The indicator of a Boolean function

Question

In the paper "Componentwise APNness, Walsh uniformity of APN functions and cyclic difference sets" by Claude Carlet, it is written that: Let F be any power function on $F_{2^n}$ and $\Delta _{F}=\{F(x)+F(x+1)+1 , x\in F_{2^n}\}$. In Remark 4.9, the indicator of $\Delta _{F}$ is denoted by $1_{\Delta _{F}}$ and in Remark 4.12, the binary sequence $a_t$ is defined as $a_t=1_{\Delta _{F}}(\alpha^{t})$ where $\alpha$ is primitive over $F_{2^n}$ and $t=0...2^n-2$. Since there is no detailed definition of $1_{\Delta _{F}}$, I do not understand what is the meaning of indicator of such a Boolean function. When I searche google, I see the definition of indicator of subsets/subspaces of finite fields. Thanks for replys.

Answer 1

With regards to your second to last sentence, $\Delta_F$ is a subset of $F$, so $1_{\Delta_F}$ is an indicator of a subset of a finite field.

$1_{\Delta_F}$ is a function from $F$ to $\{0,1\}$, where $$1_{\Delta_F}(x)=\begin{cases}1 & x\in \Delta_F,\\0 & x\notin\Delta_F.\end{cases}$$