Concrete constructions of unbalanced bipartite expander graphs and generalized conductors

Abstract

Bipartite expander graphs are bipartite graphs for which the left vertex set has a guaranteed expansion parameter to the right vertex set, or more formally, a bipartite graph G = (V1,V2, E) has expansion parameter γ if for every set X ⊂ V1 (with bounded size) with its neighbors Γ(X ) ⊂ V2, we have |Γ(X )| ≥ γ · |X |. We are interested in the unbalanced case, where the set V1 is much larger than V2. If the neighbors of each vertex in V1 can be e ciently computed, the expander graph is additionally called explicit. In this thesis, we introduce a generalized notion of conductors, which are functions of the form {0, 1}n × {0, 1}d → {0, 1}m which take as rst input a string with certain min-entropy and as second input some few truly random bits, and provide some entropy guarantees on the output. These generalized conductors are usually interpreted as explicit unbalanced bipartite expander graphs with stronger properties. In particular, we provide strong composition theorems for such conductors which allow us to obtain explicit unbalanced bipartite expander graphs with good expansion parameters and su ciently small left-degree, and to study the concrete values of these parameters. Explicit unbalanced bipartite expander graphs with good expansion parameters can be used in cryptographic schemes (like the domain extender of public random functions due to Maurer and Tessaro). In particular, a small left-degree of the expander graph is crucial for the e ciency of the protocols using such expander graphs. Therefore, we focus on nding a construction of an explicit expander graph with small left-degree: We show non-constructively that such expander graphs (as well as other types of conductors) with small left-degree and good expansion must exist and try to nd an explicit construction of such a good expander graph. In particular, we analyze an expander graph construction which leads to a small leftdegree of the expander graph in complexity-theoretic terms and we investigate the concrete value of this left-degree. We show that even though the left-degree is polynomial, the actual degree of the polynomial makes it not feasible to use the construction in practice. This give us the motivation to analyze an unbalanced bipartite expander graph construction which is based on selecting (according to some rule) substrings of length n as the neighbors of a string which has length multiple of n: Although this construction has a small left-degree and promises a good expansion of the left vertices, we show that it is impossible to construct an expander graph with good expansion on the basis of substring selection.