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Boaz Barak works on computational complexity theory, cryptography and the mathematical foundations of machine learning. His research homepage describes these interests and identifies his textbooks and lecture notes.^[ref] His Introduction to Theoretical Computer Science by Boaz Barak develops computation through explicit models and resource bounds.

One example is pseudorandomness: a generated distribution can look random to efficient observers even though an unrestricted observer distinguishes it. Specifying the observer turns “looks random” into a mathematical guarantee, and connects a pseudorandom generator with the randomized algorithms it can safely simulate.^[ref]

Two indexed families of probability distributions $X_n$ and $Y_n$ are computationally indistinguishable when every allowed efficient test has only negligible distinguishing advantage:Here negligible means eventually smaller than $n^{-k}$ for every fixed positive $k$. The probabilities include the sampled input and the test's randomness; the model must specify whether tests are uniform algorithms or nonuniform Boolean circuits.

The claim limits an observer's power. Two distributions can be far apart for an unrestricted observer yet indistinguishable to polynomial-time observers. This is the security goal for a pseudorandom generator: its output should look uniform even though a short seed reaches only a small fraction of all possible outputs.^[ref]

An oracle Turing machine is a Turing machine with an additional query operation for a fixed formal language $A$. It writes a finite query $x$ on a designated tape and receives the membership answer $x\in A$ in one oracle step. Writing the query still takes ordinary machine work. The oracle's internal solution procedure is excluded from the cost model.

For example, a machine with an oracle for Boolean satisfiability can ask whether a given Boolean formula has a satisfying assignment. By fixing variables one at a time and querying satisfiability again, it can recover an assignment with at most one additional query per variable. The example turns decision access into search access; it does not provide an ordinary fast satisfiability solver. Oracles let computability theory and computational complexity theory compare problems while making the granted information explicit.^[ref]

A pseudorandom generator is a deterministic, efficiently computable function $G:\{0,1{\}}^n\to \{0,1{\}}^{m(n)}$ with $m(n)>n$, whose output on a uniform seed is computationally indistinguishable from a uniform $m(n)$-bit string. Its security is relative to a specified class of tests.

For example, suppose a generator maps 128 seed bits to 256 output bits. Its range contains at most $2^{128}$ strings out of $2^{256}$ possibilities. Enumerating the range would distinguish it almost perfectly, but requires infeasible work at this size. A secure construction must resist efficient tests that avoid enumeration as well. Merely copying the seed twice fails immediately: a test compares the two halves. Useful generators enable randomized algorithms or cryptographic protocols to expand a limited supply of random seed material.^[ref]

Pseudorandomness means that specified tests cannot effectively distinguish a generated distribution from a genuinely random reference distribution. The allowed tests and their advantage are part of the definition. Passing a few frequency tests does not imply security against every feasible computation.

A pseudorandom generator expands a uniformly chosen $s$-bit seed to $m>s$ bits. For a test $D$, its distinguishing advantage isHere $U_k$ denotes a uniformly sampled $k$-bit string. There are at most $2^s$ generated strings, so an unrestricted test for membership in the generator's range distinguishes with advantage at least $1-2^{s-m}$. The useful claim is therefore computational indistinguishability for a restricted test class, not equality of the two distributions. Such a generator can replace random choices in an algorithm when the relevant computation is among the tests it fools.^[ref]