12,073 research outputs found
On Polynomial Multiplication in Chebyshev Basis
In a recent paper Lima, Panario and Wang have provided a new method to
multiply polynomials in Chebyshev basis which aims at reducing the total number
of multiplication when polynomials have small degree. Their idea is to use
Karatsuba's multiplication scheme to improve upon the naive method but without
being able to get rid of its quadratic complexity. In this paper, we extend
their result by providing a reduction scheme which allows to multiply
polynomial in Chebyshev basis by using algorithms from the monomial basis case
and therefore get the same asymptotic complexity estimate. Our reduction allows
to use any of these algorithms without converting polynomials input to monomial
basis which therefore provide a more direct reduction scheme then the one using
conversions. We also demonstrate that our reduction is efficient in practice,
and even outperform the performance of the best known algorithm for Chebyshev
basis when polynomials have large degree. Finally, we demonstrate a linear time
equivalence between the polynomial multiplication problem under monomial basis
and under Chebyshev basis
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Polynomial evaluation over finite fields: new algorithms and complexity bounds
An efficient evaluation method is described for polynomials in finite fields.
Its complexity is shown to be lower than that of standard techniques when the
degree of the polynomial is large enough. Applications to the syndrome
computation in the decoding of Reed-Solomon codes are highlighted.Comment: accepted for publication in Applicable Algebra in Engineering,
Communication and Computing. The final publication will be available at
springerlink.com. DOI: 10.1007/s00200-011-0160-
Conditionals in Homomorphic Encryption and Machine Learning Applications
Homomorphic encryption aims at allowing computations on encrypted data
without decryption other than that of the final result. This could provide an
elegant solution to the issue of privacy preservation in data-based
applications, such as those using machine learning, but several open issues
hamper this plan. In this work we assess the possibility for homomorphic
encryption to fully implement its program without relying on other techniques,
such as multiparty computation (SMPC), which may be impossible in many use
cases (for instance due to the high level of communication required). We
proceed in two steps: i) on the basis of the structured program theorem
(Bohm-Jacopini theorem) we identify the relevant minimal set of operations
homomorphic encryption must be able to perform to implement any algorithm; and
ii) we analyse the possibility to solve -- and propose an implementation for --
the most fundamentally relevant issue as it emerges from our analysis, that is,
the implementation of conditionals (requiring comparison and selection/jump
operations). We show how this issue clashes with the fundamental requirements
of homomorphic encryption and could represent a drawback for its use as a
complete solution for privacy preservation in data-based applications, in
particular machine learning ones. Our approach for comparisons is novel and
entirely embedded in homomorphic encryption, while previous studies relied on
other techniques, such as SMPC, demanding high level of communication among
parties, and decryption of intermediate results from data-owners. Our protocol
is also provably safe (sharing the same safety as the homomorphic encryption
schemes), differently from other techniques such as
Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical
section on polynomial approximatio
New Algorithms for Computing a Single Component of the Discrete Fourier Transform
This paper introduces the theory and hardware implementation of two new
algorithms for computing a single component of the discrete Fourier transform.
In terms of multiplicative complexity, both algorithms are more efficient, in
general, than the well known Goertzel Algorithm.Comment: 4 pages, 3 figures, 1 table. In: 10th International Symposium on
Communication Theory and Applications, Ambleside, U
A hybrid approach to Fermi operator expansion
In a recent paper we have suggested that the finite temperature density
matrix can be computed efficiently by a combination of polynomial expansion and
iterative inversion techniques. We present here significant improvements over
this scheme. The original complex-valued formalism is turned into a purely real
one. In addition, we use Chebyshev polynomials expansion and fast summation
techniques. This drastically reduces the scaling of the algorithm with the
width of the Hamiltonian spectrum, which is now of the order of the cubic root
of such parameter. This makes our method very competitive for applications to
ab-initio simulations, when high energy resolution is required.Comment: preprint of ICCMSE08 proceeding
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