Hypergraph expanders from Cayley graphs

We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.Comment: 13 page

Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs

A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood

Proof of Space from Stacked Expanders

Recently, proof of space (PoS) has been suggested as a more egalitarian alternative to the traditional hash-based proof of work. In PoS, a prover proves to a verifier that it has dedicated some specified amount of space. A closely related notion is memory-hard functions (MHF), functions that require a lot of memory/space to compute. While making promising progress, existing PoS and MHF have several problems. First, there are large gaps between the desired space-hardness and what can be proven. Second, it has been pointed out that PoS and MHF should require a lot of space not just at some point, but throughout the entire computation/protocol; few proposals considered this issue. Third, the two existing PoS constructions are both based on a class of graphs called superconcentrators, which are either hard to construct or add a logarithmic factor overhead to efficiency. In this paper, we construct PoS from stacked expander graphs. Our constructions are simpler, more efficient and have tighter provable space-hardness than prior works. Our results also apply to a recent MHF called Balloon hash. We show Balloon hash has tighter space-hardness than previously believed and consistent space-hardness throughout its computation