10,889 research outputs found
Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2
In this paper various representations of the exceptional Lie algebra G_2 are
investigated in a purely algebraic manner, and multi-boson/multi-fermion
realizations are obtained. Matrix elements of the master representation, which
is defined on the space of the universal enveloping algebra of G_2, are
explicitly determined. From this master representation, different
indecomposable representations defined on invariant subspaces or quotient
spaces with respect to these invariant subspaces are discussed. Especially, the
elementary representations of G_2 are investigated in detail, and the
corresponding six-boson realization is given. After obtaining explicit forms of
all twelve extremal vectors of the elementary representation with the highest
weight {\Lambda}, all representations with their respective highest weights
related to {\Lambda} are systematically discussed. For one of these
representations the corresponding five-boson realization is constructed.
Moreover, a new three-fermion realization from the fundamental representation
(0,1) of G_2 is constructed also.Comment: 29 pages, 4 figure
A voting scheme with post-quantum security based on physical laws
Traditional cryptography is under huge threat along of the evolution of
quantum information and computing. In this paper, we propose a new post-quantum
voting scheme based on physical laws by using encrypted no-key protocol to
transmit message in the channel, which ensures the post-quantum security.
Unlike lattice-based and multivariate-based electronic voting schemes, whose
security is based on the computational problems assumption that has not been
solved by effective quantum algorithms until now, the security of the voting
scheme based on the physical laws is depended on inherent limitations of
quantum computers and not influenced by the evolution of new quantum
algorithms. In detail, we also rigorously demonstrate that the scheme achieves
the post-quantum security and all properties necessary for voting scheme such
as the completeness, robustness, privacy, eligibility, unreusability, fairness,
and verifiability.Comment: 23pages,1figure,5table
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