95,393 research outputs found
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
All the shapes of spaces: a census of small 3-manifolds
In this work we present a complete (no misses, no duplicates) census for
closed, connected, orientable and prime 3-manifolds induced by plane graphs
with a bipartition of its edge set (blinks) up to edges. Blinks form a
universal encoding for such manifolds. In fact, each such a manifold is a
subtle class of blinks, \cite{lins2013B}. Blinks are in 1-1 correpondence with
{\em blackboard framed links}, \cite {kauffman1991knots, kauffman1994tlr} We
hope that this census becomes as useful for the study of concrete examples of
3-manifolds as the tables of knots are in the study of knots and links.Comment: 31 pages, 17 figures, 38 references. In this version we introduce
some new material concerning composite manifold
Using ACL2 to Verify Loop Pipelining in Behavioral Synthesis
Behavioral synthesis involves compiling an Electronic System-Level (ESL)
design into its Register-Transfer Level (RTL) implementation. Loop pipelining
is one of the most critical and complex transformations employed in behavioral
synthesis. Certifying the loop pipelining algorithm is challenging because
there is a huge semantic gap between the input sequential design and the output
pipelined implementation making it infeasible to verify their equivalence with
automated sequential equivalence checking techniques. We discuss our ongoing
effort using ACL2 to certify loop pipelining transformation. The completion of
the proof is work in progress. However, some of the insights developed so far
may already be of value to the ACL2 community. In particular, we discuss the
key invariant we formalized, which is very different from that used in most
pipeline proofs. We discuss the needs for this invariant, its formalization in
ACL2, and our envisioned proof using the invariant. We also discuss some
trade-offs, challenges, and insights developed in course of the project.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Quantum graphs whose spectra mimic the zeros of the Riemann zeta function
One of the most famous problems in mathematics is the Riemann hypothesis:
that the non-trivial zeros of the Riemann zeta function lie on a line in the
complex plane. One way to prove the hypothesis would be to identify the zeros
as eigenvalues of a Hermitian operator, many of whose properties can be derived
through the analogy to quantum chaos. Using this, we construct a set of quantum
graphs that have the same oscillating part of the density of states as the
Riemann zeros, offering an explanation of the overall minus sign. The smooth
part is completely different, and hence also the spectrum, but the graphs pick
out the low-lying zeros.Comment: 8 pages, 8 pdf figure
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