In Provably Secure Steganography by Hopper, et al, we have the following definition

Cryptographic notions

Let $F:\{0,1\}^k \times \{0,1\}^L \rightarrow \{0,1\}^l$ denote a family of functions. Let $A$ be an oracle probabilistic adversary:

Define the $prf$-advantage of $A$ over $F$ as

$$Adv^{prf}_F(A)= | Pr_{K \leftarrow U(k), r\leftarrow \{0,1\}^*}[A_r^{F_k(.)}=1]- Pr_{g \leftarrow U(L,l), r\leftarrow \{0,1\}^*}[A_r^{g}=1] |.$$

where $r$ is the string of random bits used by adversary $A$. Define the insecurity of $F$ as

$$InSec^{prf}_{F}(t,q) = max_{A\in \mathcal{A}(t,q)} \{ Adv_F^{prf}(A)\}$$

where $U(k)$ is a uniform distribution over $k$ bits, $U(L,l)$ is a uniform distribution on functions from $\{0,1\}^L$ to $\{0,1\}^l$, and $F_K$ is a function $F_K:\{0,1\}^k\times\{0,1\}^L\to\{0,1\}^\ell$. But honestly, I don't know how I'm supposed to understand or parse the definition. Intuitively, I get the adversary $A$ is outputting $1$ if it decides that $F_K$ is not sufficiently different from a sample $r$ drawn from the channel distribution. But can someone walk me through, like I'm an idiot, as to what role each part of the definition plays in the overall mechanism of prf-advantage?

• It would be more common to see $\: A^{F_k}\hspace{-0.03 in}(r) \:$ and $\: A^g\hspace{-0.02 in}(r) \:$ than $\: A_r^{F_k(.)} \:$ and $\: A_r^g \;$. $\hspace{1.49 in}$