2,150 research outputs found
An Alternative View of the Graph-Induced Multilinear Maps
In this paper, we view multilinear maps through the lens of ``homomorphic obfuscation . In specific, we show how to homomorphically obfuscate the kernel-test and affine subspace-test functionalities of high dimensional matrices. Namely, the evaluator is able to perform additions and multiplications over the obfuscated matrices, and test subspace memberships on the resulting code. The homomorphic operations are constrained by the prescribed data structure, e.g. a tree or a graph, where the matrices are stored. The security properties of all the constructions are based on the hardness of Learning with errors problem (LWE). The technical heart is to ``control the ``chain reactions\u27\u27 over a sequence of LWE instances.
Viewing the homomorphic obfuscation scheme from a different angle, it coincides with the graph-induced multilinear maps proposed by Gentry, Gorbunov and Halevi (GGH15). Our proof technique recognizes several ``safe modes of GGH15 that are not known before, including a simple special case: if the graph is acyclic and the matrices are sampled independently from binary or error distributions, then the encodings of the matrices are pseudorandom
Quilted strips, graph associahedra, and A-infinity n-modules
We consider moduli spaces of quilted strips with markings and their
compactifications. Using the theory of moment maps of toric varieties we
identify the compactified moduli spaces with certain graph associahedra. We
demonstrate how these moduli spaces govern the combinatorics of A-infinity
n-modules, which are natural generalizations of A-infinity modules (n=1) and
bimodules (n=2).Comment: 15 pages, 10 figure
Combinatorics and formal geometry of the master equation
We give a general treatment of the master equation in homotopy algebras and
describe the operads and formal differential geometric objects governing the
corresponding algebraic structures. We show that the notion of Maurer-Cartan
twisting is encoded in certain automorphisms of these universal objects
L-infinity maps and twistings
We give a construction of an L-infinity map from any L-infinity algebra into
its truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity
analogues. This map fits with the inclusion into the full Chevalley-Eilenberg
complex (or its respective analogues) to form a homotopy fiber sequence of
L-infinity-algebras. Application to deformation theory and graph homology are
given. We employ the machinery of Maurer-Cartan functors in L-infinity and
A-infinity algebras and associated twistings which should be of independent
interest.Comment: 16 pages, to appear in Homology, Homotopy and Applications. This
version contains many corrections of technical nature and minor improvement
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