46,213 research outputs found
A note on the implementation of the Number Theoretic Transform
The Number Theoretic Transform (NTT) is the time critical function required by cryptographic protocols based on the Ring Learning With Errors problem (RLWE),a popular choice for post-quantum cryptography.
Here we apply a simple methodology to convert the NTT and its inverse from a mathematically correct (but side-channel vulnerable) description, to an efficient constant-time side-channel resistant version
Optimal Discrete Uniform Generation from Coin Flips, and Applications
This article introduces an algorithm to draw random discrete uniform
variables within a given range of size n from a source of random bits. The
algorithm aims to be simple to implement and optimal both with regards to the
amount of random bits consumed, and from a computational perspective---allowing
for faster and more efficient Monte-Carlo simulations in computational physics
and biology. I also provide a detailed analysis of the number of bits that are
spent per variate, and offer some extensions and applications, in particular to
the optimal random generation of permutations.Comment: first draft, 22 pages, 5 figures, C code implementation of algorith
Fast Digital Convolutions using Bit-Shifts
An exact, one-to-one transform is presented that not only allows digital
circular convolutions, but is free from multiplications and quantisation errors
for transform lengths of arbitrary powers of two. The transform is analogous to
the Discrete Fourier Transform, with the canonical harmonics replaced by a set
of cyclic integers computed using only bit-shifts and additions modulo a prime
number. The prime number may be selected to occupy contemporary word sizes or
to be very large for cryptographic or data hiding applications. The transform
is an extension of the Rader Transforms via Carmichael's Theorem. These
properties allow for exact convolutions that are impervious to numerical
overflow and to utilise Fast Fourier Transform algorithms.Comment: 4 pages, 2 figures, submitted to IEEE Signal Processing Letter
Role of anticausal inverses in multirate filter-banks. I. System-theoretic fundamentals
In a maximally decimated filter bank with identical decimation ratios for all channels, the perfect reconstructibility property and the nature of reconstruction filters (causality, stability, FIR property, and so on) depend on the properties of the polyphase matrix. Various properties and capabilities of the filter bank depend on the properties of the polyphase matrix as well as the nature of its inverse. In this paper we undertake a study of the types of inverses and characterize them according to their system theoretic properties (i.e., properties of state-space descriptions, McMillan degree, degree of determinant, and so forth). We find in particular that causal polyphase matrices with anticausal inverses have an important role in filter bank theory. We study their properties both for the FIR and IIR cases. Techniques for implementing anticausal IIR inverses based on state space descriptions are outlined. It is found that causal FIR matrices with anticausal FIR inverses (cafacafi) have a key role in the characterization of FIR filter banks. In a companion paper, these results are applied for the factorization of biorthogonal FIR filter banks, and a generalization of the lapped orthogonal transform called the biorthogonal lapped transform (BOLT) developed
Efficient Information Theoretic Clustering on Discrete Lattices
We consider the problem of clustering data that reside on discrete, low
dimensional lattices. Canonical examples for this setting are found in image
segmentation and key point extraction. Our solution is based on a recent
approach to information theoretic clustering where clusters result from an
iterative procedure that minimizes a divergence measure. We replace costly
processing steps in the original algorithm by means of convolutions. These
allow for highly efficient implementations and thus significantly reduce
runtime. This paper therefore bridges a gap between machine learning and signal
processing.Comment: This paper has been presented at the workshop LWA 201
Euler-Bessel and Euler-Fourier Transforms
We consider a topological integral transform of Bessel (concentric
isospectral sets) type and Fourier (hyperplane isospectral sets) type, using
the Euler characteristic as a measure. These transforms convert constructible
\zed-valued functions to continuous -valued functions over a vector
space. Core contributions include: the definition of the topological Bessel
transform; a relationship in terms of the logarithmic blowup of the topological
Fourier transform; and a novel Morse index formula for the transforms. We then
apply the theory to problems of target reconstruction from enumerative sensor
data, including localization and shape discrimination. This last application
utilizes an extension of spatially variant apodization (SVA) to mitigate
sidelobe phenomena
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