Superscript vs subscript notation in cryptographic formulation

Question

I'm currently reading this paper [PDF]. On page 4, I bumped into these notations :

\begin{equation} \text { Experiment } \operatorname{Exp}_{\mathcal{F} \mathcal{E}, A}^{\text {ind-mode }}(k) \text { : } \end{equation}

\begin{equation} A_{1}^{\mathrm{KDer}\left(s k_{i}\right)}(p k) \end{equation}

I tried searching online and resolved most of the other notations involved, like the $ \stackrel{\\\$}{\leftarrow}$, but no one provided a link to a source for resolving similar problems. I couldn't resolve any of these. This is the definition where these notations belong to. For more information you can refer to the paper itself.

\begin{array}{l} \text { Experiment } \operatorname{Exp}_{\mathcal{F} \mathcal{E}, A}^{\text {ind-mode }}(k): \\ b \stackrel{\\\$}{\leftarrow}\{0,1\} \\ (p k, s k) \\\$ \operatorname{Setup}\left(1^{k}\right) \\ \left(m_{0}, m_{1}, s t\right) \stackrel{\\\$}{\leftarrow} A_{1}^{\mathrm{KDer}(s k, \cdot)}(p k) \\ c \leftarrow{E n c}\left(p k, m_{b}\right) \\ b^{\prime} \stackrel{\\\$}{\leftarrow} A_{2}^{\mathcal{O}(s k, \cdot)}(p k, c, s t) \\ \text { If } b=b^{\prime} \text { return } 1 \text { else return } 0 \end{array}

Answer 1

$\textrm{Exp}^{\textrm{ind-mode}}_{\mathcal{FE},A}$ is just the name given to the interaction. The "exponent" $\textrm{ind-mode}$ is part of that name. There is not really a standard, universal way of giving names to these kinds of games. But usually the author has to specify: what game is it? what scheme is being attacked? what is the attacker? and maybe other parameters too. Since there is a lot of information to include, we often use both subscripts and superscripts to include it.

$A^{\textrm{KDer}(sk,\cdot)}(pk)$ is referring to an adversary program $A$. The adversary is given $pk$ as its input. It is also given oracle access to $\textrm{KDer}(sk,\cdot)$. Oracle access means: at any time, $A$ can ask a question $x$ and receive the answer $\textrm{KDer}(sk,x)$. It can ask many such questions. Writing an oracle as a superscript is very standard in cryptography and other areas of computer science (especially computational complexity).

From what I can tell, the convention of writing an oracle as a superscript is from as far back as a 1954 paper by Kleene & Post.