Using Hard Problems to Create Pseudorandom Generators (ACM Doctoral Dissertation Award) by Noam Nisan (PDF)

Description

Randomization is an important tool in the design of algorithms, and the ability of randomization to provide enhanced power is a major research topic in complexity theory. Noam Nisan continues the investigation into the power of randomization and the relationships between randomized and deterministic complexity classes by pursuing the idea of emulating randomness, or pseudorandom generation. Pseudorandom generators reduce the number of random bits required by randomized algorithms, enable the construction of certain cryptographic protocols, and shed light on the difficulty of simulating randomized algorithms by deterministic ones. The research described here deals with two methods of constructing pseudorandom generators from hard problems and demonstrates some surprising connections between pseudorandom generators and seemingly unrelated topics such as multiparty communication complexity and random oracles. Nisan first establishes a precise connection between computational complexity and pseudorandom number generation, revealing that efficient deterministic simulation of randomized algorithms is possible under much weaker assumptions than was previously known, and bringing to light new consequences concerning the power of random oracles. Using a remarkable argument based on multiparty communication complexity, Nisan then constructs a generator that is good against all tests computable in logarithmic space. A consequence of this result is a new construction of universal traversal sequences.ContentsIntroduction • Hardness vs. Randomness • Pseudorandom Generators for Logspace and Multiparty Protocols

User’s Reviews

Editorial Reviews: About the Author Noam Nisan is Dean of the School of Computer Science and Engineering at the Hebrew University of Jerusalem.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is an elaboration of the results of the author’s Phd thesis on the connnection between pseudorandom generators and computational hardness. As such it requires the reader to have a background in this area. The author does not attempt to define the notation and concepts needed for the results outlined. Randomization can of course be very useful in the design of efficient algorithms, and are often simpler for a given problem. Monte Carlo simulation is an example of this, and is used extensively in physics, chemistry, biology, and finance. In primality testing and some counting problems, the only algorithms known are randomized ones. Some problems, like the approximate computation of the volume of a convex body, can only be solved by a randomized algorithm. In addition, the simulation of a randomized algorithm can be a powerful tool: the computing of the convex hull of n-points in d-dimensional space is obtained by simulating a randomized algorithm. In linear programming, randomized algorithms overtake the best deterministic algorithms by a factor exponential in the dimension d. In spite of the fact that there are no explicit concrete examples in the book, it is interesting reading, and the author outlines some ideas on how to efficiently simulate (deterministically) random algorithms using assumptions that are not as strong as ones known at that time. The book could be viewed as an attempt to answer whether or not randomization is really of any assistance in making computations efficient, i.e. whether P = BPP. There has been a considerable amount of work done on resolving this question since this thesis was written. It has been shown that if there exists a sparse “efficiently enumerable” language of sufficiently high circuit complexity, then P = BPP. In addition, it has been shown that either all the decision problems solvable in time 2^O(n) are solvable by circuits of size 2^O(n) or B = BPP. The author also resolves the open question as to whether NP with a random oracle is identical to the class of Arthur-Merlin (AM) languages accepted by some AM machine that runs in polynommial time. In addition, he also describes the construction of a pseudorandom generator for Logspace using multiparty communication games. Although the book is somewhat dated, it still serves as a good introduction to an important area, and is worth perusing as preparation for further research.

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