Improved time bounds for near-optimal sparse Fourier representations

Gilbert, Anna C.; Muthukrishnan, S.; Strauss, Martin J.

doi:10.1117/12.615931

Cited by 173 publications

(208 citation statements)

References 14 publications

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“…In particular, in the Z N case -which is enough for our purposes -they prove that Theorem 1. There is an algorithm which, given query access to g : Another algorithm to the same purpose was given by Strauss and Mutukrishnan [5], resulting in a running time with improved dependence in 1/τ . This theorem is used in [1] to prove the following…”

Section: Definition 9 a Code C Is Recoverable If There Exists A Recmentioning

confidence: 99%

The Security of All Bits Using List Decoding

Morillo

Ràfols

2009

Public Key Cryptography – PKC 2009

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Abstract. The relation between list decoding and hard-core predicates has provided a clean and easy methodology to prove the hardness of certain predicates. So far this methodology has only been used to prove that the O(log log N ) least and most significant bits of any function with multiplicative access -which include the most common number theoretic trapdoor permutations-are secure. In this paper we show that the method applies to all bits of any function defined on a cyclic group of order N with multiplicative access for cryptographically interesting N . As a result, in this paper we reprove the security of all bits of RSA, the discrete logarithm in a group of prime order or the Paillier encryption scheme.

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Section: Definition 9 a Code C Is Recoverable If There Exists A Recmentioning

confidence: 99%

The Security of All Bits Using List Decoding

Morillo

Ràfols

2009

Public Key Cryptography – PKC 2009

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“…2 Noise is out of scope in the analysis of the universal algorithms [12,10,11]. These SFT algorithms [12,6,2,7,10,11] are insufficient for our result solving HNP. Both universality as well as handling functions that are neither compressible nor Fourier sparse are crucial for our algorithm solving HNP.…”

Section: New Tool: Universally Finding Significant Fourier Coefficientsmentioning

confidence: 99%

“…For functions over Z p , prior SFT algorithms [6,2,7] are not universal. In concurrent works [10,11] gave a universal SFT algorithm for a restricted class of functions over Z p : compressible or Fourier sparse functions.…”

Section: New Tool: Universally Finding Significant Fourier Coefficientsmentioning

confidence: 99%

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Solving Hidden Number Problem with One Bit Oracle and Advice

Akavia

2009

Advances in Cryptology - CRYPTO 2009

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Abstract. In the Hidden Number Problem (HNP), the goal is to find a hidden number s, when given p, g and access to an oracle that on query a returns the k most significant bits of s · g a mod p.We present an algorithm solving HNP, when given an advice depending only on p and g; the running time and advice length are polynomial in log p. This algorithm improves over prior HNP algorithms in achieving:(1) optimal number of bits k ≥ 1 (compared with k ≥ Ω(log log p)); (2) robustness to random noise; and (3) handling a wide family of predicates on top of the most significant bit.As a central tool we present an algorithm that, given oracle access to a function f over ZN , outputs all the significant Fourier coefficients of f (i.e., those occupying, say, at least 1% of the energy). This algorithm improves over prior works in being:-Local. Its running time is polynomial in log N and L1( f ) (for L1( f ) the sum of f 's Fourier coefficients, in absolute value). -Universal. For any N, t, the same oracle queries are asked for all functions f over ZN s.t. L1( f ) ≤ t. -Robust. The algorithm succeeds with high probability even if the oracle to f is corrupted by random noise.

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“…pursuit [74], Chaining Pursuits [76] and sub-linear Fourier transform [77] are typical examples of this class of algorithms [11,29].…”

Section: Combinatorial Algorithmsmentioning

confidence: 99%

Improving the image quality in compressed sensing MRI by the exploitation of data properties

Yang¹

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Minimizing scan time without compromising image quality has been the main thrust of magnetic resonance imaging (MRI) since the establishment of the field. Tremendous effort has been put into MRI systems in pursuit of faster MR imaging techniques, which can substantially facilitate clinical diagnoses and improve the patient experience. Traditional MRI accelerated approaches mainly focused on hardware improvements to improve the speed of scanning. These methods are still governed by Nyquist-Shannon sampling rate. Hence, when the acceleration rate is pushed beyond the theoretical limit, aliasing artefacts are introduced which degrade the reconstructed image quality and are difficult to eliminate via current methods. Recently, it has been found that a newly developed signal processing theory, compressed sensing (CS), can be used to create high-resolution signal data sets from sparse samples, despite violating the classical Nyquist-Shannon sampling criteria. The application of CS to MRI opens up an appealing avenue to offer potentially significant scan time reductions. CS reconstructs MR images from incomplete k-space measurements. Taking advantage of the fact that MR images have sparse representations under certain mathematical transformations, CS overcomes the distortions resulting from the under-sampled k-space data. Significant progress has been made in the development of CS MRI. However, there are two major remaining challenges to its full adoption in clinical applications: (1) the small amount of under-sampled data (for example, less than 1/8 of the whole k-space data) will inherently introduce artefacts in the reconstructed image and compromise diagnosis accuracy for the reconstructed images; (2) owing to hardware limits, it is difficult to realise two-dimension random under-sampling, which is favoured by CS operation.Practically, the random phase-encode under-sampling pattern has been widely implemented.However, it introduces coherent aliasing artefacts that are hard to eliminate using the CS theory.In this thesis, several techniques are proposed that can improve the quality of images reconstructed by CS MRI. These techniques have the potential to facilitate faithful reconstruction of CS MRI in clinical applications. A novel two-stage reconstruction scheme is introduced in Chapter 3. In this method, the under-sampled k-space data is segmented into low-frequency and high-frequency parts.Then, in stage one, using dense measurements, the low-frequency region of k-space data is faithfully reconstructed. The fully reconstituted low-frequency k-space data from the first stage is then combined with the under-sampled high-frequency k-space data to complete the second stage reconstruction of the whole image. With this two-stage approach, each reconstruction inherently incorporates a lower data under-sampling rate than conventional approaches. Because the restricted isometric property is easier to satisfy than conventional CS method, the reconstruction consequently ii produces lower residual errors in each step...

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Improved time bounds for near-optimal sparse Fourier representations

Cited by 173 publications

References 14 publications

The Security of All Bits Using List Decoding

The Security of All Bits Using List Decoding

Solving Hidden Number Problem with One Bit Oracle and Advice

Improving the image quality in compressed sensing MRI by the exploitation of data properties

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