Probability

Discrete Problems

Cryptography requires problems that are hard in the average case.

For our purposes, a probability space is a finite set \(\Omega = \{0,1\}^n\), and a function \(Pr:2^\Omega \rightarrow [0,1].\) such that \(Pr[F] = \Sigma_{x\in F} Pr[x]\) for all \(F \subseteq \Omega\).

(See the Wikipedia for the general definition of a probability space. Briefly, a probability space is a set \(S\), a \(\sigma\)-algebra \(X\) on \(S\), and a measure \(P\) on \(X\) such that \(P(S) = 1\).)

Type 1 Random Variables

A Type 1 RV is a function \(X : \Omega \rightarrow S\), for some finite set \(S\).

A predicate is a function \(Q(x):S\rightarrow\{0,1\}\), i.e. it maps the set to a boolean. Define \(Pr[Q(x)] = \Sigma_{y \in \Omega, Q(X(y))}Pr[y]\).

Some more notation:

  • \(X \leftarrow \Omega\) defines a uniform random variable on \(\Omega\) (\(X \leftarrow \{0,1\}^n\)). Formally, \(Pr[A] = |A|/|\Omega|\) for any \(A \subseteq \Omega\), \(X\) is the identity function, thus \(Pr[X=a] = 1/|\Omega|\) for all \(a \in \Omega\).

  • \(A(x,r)\) denotes a randomized algorithm: \(x\) is the input, and \(r \in \{0,1\}^m = \Omega\) is the randomness.

  • \(X \leftarrow A(y)\) denotes a random variable \(X\) as the output of the algorithm \(A\) on input \(y\). Output of algorithm is a probability distribution.

  • For a predicate \(Q\), define \(Pr[Q(x) | x \leftarrow A(y)]\) as the probability that \(Q(x)\) holds for output of \(A\) on input \(y\).

Example: Let \(A(n)\) be a randomized algorithm that outputs an odd prime less than or equal to \(n\). Then

\[Pr[(\frac{X}{p}) = 1 | p \leftarrow A(n), X \leftarrow \mathbb{F}_p^*] = 1/2 \]

Type 2 Random Variables

A Type 2 RV is a function \(X : \Omega \rightarrow \mathbb{R}\).

We shall always use a uniform distribution on \(\Omega\), which means that for all \(r \in \mathbb{R}\),

\[Pr[X=r] = |\{y|X(y) = r\}|/|\Omega|\]

Define

\[ E[X] = \Sigma_{y \in \Omega}Pr[y]X(y) \]

and

\[ \sigma^2(x) = E[X^2] - (E[X])^2 = E[(X - E[X])^2].\]

\(X, Y\) are independent if for all \(a,b\) we have

\[Pr[X=a \wedge Y=b] = Pr[X=a]Pr[Y=b].\]

Large Deviation Bounds

Markoff’s Inequality:

\[Pr[X \gt \lambda] \le E[|x|]/\lambda.\]

Chebyshev’s Inequality:

\[Pr[|X-E[X]|\ge\lambda\sigma(X)]\le 1/\lambda^2.\]

Chernoff Bound: For \(i=1,...,n\), let \(X_i\) be independent random variables variables such that \(Pr[X_i = 1 ] = p\), \(Pr[X_i = 0] = 1-p\), and define \(X = \Sigma^n_{i=1}X_i.\) Then

\[Pr[|X - E[X]| \ge \sqrt{n}\lambda]\le 2e^{-2\lambda^2}.\]

Ben Lynn blynn@cs.stanford.edu 💡