Expanding graphs, Ramanujan graphs, and 1-factor perturbations

We construct (k+-1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of k-regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with k-1 the order of a finite field). If k+1 = 7, our construction results in a sequence of 7-regular expanders with all spectral gaps at least about 1.52

Improved Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: (i) the codes lie close to the Singleton bound, (ii) they can be encoded in time complexity that is linear in their code length, and (iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace MDS outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size.Comment: Part of this work was presented at the 2004 IEEE Int'l Symposium on Information Theory (ISIT'2004), Chicago, Illinois (June 2004). This work was submitted to IEEE Transactions on Information Theory on January 21, 2005. To appear in IEEE Transactions on Information Theory, August 2006. 12 page

Hypergraph expanders of all uniformities from Cayley graphs

Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.Comment: 32 page

The Relativized Second Eigenvalue Conjecture of Alon

We prove a relativization of the Alon Second Eigenvalue Conjecture for all $d$-regular base graphs, $B$, with $d\ge 3$: for any $\epsilon>0$, we show that a random covering map of degree $n$ to $B$ has a new eigenvalue greater than $2\sqrt{d-1}+\epsilon$ in absolute value with probability $O(1/n)$. Furthermore, if $B$ is a Ramanujan graph, we show that this probability is proportional to $n^{-{\eta_{\rm \,fund}}(B)}$, where ${\eta_{\rm \,fund}}(B)$ is an integer depending on $B$, which can be computed by a finite algorithm for any fixed $B$. For any $d$-regular graph, $B$, ${\eta_{\rm \,fund}}(B)$ is greater than $\sqrt{d-1}$. Our proof introduces a number of ideas that simplify and strengthen the methods of Friedman's proof of the original conjecture of Alon. The most significant new idea is that of a ``certified trace,'' which is not only greatly simplifies our trace methods, but is the reason we can obtain the $n^{-{\eta_{\rm \,fund}}(B)}$ estimate above. This estimate represents an improvement over Friedman's results of the original Alon conjecture for random $d$-regular graphs, for certain values of $d$

The Zeta Function of a Hypergraph

We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Sol\'e. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.Comment: 24 pages, 6 figure

Ramanujan Coverings of Graphs

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013). Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the $r$-th matching polynomial of $G$ to be the average matching polynomial of all $r$-coverings of $G$. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$.Comment: 38 pages, 4 figures, journal version (minor changes from previous arXiv version). Shortened version appeared in STOC 201

Ramanujan graphs in cryptography

In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Pl\^{u}t proposed a cryptographic system based on Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.Comment: 33 page

Deterministic Constructions for Large Girth Protograph LDPC Codes

The bit-error threshold of the standard ensemble of Low Density Parity Check (LDPC) codes is known to be close to capacity, if there is a non-zero fraction of degree-two bit nodes. However, the degree-two bit nodes preclude the possibility of a block-error threshold. Interestingly, LDPC codes constructed using protographs allow the possibility of having both degree-two bit nodes and a block-error threshold. In this paper, we analyze density evolution for protograph LDPC codes over the binary erasure channel and show that their bit-error probability decreases double exponentially with the number of iterations when the erasure probability is below the bit-error threshold and long chain of degree-two variable nodes are avoided in the protograph. We present deterministic constructions of such protograph LDPC codes with girth logarithmic in blocklength, resulting in an exponential fall in bit-error probability below the threshold. We provide optimized protographs, whose block-error thresholds are better than that of the standard ensemble with minimum bit-node degree three. These protograph LDPC codes are theoretically of great interest, and have applications, for instance, in coding with strong secrecy over wiretap channels.Comment: 5 pages, 2 figures; To appear in ISIT 2013; Minor changes in presentatio