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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?

BlockFi CEO Responds To Rumors They Sold For $25 Million

Rumors of a BlockFi sale for a paltry $25 million are swirling around the internet. CEO Zac Prince responded to the claim… FTX closes in on a deal to buy embattled crypto lender BlockFi for $25 million in a fire sale https://t.co/9uRESw9KCt — CNBC (@CNBC) June 30, 2022 Kate Rooney of CNBC reported earlier today::Listen

Rumors of a BlockFi sale for a paltry $25 million are swirling around the internet. CEO Zac Prince responded to the claim…

Kate Rooney of CNBC reported earlier today that FTX is buying BlockFi for the super low price of $25 million USD. According to Rooney, three sources familiar with the matter — who wished to remain unnamed — divulged the details of the deal.

Recommended: Grayscale Sues The SEC

  • The term sheet is not finalized but it’s set to finalize by the end of the week. Rooney didn’t clarify whether a BlockFi sale should be expected by Friday or Sunday.
  • Both BlockFi and FTX declined to comment. However, CEO of BlockFI Zac Prince responded to the rumors via Twitter: Lots of market rumors out there – I can 100% confirm that we aren’t being sold for $25M. I encourage everyone to trust only details that you hear directly from @BlockFi. We will share more w you as soon as we can.”
  • The obscenely low number being reported is quite different than the last valuation of BlockFi, which valued the crypto savings and lender at 4.8 billion.
  • We previously reported on BlockFi and FTX and how early investor Morgan Creek Digital was trying to raise enough money to beat an offer by FTX. An FTX purchase would possibly leave Morgan Creek with nothing.
  • Reportedly, the $250 million dollar credit line extended to BlockFi doesn’t hamper its ability to sell to the highest bidder. BlockFi is apparently entertaining several offers.

 

The post BlockFi CEO Responds To Rumors They Sold For $25 Million appeared first on CryptosRus.

Probability conventions in cryptography

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