CS 225 Syllabus Spring 2015

CS 225: Pseudorandomness
Spring 2015 


Course Content | Topics | Format and Goals | Prerequisites | Grading | Textbook | Related Courses

Course meetings:  Tue-Thu 10-11:30, Maxwell-Dworkin 319 (33 Oxford Street).

Lecturer: Salil Vadhan
Office: Maxwell-Dworkin 337
Shopping week office hours: 1/26 9:30-11:30, 1/28 12:30-2:30, 1/29 12:30-2:30, 4:30-5:30 (all in MD 337) and 1/30 1-3 (at AM study card party in Pierce 307b)

Teaching Fellow: Thomas Steinke
Office: Maxwell-Dworkin 138
Office Hours/Section: Wednesdays 4pm Pierce 301 or by request.

E-mail address for submitting homeworks: cs225-hw@seas.harvard.edu
Questions should be posted on Piazza.
Course website: http://www.courses.fas.harvard.edu/4869

Course Content

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.


  • 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 (2000)
    • Reingoldís logspace algorithm for undirected s-t connectivity (2005)
  • 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 (2005)
    • List-decoding view of expanders
    • Unbalanced expanders from Parvaresh-Vardy codes (2006)
  • 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 and list-decodable codes.
    • Extraction from block sources
    • Condensers 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
    • Relation to extractors: Trevisanís extractor (1999)
    • Applications

Format and Goals

The main components of the course are as follows:

  • Reading and commenting: For every class meeting, we will assign reading for you to do in advance. You will be expected to read and comment on this material prior by midnight before lecture, using the online forum NB. See the guidelines on reading and commenting.
  • Class participation: Our class meetings will be very interactive, with you collectively bringing out the key concepts, ideas, and intuition, as well as working through the difficult technical material together (with our guidance, of course). This will demand more of you than a standard lecture-based course, but the hope is that you will come away with a much deeper understanding of the material.
  • Problem Sets: There will be 6-7 problems sets (including a take-home exam) during the course. Problem sets will typically be assigned on Thursdays and due back roughly two weeks later on Friday no later than 5pm (unless using late days).

By the end of the course, I hope that you will all be able to:

  • Comfortably work with a variety of "pseudorandom objects" and related topics (e.g. spectral graph theory, entropy measures, the probabilistic method, universal hashing) that are interesting in their own right and useful in many different areas,
  • Fluidly translate between and take advantage of very different ways of viewing the same mathematical and computational objects,
  • Extract both the high-level ideas and low-level details when reading a text and identify interesting questions that are not answered,
  • Explain and collaboratively work through an advanced subject with your peers,
  • Understand the state of the art in the theory of pseudorandomness as needed to engage in research in pseudorandomness and/or apply it to other areas.


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: Exposure to some randomized algorithms, as in CS 124, 223, or 224.
  • Algebra:  The basics of groups, (finite) fields, vector spaces, eigenvectors/eigenvalues.  Any of CS 226, Math 122-123, AM 106 should be sufficient.
  • Other: Basic discrete probability, graph theory & combinatorics.

Grading & Problem Sets

The requirements of the course:

  • Reading and commenting on the reading before lecture.
  • Participation in class, section, office hours, and the online fora (NB and Piazza).
  • Biweekly problem sets.
  • Take-home final exam.


The biweekly problem sets will typically be due on Friday by 5 PM.  Your problem set solutions must be typed and submitted electronically to cs225-hw@seas.harvard.edu. You are allowed 12 late days for the semester, of which at most 5 can be used on any individual problem set. (1 late day = 24 hours exactly).

The problem sets may require a lot of thought, 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.


We will be following my monograph Pseudorandomness. When you post your comments on the reading, please include any corrections or suggestions for improving it!


You may also find the following references useful.  Most of them should be in the libraries, on reserve.

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