14,170 research outputs found
On shuffle ideals of general algebras
We extend a word language concept called shuffle ideal to general algebras. For this purpose, we introduce the relation SH and show that there exists a natural connection between this relation and the homeomorphic embedding order on trees. We establish connections between shuffle ideals, monotonically ordered algebras and automata, and piecewise testable tree languages
Quillen homology for operads via Gr\"obner bases
The main goal of this paper is to present a way to compute Quillen homology
of operads. The key idea is to use the notion of a shuffle operad we introduced
earlier; this allows to compute, for a symmetric operad, the homology classes
and the shape of the differential in its minimal model, although does not give
an insight on the symmetric groups action on the homology. Our approach goes in
several steps. First, we regard our symmetric operad as a shuffle operad, which
allows to compute its Gr\"obner basis. Next, we define a combinatorial
resolution for the "monomial replacement" of each shuffle operad (provided by
the Gr\"obner bases theory). Finally, we explain how to "deform" the
differential to handle every operad with a Gr\"obner basis, and find explicit
representatives of Quillen homology classes for a large class of operads. We
also present various applications, including a new proof of Hoffbeck's PBW
criterion, a proof of Koszulness for a class of operads coming from commutative
algebras, and a homology computation for the operads of Batalin-Vilkovisky
algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint
arXiv:0912.4895. Final version, to appear in Documenta Mat
AND Protocols Using Only Uniform Shuffles
Secure multi-party computation using a deck of playing cards has been a
subject of research since the "five-card trick" introduced by den Boer in 1989.
One of the main problems in card-based cryptography is to design
committed-format protocols to compute a Boolean AND operation subject to
different runtime and shuffle restrictions by using as few cards as possible.
In this paper, we introduce two AND protocols that use only uniform shuffles.
The first one requires four cards and is a restart-free Las Vegas protocol with
finite expected runtime. The second one requires five cards and always
terminates in finite time.Comment: This paper has appeared at CSR 201
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