110 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
A Note on Koblitz Curves over Prime Fields
Besides the well-known class of Koblitz curves over binary fields, the class of
Koblitz curves over prime fields with is also
of some practical interest. By refining a classical result of Rajwade for the cardinality of , we obtain a simple formula of in terms of the norm on the ring of Eisenstein integers, that is, for some with and some unit ,
holds. This establishes an interesting relation between the number of points on this class of curves and the number of elements of their underlying fields, they are given by the norm of two integers of whose difference is just a unit. It is also interesting to note that such relationship has already been derived for the case of Koblitz curves over binary fields. Some tools that are useful in the computation of cubic residues are also developed
Pre-Computation Scheme of Window NAF for Koblitz Curves Revisited
Let be a Koblitz curve. The window -adic non-adjacent form (window NAF) is currently the standard representation system to perform scalar multiplications on utilizing the Frobenius map . This work focuses on the pre-computation part of scalar multiplication. We first introduce -operations where and is the complex conjugate of . Efficient formulas of -operations are then derived and used in a novel pre-computation scheme. Our pre-computation scheme requires {\bf M}{\bf S}, {\bf M}{\bf S}, {\bf M}{\bf S}, and {\bf M}{\bf S} () and {\bf M}{\bf S}, {\bf M}{\bf S}, {\bf M}{\bf S}, and {\bf M}{\bf S} () for window NAF with widths from to respectively. It is about two times faster, compared to the state-of-the-art technique of pre-computation in the literature. The impact of our new efficient pre-computation is also reflected by the significant improvement of scalar multiplication. Traditionally, window NAF with width at most is used to achieve the best scalar multiplication. Because of the dramatic cost reduction of the proposed pre-computation, we are able to increase the width for window NAF to for a better scalar multiplication. This indicates that the pre-computation part becomes more important in performing scalar multiplication. With our efficient pre-computation and the new window width, our scalar multiplication runs in at least 85.2\% the time of Kohel\u27s work (Eurocrypt\u272017) combining the best previous pre-computation. Our results push the scalar multiplication of Koblitz curves, a very well-studied and long-standing research area, to a significant new stage
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