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Multiplication hits the speed limit

2019
*
Communications of the ACM
*

Schönhage and Strassen's

doi:10.1145/3371387
fatcat:atsxey72hrfuhbghrkemx4l6xu
*algorithm*, which laid the groundwork*for*the*new**algorithm*announced this past March, leverages the fast Fourier transform, a procedure*for*sampling and reconstructing*polynomials*...*Multiplication*Hits the Speed Limit A problem "around since antiquity" may have been resolved by a*new**algorithm*.*news*"Now we know that all these*algorithms*that depend on*multiplication*are the time ...##
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On square-free factorization of multivariate polynomials over a finite field

1997
*
Theoretical Computer Science
*

In this paper we present a

doi:10.1016/s0304-3975(97)00059-5
fatcat:3zucsxzip5dchdpeayu5sijx2m
*new*deterministic*algorithm**for*computing the square-free decomposition of multivariate*polynomials*with coefficients from a finite field. ... Our*algorithm*is based on Yun's square-free factorization*algorithm**for*characteristic 0. ... Our example is a*polynomial*with only one square-free factor of high*multiplicity*. As seen above, this is the worst case*for*our*algorithm*if the characteristic is larger than the*multiplicity*. ...##
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Some new results on binary polynomial multiplication

2015
*
Journal of Cryptographic Engineering
*

This paper presents several methods

doi:10.1007/s13389-015-0101-6
fatcat:zywrelcyhbcednovkq5rfquxp4
*for*reducing the number of bit operations*for**multiplication*of*polynomials*over the binary field. ... This*new**algorithm*uses three*multiplications*of one-third size*polynomials*over the binary field and one*multiplication*of one-third size*polynomials*over the finite field with four elements. ... The authors would like to thank undergraduate research assistant Ryan Young, who wrote a C-code*for*us to automate the generation of a part of the data included in Table 2 in Appendix A. ...##
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On the Number of Multiplications for the Evaluation of a Polynomial and Some of Its Derivatives

1974
*
Journal of the ACM
*

We give a

doi:10.1145/321796.321810
fatcat:scewcz2rgbhcdbee3e6c744pbe
*new**algorithm*which computes all the derivatives of a*polynomial*in 3n-2*multiplications*or divisions (M/D). ... The general specification of the*algorithm*requires n additions and 2n*multiplications*; however, a*new**algorithm**for*computing P(x) and P f (x). It requires 2n-l additions and n-1 + 2 v^n+l M/D. ...##
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New Algorithms for Computing a Single Component of the Discrete Fourier Transform
[article]

2015
*
arXiv
*
pre-print

This paper introduces the theory and hardware implementation of two

arXiv:1503.02577v1
fatcat:fsk2j66l5vgsbber26kxzlviqa
*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*. ... In this paper, a*new**algorithm**for*computing a single DFT component, which is based on cyclotomic*polynomials*, is introduced in Section II. ...##
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Achieving NTRU with montgomery multiplication

2003
*
IEEE transactions on computers
*

In this paper, we propose a

doi:10.1109/tc.2003.1190585
fatcat:kuybatdgfbg3jjfr6hndvaiwma
*new*unified architecture that utilizes the Montgomery*Multiplication**algorithm*to perform a modular*multiplication**for*both integers and binary*polynomials*and NTRU's*polynomial*...*multiplications*. ... ACKNOWLEDGMENTS The authors thank Gunnar Gaubatz*for*his contributions to the unified Montgomery Multiplier core design. ...##
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An improved Šiljak's algorithm for solving polynomial equations converges quadratically to multiple zeros

1995
*
Journal of Computational and Applied Mathematics
*

We shall provide time comparisons of the

doi:10.1016/0377-0427(94)00114-6
fatcat:nz5pt34gang6vdy7j4sai7nwcm
*new**algorithm*with the standard zero inclusion*algorithms*to show that it is as fast or faster*for*general*polynomial*problems, and significantly faster*for**polynomials*... ~iljak's method provides a globally convergent*algorithm**for*inclusion of*polynomial*zeros. ... We note that the convergence is indeed quadratic*for*this*multiple*zero example. Example 5.4. One additional feature of the*new**algorithm*is that it is extremely efficient*for*poor initial estimates. ...##
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Automatic Library Generation for Modular Polynomial Multiplication
[article]

2016
*
arXiv
*
pre-print

, using

arXiv:1609.01010v1
fatcat:73syghptxja5tjb5ifmfomv7uu
*new*and existing*algorithms*. ...*Polynomial**multiplication*is a key*algorithm*underlying computer algebra systems (CAS) and its efficient implementation is crucial*for*the performance of CAS. ...*algorithm**for*computing the circular convolution, and thus*for**polynomial**multiplication*mod x N −1. ...##
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Page 3129 of Mathematical Reviews Vol. , Issue 82g
[page]

1982
*
Mathematical Reviews
*

*Multiplicative*

*algorithmic*complexity is assigned as the number of

*multiplication*and partition operations forming a part of the given

*algorithm*. ...

*Multiplicative*complexity is also determined

*for*any sequence of elements of the field using the notion of

*algorithms*computing the elements of this sequence. Such sequences are called systems. ...

##
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Roots multiplicity without companion matrices
[article]

2017
*
arXiv
*
pre-print

We show a method

arXiv:1703.06120v1
fatcat:yc7fgi64t5cm5li52iboexryg4
*for*constructing a*polynomial*interpolating roots'*multiplicities*of another*polynomial*, that does not use companion matrices. ... This leads to a modification to Guersenzvaig--Szechtman square-free decomposition*algorithm*that is more efficient both in theory and in practice. ... Szechtman invented a completely*new**algorithm*(see [2]). They associate to a given*polynomial*f its roots-*multiplicity**polynomial*Mf . ...##
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Spectral transformations for two-dimensional filters via FFT

2002
*
IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications
*

In this paper, a

doi:10.1109/tcsi.2002.1010037
fatcat:6ifvdwgb7jh43dhxw34sysjwsu
*new*fast*algorithm**for*spectral transformations*for*two-dimensional digital filters is presented. The*algorithm*is based on the use of the fast Fourier transform. ... The computational complexity of this*algorithm*is evaluated. The simplicity and efficiency of the*algorithm*is illustrated by a numerical example. ... In the present paper, a*new**algorithm**for*the same problem is proposed. The*algorithm*is based on the discrete Fourier transform (DFT). ...##
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Reconstructing a Linear Scrambler With Improved Detection Capability and in the Presence of Noise

2012
*
IEEE Transactions on Information Forensics and Security
*

*For*both cases, factors which affect the performance of the reconstruction

*algorithm*are discussed. ... In this paper, the problem of reconstruction of the feedback

*polynomial*in a linear scrambler is studied. Our work contains two parts. ... Instead of searching

*for*the feedback

*polynomial*directly, Cluzeau's

*algorithm*searched

*for*sparse

*multiples*of with the degree of the sparse

*multiples*varying from low to high. ...

##
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All the Polynomial Multiplication You Need on RISC-V
[article]

2021
*
IACR Cryptology ePrint Archive
*

We re-designed expensive implementations of

dblp:journals/iacr/SeoKEJKKSSL21
fatcat:mhb5xsyhlnbi5kuo6fxkltslru
*polynomial**multiplication*on legacy microcontrollers (e.g. 8-bit AVR, 16-bit MSP, and 32-bit ARM)*for**new*instruction sets of 32bit RISC-V processors. ... Third, we propose instruction set extensions*for*the optimal implementation of*polynomial**multiplication*on 32-bit RISC-V processors. This*new*feature introduces significant performance enhancements. ... This*new*feature can accelerate the performance of*polynomial**multiplication*by reducing clock cycles. ...##
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Fast Operations on Linearized Polynomials and their Applications in Coding Theory
[article]

2017
*
arXiv
*
pre-print

We propose a

arXiv:1512.06520v3
fatcat:ea54m3ztzff6ri7mjdt3hexkzq
*new**multiplication**algorithm**for*skew*polynomials*(a generalization of linearized*polynomials*) which has sub-quadratic complexity in the*polynomial*degree s, independent of the underlying ... Using the*new*fast*algorithm**for*the q-transform, we show how matrix*multiplication*over a finite field can be implemented by multiplying linearized*polynomials*of degrees at most s=m if an elliptic normal ... Acknowledgement The authors would like to thank Johan Rosenkilde né Nielsen*for*the valuable discussions, his idea leading to*Algorithm*4 and several comments improving the readability of the paper. ...##
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Studying Software Implementations of Elliptic Curve Cryptography

2006
*
Third International Conference on Information Technology: New Generations (ITNG'06)
*

We identified a set of

doi:10.1109/itng.2006.127
dblp:conf/itng/YanS06
fatcat:nc23cotjpvhdzhf352vshehuti
*algorithms**for*ECC implementation*for*low-end processors. ... Elliptic Curve Cryptography (ECC) is a promising alternative*for*public-key*algorithms*in resource-constrained systems because it provides a similar level of security with much shorter keys than conventional ... When*new*hardware is implemented, the performance of*multiplication*and inversion should be evaluated to choose the best point representations*for*better performances. ...
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