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Construction of Capacity-Achieving Lattice Codes: Polar Lattices
In this paper, we propose a new class of lattices constructed from polar
codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR)
of the additive white Gaussian-noise (AWGN) channel. Our construction follows
the multilevel approach of Forney \textit{et al.}, where we construct a
capacity-achieving polar code on each level. The component polar codes are
shown to be naturally nested, thereby fulfilling the requirement of the
multilevel lattice construction. We prove that polar lattices are
\emph{AWGN-good}. Furthermore, using the technique of source polarization, we
propose discrete Gaussian shaping over the polar lattice to satisfy the power
constraint. Both the construction and shaping are explicit, and the overall
complexity of encoding and decoding is for any fixed target error
probability.Comment: full version of the paper to appear in IEEE Trans. Communication
Worst case QC-MDPC decoder for McEliece cryptosystem
McEliece encryption scheme which enjoys relatively small key sizes as well as
a security reduction to hard problems of coding theory. Furthermore, it remains
secure against a quantum adversary and is very well suited to low cost
implementations on embedded devices.
Decoding MDPC codes is achieved with the (iterative) bit flipping algorithm,
as for LDPC codes. Variable time decoders might leak some information on the
code structure (that is on the sparse parity check equations) and must be
avoided. A constant time decoder is easy to emulate, but its running time
depends on the worst case rather than on the average case. So far
implementations were focused on minimizing the average cost. We show that the
tuning of the algorithm is not the same to reduce the maximal number of
iterations as for reducing the average cost. This provides some indications on
how to engineer the QC-MDPC-McEliece scheme to resist a timing side-channel
attack.Comment: 5 pages, conference ISIT 201
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