Linear Boolean function ($f_a(x)=a\cdot x$)

A linear Boolean function is a linear map from ${\mathbb{F}}_2^n$ to ${\mathbb{F}}_2$. Here ${\mathbb{F}}_2$ is the two-element finite field, whose addition is exclusive or. Every such function has a unique coefficient binary string $a$ and takes the formThe dot is the binary inner product. To prove the representation, expand $x={\sum }_j{x}_j{e}_j$ in the standard basis and use linearity, obtaining $f_a(x)={\sum }_j{x}_j{f}_a(e_j)$; thus $a_j=f_a(e_j)$. For example, $a=101$ gives $f_a(x)=x_1\oplus x_3$. Adding a nonzero constant produces an affine map, not a linear map.