One of the main contribution of his work is a new type of graph product, called the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits its size from the large one, its degree from the small one, and its expansion properties from both. Iteration yields simple explicit constructions of constant-degree expanders of every size, starting from one constant-size expander. Crucial to the intuition and simple analysis of the properties of the zig-zag product is the view of expanders as functions which act as “entropy wave” propagators — they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves. A variant of this product can be applied to extractors, giving the first explicit extractors whose seed length depends logarithmically on only the entropy deficiency of the source and that extract almost all the entropy of high min-entropy sources. These high min-entropy extractors have several interesting applications, including the first constant-degree explicit expanders which beat the “eigenvalue bound.” Vadhan also came up with another simplified approach to the undirected ST-connectivity problem following Reingold's breakthrough result. Also the zig-zag product was useful in Omer Reingold's proof that SL=L.
Zero-knowledge proofs
His work in this area is to use complexity-theoretic methods to understand the power and limitations of zero-knowledge proofs. In a series of papers with Oded Goldreich and Amit Sahai, they gained thorough understanding of the class SZK of problems possessing statistical zero-knowledge proofs, characterized the class SZK and proved that SZK is closed under various operations. Recently his work was trying to work on the zero-knowledge proof beyond the confines of SZK class.
Randomness extractors
With Lu, Omer Reingold, and Avi Wigderson, he gave the first construction of randomness extractors that are “optimal up to constant factors,” reaching a milestone in a decade of work on the subject. With Trevisan, Zuckerman, Kamp, and Rao, he developed a theory of randomness extraction from samplable sources, which are random sources generated by an efficient algorithm.
