CS 225: Pseudorandomness
Spring 2007 


Description | Topics | Prerequisites | Grading | Readings | Related Courses

Course meetings:  Tue-Thu 1-2:30, Maxwell Dworkin 135 (33 Oxford Street)

Lecturer: Salil Vadhan
Office: Maxwell-Dworkin 337
Shopping period office hours: Thu 2/1 2:30-3, Fri 2/2 1-2, Mon 2/5 4-5, Tue 2/6 2:30-4, Wed 2/7 1:30-2.

Teaching Fellows: Kai-Min Chung and Shien Jin Ong
Office: Maxwell-Dworkin 138

E-mail address for questions: cs225@eecs.harvard.edu
E-mail address for submitting homeworks: cs225-hw@eecs.harvard.edu
Course website: http://www.courses.fas.harvard.edu/4869

Course Description

Over the past few decades, randomization has become one of the most pervasive paradigms in computer science.  Its widespread use includes:

  • Algorithm Design: For a number of important algorithmic problems (including problems in algebra, statistical physics, and approximate counting), the only efficient algorithms known are randomized.
  • Cryptography: Randomness is woven into the very way we define security.
  • Combinatorial Constructions: Many useful combinatorial objects, such as error-correcting codes and expander graphs (see below), can be constructed simply by generating them at random.
  • Interactive Proofs: Randomization, together with interactive communication,  can also add dramatic efficiency improvements and novel properties (such as "zero knowledge") to classical "written" mathematical proofs.

So randomness appears to be extremely useful in these settings, but we still do not know to what extent it is really necessary.  Thus, in this course we will ask:

Main Question: Can we reduce or even eliminate the need for randomness in the above settings?

Why do we want to do this?    First, essentially all of the applications of randomness assume we have a source of perfect randomness ≠ one that gives "coin tosses" that are completely unbiased and independent of each other.  It is unclear whether physical sources of perfect randomness exist and are inexpensive to access.  Second, randomized constructions of objects such as error-correcting codes and expander graphs often do not provide us with efficient algorithms for using them; indeed, even writing down a description of a randomly selected object can be infeasible.  Finally, and most fundamentally, our understanding of computation would be incomplete without understanding the power that randomness provides.

In this course, we will address the Main Question via a powerful paradigm known as pseudorandomnessThis is the theory of efficiently generating objects that "look random", despite being constructed using little or no randomness.  Specifically, we will study several kinds of "pseudorandom" objects, such as:

  • Pseudorandom Generators: These are procedures which stretch a short "seed" of truly random bits into a long string of "pseudorandom" bits which cannot be distinguished from truly random by any efficient algorithm.  They can be used to reduce and even eliminate the randomness used by any efficient algorithm.  They are also a fundamental tool in cryptography.
  • Randomness Extractors: These are procedures which extract almost uniformly distributed bits from sources of biased and correlated bits.  Their original motivation was to allow us to use randomized algorithms even with imperfect physical sources of randomness, but they have also turned out to have a wide variety of other applications.
  • Expander Graphs:  These are graphs which are sparse but nevertheless highly connected.  They have been used to address many fundamental problems in computer science, on topics such as network design, complexity theory, coding theory, cryptography, and computational group theory.
  • Error-Correcting Codes: These are methods for encoding messages so that even if many of the symbols are corrupted, the original message can still be decoded.  We will focus on "list decoding", where there are so many corruptions that uniquely decoding the original message is impossible, but it is still possible to produce a short list of possible candidates.

Each of the above objects has been the center of a large and beautiful body of research, and until recently these corpora were largely distinct. An exciting recent development has been the realization that all four of these objects are almost the same when interpreted appropriately.  Their intimate connections will be a major focus of the course, tying together the variety of constructions and applications of these objects we will cover.

The course will reach the cutting-edge of current research in this area, covering some results from within the last year.  At the same time, the concepts we will cover are general and useful enough that hopefully anyone with an interest in the theory of computation or combinatorics could find the material appealing.

Tentative Outline

  • The Power of Randomness
    • Examples of randomized algorithms
    • Complexity classes (BPP, RP, RL,...)
    • Basic properties, e.g. error reduction
  • Basic Derandomization Techniques
    • Enumeration
    • Nonuniformity
    • Nondeterminism
    • Method of conditional probabilities
    • Pairwise and k-wise independence
  • Expander Graphs
    • Measures of expansion
    • Probabilistic existence
    • Random walks on expanders
    • Explicit constructions: the zig-zag product
    • Reingoldís logspace algorithm for undirected s-t connectivity (2005)
  • Randomness Extractors
    • Weak random sources, entropy measures, impossibility of deterministic extraction
    • Probabilistic existence
    • Simulating BPP with a weak random source
    • The Leftover Hash Lemma
    • Relation to expanders
    • Extraction from block sources
  • List-Decodable Error-Correcting Codes
    • Probabilistic existence
    • The Johnson Bound
    • Reed-Solomon, Reed-Muller, and Hadamard Codes
    • Guruswami-Sudan decoding algorithm
    • Parvaresh-Vardy and Guruswami-Rudra codes
    • Relation to extractors
    • Expanders and extractors from Parvaresh-Vardy codes (2006)
  • Pseudorandom Generators
    • Blum-Micali-Yao and Nisan-Wigderson definitions
    • Survey of BMY-type pseudorandom generators from one-way functions
    • The Nisan-Wigderson generator (from average-case hardness)
    • Unconditional derandomization of constant-depth circuits
    • Worst-case/average-case connections from locally list-decodable codes
    • Evidence that BPP=P and AM=NP
    • Relation to extractors: Trevisanís extractor
    • Applications
  • Potential Additional Topics
    • Extractors and pseudorandom generators from Reed-Muller codes (2001)
    • Pseudorandom generators from space-bounded computation
    • Deterministic extractors: bit-fixing sources, multiple sources, Ramsey graphs
    • Are circuit lower bounds necessary for derandomization?
    • Small bias spaces, Cayley expanders, and Ramanujan graphs
    • Hardness amplification: Yaoís XOR lemma and generalizations


This is an advanced graduate course, so I will be assuming that you have general "mathematical maturity" and a good undergraduate background in the theory of computation.  One concrete guideline is that you should have had a minimum of two other courses in the theory of computation, including at least one graduate course.  If you have particularly strong math background, then there can be a bit more flexibility with this.  But if you haven't had a prior graduate course in the theory of computation (numbered CS 22x at Harvard), you must come speak to me at office hours before registering for the class.

In terms of topics, I will be assuming familiarity with the following.  In all cases (especially complexity theory), the more background you have, the better.

  • Complexity Theory:  P, NP, NP-completeness, reductions (as in CS 121).
  • Randomized Algorithms: You should have seen several examples of randomized algorithms, as in CS 124, 223, or 224.
  • Algebra:  The basics of groups, (finite) fields, vector spaces, eigenvectors/eigenvalues.  Any of CS 224, Math 122-123, AM 106 should be sufficient.
  • Other: Basic discrete probability, graph theory & combinatorics.

Grading & Problem Sets

The requirements of the course:

  • Biweekly problem sets.
  • Take-home final exam.
  • Class participation (including providing comments/corrections to the lecture notes).

The biweekly problem sets will typically be due on Wednesday by 1 PM.  Your problem set solutions must be typed and submitted electronically to cs225-hw@eecs.harvard.edu. You are allowed 12 late days for the semester, of which at most 7 can be used on any individual problem set. (1 late day = 24 hours exactly).

The problem sets will be challenging, so be sure to start them early.  You are encouraged to discuss the course material and the homework problems with each other in small groups (2-3 people).   Discussion of homework problems may include brainstorming and verbally walking through possible solutions, but should not include one person telling the others how to solve the problem.  In addition, each person must write up their solutions independently, and these write-ups should not be checked against each other or passed around.


There is no textbook for the course, but I will be handing out lecture notes.The scribe notes from Spring `04 are here.You may also find the following references useful.  Most of them should be in the libraries, on reserve.

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