into the other through invertible affine transformations over the finite field $\mathbb{F}_2^{8}$.
Formally, two functions ( f ) and ( g ) are affine equivalent if there exist invertible affine mappings ( A ), ( B ), and constants ( a ), ( b ) such that:
$$ f(x) = (B \circ g \circ A)(x \oplus a) \oplus b $$
$$ \text{where } x,a,b\in \{0,1\}^8, \quad A,B\in\mathbb{F}_2^{8\times 8}. $$
- ( f ) is the CLEFIA S-box
- ( g ) is the Rijndael (AES) S-box
- ( A ) and ( B ) are invertible binary matrices
- ( a ) and ( b ) are affine offsets in $\mathbb{F}_2^8$
🧮 Transformation Matrices
Matrix ( A )
$$ A = \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1\\ 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \end{bmatrix} ,\quad a = 90 = \text{0x5A}. $$
Matrix ( B )
$$ B = \begin{bmatrix} 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1\\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1\\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1\\ 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 \end{bmatrix} ,\quad b = 221 = \text{0xDD}. $$
🧩 Interpretation
The relation shows that CLEFIA’s S-box can be obtained directly from AES’s S-box through linear transformations ( A, B ) and translations ( a, b ).
This equivalence means their algebraic properties—such as nonlinearity, differential uniformity, and fixed-point structure—are inherently preserved.
From an implementation viewpoint:
- Designers can reuse hardware structures optimized for AES S-boxes to build CLEFIA’s equivalent (such as SIMD)