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### Multiplication hits the speed limit

Erica Klarreich
2019 Communications of the ACM
Schönhage and Strassen's 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  ...

### On square-free factorization of multivariate polynomials over a finite field

Laurent Bernardin
1997 Theoretical Computer Science
In this paper we present a 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.  ...

### Some new results on binary polynomial multiplication

Murat Cenk, M. Anwar Hasan
2015 Journal of Cryptographic Engineering
This paper presents several methods 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.  ...

### On the Number of Multiplications for the Evaluation of a Polynomial and Some of Its Derivatives

Mary Shaw, J. F. Traub
1974 Journal of the ACM
We give a 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.  ...

### New Algorithms for Computing a Single Component of the Discrete Fourier Transform [article]

G. Jerônimo da Silva Jr., R.M. Campello de Souza, H.M. de Oliveira
2015 arXiv   pre-print
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.  ...  In this paper, a new algorithm for computing a single DFT component, which is based on cyclotomic polynomials, is introduced in Section II.  ...

### Achieving NTRU with montgomery multiplication

C. O'Rourke, B. Sunar
2003 IEEE transactions on computers
In this paper, we propose a 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.  ...

### An improved Šiljak's algorithm for solving polynomial equations converges quadratically to multiple zeros

J.A. Stolan
1995 Journal of Computational and Applied Mathematics
We shall provide time comparisons of the 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.  ...

### Automatic Library Generation for Modular Polynomial Multiplication [article]

Lingchuan Meng
2016 arXiv   pre-print
, using 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.  ...

### 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.  ...

### Roots multiplicity without companion matrices [article]

Przemysław Koprowski
2017 arXiv   pre-print
We show a method 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 ). They associate to a given polynomial f its roots-multiplicity polynomial Mf .  ...

### Spectral transformations for two-dimensional filters via FFT

N.E. Mastorakis, M.N.S. Swamy
2002 IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications
In this paper, a 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).  ...

### Reconstructing a Linear Scrambler With Improved Detection Capability and in the Presence of Noise

Xiao-Bei Liu, Soo Ngee Koh, Xin-Wen Wu, Chee-Cheon Chui
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.  ...

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

Hwajeong Seo, Hyeokdong Kwon, Siwoo Eum, Kyungbae Jang, Hyunjun Kim, Hyunji Kim, Minjoo Sim, Gyeongju Song, Wai-Kong Lee
2021 IACR Cryptology ePrint Archive
We re-designed expensive implementations of 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.  ...

### Fast Operations on Linearized Polynomials and their Applications in Coding Theory [article]

Sven Puchinger, Antonia Wachter-Zeh
2017 arXiv   pre-print
We propose a 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.  ...

### Studying Software Implementations of Elliptic Curve Cryptography

Hai Yan, Zhijie Jerry Shi
2006 Third International Conference on Information Technology: New Generations (ITNG'06)
We identified a set of 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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