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Probability conventions in cryptography

I am working on Victor Shoup’s tutorial on game-based security proof and want to figure out some notions from the perspective of probability theory.

Consider the following PRF advantage defined on Page 11:
$$
bigl|,Pr[s leftarrow S: A^{F_s}() = 1] – Pr[f leftarrow Gamma_{ell_1, ell_2}: A^f() = 1],bigr|
$$

where ${ F_s }_{s in S}$ is a family of keyed functions with key space $S$, domain ${ 0,1 }^{ell_1}$ and range ${ 0,1 }^{ell_2}$, and $Gamma_{ell_1, ell_2}$ denotes the set of all functions from ${ 0,1 }^{ell_1}$ to ${ 0,1 }^{ell_2}$.

I have the following questions:

  1. Does the probability notion $Pr[s leftarrow S: cdot]$ (or, $Pr[cdot | s leftarrow S]$ ) has the same meaning as the notion $Pr_{s leftarrow S}[cdot]$ that is also commonly used in crypto contexts (e.g., this)?
    For me, $Pr[s leftarrow S: cdot]$ and $Pr[cdot | s leftarrow S]$ seem to be two conditional probabilities, and $Pr_{s leftarrow S}[cdot]$ is like a conditional probability measure.

  2. How do we interpret a probability $Pr[s leftarrow S: A^{F_s}() = 1]$?
    Literally, I know that this captures the probability that the distinguisher $A$ outputs $1$ if it is given oracle access to a function $F_s$ keyed by $s in S$, and the probability is taken over the random choice of $s$.
    However, from probability theory, what is the probability space $(Omega, mathcal{F}, Pr)$ where the "event" that "the distinguisher $A$ with oracle access to $F_s$ outputs $1$" is defined? Does the sample space $Omega$ contains all outcomes of $s leftarrow S$ (so that we can say "the probability is taken over the random choice of $s$")?
    If so, does this implies that the two probabilities in the PRF advantage comes from two different probability spaces?

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#Gijs Gulik #TheSupremeTeam
Artist Name: Gijs Gulik

Who are you?

Im a beginner dj, i like to improve myself again en again

Where are you from?

Im from the netherlands, most of the people here listen to adm techno and house

How can we follow you?

https://soundcloud.app.goo.gl/H5HBvZWSDAJn9s776

Song Title: Techhouse mix (Gijs Gulik)

Listen to Gijs Gulik:

https://soundcloud.app.goo.gl/H5HBvZWSDAJn9s776

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Probability conventions in cryptography

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