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103 lines
3.7 KiB
Markdown
103 lines
3.7 KiB
Markdown
---
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title: "Cryptographic Games"
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date: 2020-01-13T21:35:09-05:00
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draft: false
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math: true
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tags: ["Security"]
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medium_enabled: true
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---
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When analyzing cryptographic algorithms, we characterize the strength of the crypto-system by analyzing what happens in various crypto games. Below are a couple examples of crypto games used in literature.
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Actors typically involved in crypto games:
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| Actor | Role |
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| ---------- | ---------------------------------------------------------- |
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| Oracle | Encrypts a given message. |
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| Challenger | Sets up the game between the oracle and adversary. |
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| Adversary | Sends messages to the oracle and makes a guess at the end. |
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## Left-Right Game
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Here we will have two oracle's `left` and `right` both with a random key $k$.
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The challenger will create a random bit $b$ which is either $0$ or $1$.
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If the bit is $0$, then the adversary will send messages to `left`. Otherwise, the adversary will send messages to `right`.
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After a certain number of interactions the adversary needs to guess which oracle it is talking to.
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We define the adversary's advantage by the distance the probability is away from $\frac{1}{2}$.
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$$
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advantage = |p - \frac{1}{2}|
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$$
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Why the absolute value? Well if we only guess correctly $10\%$ of the time, then we just need to invert our guess to be correct $90\%$ of the time.
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```
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Generate key for both oracles
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Generate random bit b
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while game not done:
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advesary generates message m
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if b = 0:
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send m to left oracle
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receive c from left oracle
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else:
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send m to right oracle
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receive c from right oracle
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Adversary guesses whether b = 0 or 1
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```
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## Real-Random Game
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This game is very similar to the last one except one of the oracle's only produces random bitstrings.
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An oracle is initialized with a random key $k$ and the challenger creates a random bit $b$.
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If $b = 0$, then the adversary will send messages to the oracle with the proper encryption function and will receive ciphertexts. Otherwise, the adversary will send messages to the random bitstring generator.
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At the end of the game, the adversary needs to guess whether its talking to a proper oracle or a random bitstring generator.
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The same metric `adversary's advantage` is used to characterize the strength of a crypto-system as the last one.
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```
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Generate key for both oracles
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Generate random bit b
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while game not done:
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advesary generates message m
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if b = 0:
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send m to proper oracle
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receive c from proper oracle
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else:
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send m to random oracle
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receive random bitstring c from random oracle
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Adversary guesses whether b = 0 or 1
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```
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## Random Function - Random Permutation Trick
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One trick that we can use in proofs is to say that two procedures are the same up to a certain point when it diverges.
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Let us define $f$ to be some function that given a bitstring $X$ produces a random bitstring $Y$.
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Now, $image(f)$ is initially undefined since we don't know which bitstrings will be randomly generated.
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However, as we calculate $Y$ we will store it to build up $image(f)$.
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At some point, we'll generate $Y$ that already exists in $image(f)$. In that moment we will split based off of whether the procedure is a random function or random permutation.
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If random function, then just return the same $Y$ and end the procedure. Otherwise, return a random bitstring outside of the $range(f)$ and end the procedure.
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```
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Initialize function f with range(f) empty
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Generate different random bitstring X
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While f(X) not in range(f):
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Add f(X) to range(f)
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Generate different random bitstring X
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If random function:
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return f(X)
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Else:
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return random bitstring Y not in range(f)
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```
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