108 research outputs found
A Remark on Fourier Transform
In this note, we describe an interpretation of the (continuous) Fourier
transform from the perspective of the Chinese Remainder Theorem. Some related
issues are discussed
On Recovery of Sparse Signals via Minimization
This article considers constrained minimization methods for the
recovery of high dimensional sparse signals in three settings: noiseless,
bounded error and Gaussian noise. A unified and elementary treatment is given
in these noise settings for two minimization methods: the Dantzig
selector and minimization with an constraint. The results of
this paper improve the existing results in the literature by weakening the
conditions and tightening the error bounds. The improvement on the conditions
shows that signals with larger support can be recovered accurately. This paper
also establishes connections between restricted isometry property and the
mutual incoherence property. Some results of Candes, Romberg and Tao (2006) and
Donoho, Elad, and Temlyakov (2006) are extended
New Bounds for Restricted Isometry Constants
In this paper we show that if the restricted isometry constant of
the compressed sensing matrix satisfies then -sparse
signals are guaranteed to be recovered exactly via minimization when
no noise is present and -sparse signals can be estimated stably in the noisy
case. It is also shown that the bound cannot be substantively improved. An
explicitly example is constructed in which ,
but it is impossible to recover certain -sparse signals
Fast Arithmetics Using Chinese Remaindering
In this paper, some issues concerning the Chinese remaindering representation
are discussed. Some new converting methods, including an efficient
probabilistic algorithm based on a recent result of von zur Gathen and
Shparlinski \cite{Gathen-Shparlinski}, are described. An efficient refinement
of the NC division algorithm of Chiu, Davida and Litow
\cite{Chiu-Davida-Litow} is given, where the number of moduli is reduced by a
factor of
Enhancing the Dual Attack against MLWE: Constructing More Short Vectors Using Its Algebraic Structure
Primal attack, BKW attack, and dual attack are three well-known attacks to LWE. To build efficient post-quantum cryptosystems in practice, the structured variants of LWE (i.e. MLWE/RLWE) are often used. Some efforts have been spent on addressing concerns about additional vulnerabilities introduced by algebraic structures and no effective attack method based on ideal lattices or module lattices has been proposed so far; these include refining primal attack and BKW attack to MLWE/RLWE. It is thus an interesting problem to consider how to enhance the dual attack against LWE with the rich algebraic structure of MLWE (including RLWE). In this paper, we present the first attempt to this problem by observing that each short vector found by BKZ generates another n β 1 vectors of the same length automatically and all of these short vectors can be used to distinguish. To this end, an interesting property which indicates the rotations are consistent with certain linear transformations is proved, and a new kind of intersection lattice is constructed with some tricks. Moreover, we notice that coefficient vectors of different rotations of the same polynomial are near-orthogonal in high-dimensional spaces. This is validated by extensive experiments and is treated as an extension to the assumption under the original dual attack against LWE. Taking Newhope512 as an example, we show that by our enhanced dual attack method, the required blocksize and time complexity (in both classical and quantum cases) all decrease. It is remarked that our improvement is not significant and its limitation is also touched on. Our results do not reveal a severe security problem for MLWE/RLWE compared to that of a general LWE, this is consistent with the findings by the previous work for using primal and BKW attacks to MLWE/RLWE
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