We derive the optimality results for key pre distribution scheme for distributed sensor networks, and relations between interesting parameters. Namely, given a key-pool of size n we derive the optimal value that is jointly achievable for parameters like, Size optimality: using less memory per node while supporting large network, Connectivity optimality: possibility of establishing secure commun...

We use expander graphs to provide eÆcient new constructions for two security applications: authentication of long digital streams over lossy networks and building scalable, r obust overlay networks.Here is a summary of our contributions: (1) To authenticate long digital streams over lossy networks, we provide a construction with a provable lower bound on the ability to authenticate a packet | a...

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π :H →G. It is not hard to see that all eigenvalues of G are also eigenvalues of H . In addition, H has n “new” eigenvalues. We conjecture that every...

We construct a randomness-efficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC0[⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC. For example, we obtain the following results: • Randomness-efficient error-reduct...

Last time, we described the properties of expander graphs and showed that they have several “extremal” properties. Before that, we described a vanilla spectral partitioning algorithms, which led to the statement and proof of Cheeger’s Inequality. Recall that one direction viewed λ2 as a relaxation of the conductance or expansion problem; while the other direction gave a “quadratic” bound as wel...

Let G = (V,E) be an undirected d-regular graph, here, |V | = n, deg(u) = d for all u ∈ V . We will typically interpret the properties of expander graphs in an asymptotic sense. That is, there will be an infinite family of graphs G, with a growing number of vertices n. By “sparse”, we mean that the degree d of G should be very slowly growing as a function of n. When n goes to infinity (n → ∞), d...

Low-density parity-check (LDPC) codes were introduced in 1962, but were almost forgotten. The introduction of turbo-codes in 1993 was a real breakthrough in communication theory and practice, due to their practical effectiveness. Subsequently, the connections between LDPC and turbo codes were considered, and it was shown that the latter can be described in the framework of LDPC codes. In recent...

We derive several upper bounds on the spectral gap of Laplacian with standard or Dirichlet vertex conditions compact metric graphs. In particular, we obtain estimates based length a shortest cycle (girth), diameter, total graph, as well further quantities introduced here for first time, such avoidance diameter. Using known results about Ramanujan graphs, class expander also prove that some thes...

We are especially interested in graphs – or better, in sequences of graphs – whose first eigenvalues λ1 are relatively small. In particular, for any finite graph, define the spectral gap ω(G) = λ0(G) − λ1(G). Define also the isoperimetric constant h(G) to be the infimum #E(V1, V2) min{#V1, #V2} over all partitions of the vertex set into two subsets V1, V2; here E(V1, V2) is the set of edges con...

2 The Construction Fix n = m2 for a natural m and let An =Zm Zm,Zm being the group of integers modulo m. An may be thought of a combinatorial torus. Consider the following 5 bijections on An: 1. σ0 : (x;y) 7! (x;y), 2. σ1 : (x;y) 7! (x;x+ y), 3. σ2 : (x;y) 7! (x;x+ y+1), 4. σ3 : (x;y) 7! (x+ y;y), and 5. σ4 : (x;y) 7! (x+ y+1;y), addition modulo m. Now define Gn = (Un;Vn; En) as follows: Un = V...