Related papers: Revisiting Fast Fourier multiplication algorithms …
The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is…
We describe an algorithm for fast multiplication of skew polynomials. It is based on fast modular multiplication of such skew polynomials, for which we give an algorithm relying on evaluation and interpolation on normal bases. Our…
Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors.…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…
In this work a rationalized algorithm for calculating the quotient of two complex numbers is presented which reduces the number of underlying real multiplications. The performing of a complex number division using the naive method takes 4…
This paper shows that it is possible to improve the computational cost, the memory requirements and the accuracy of Quick Fourier Transform (QFT) algorithm for power-of-two FFT (Fast Fourier Transform) just introducing a slight modification…
We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate…
In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two…
We present an algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations. Previously, the best known bound was O(n log n 6^{log^* n}). We also prove that for a fixed prime p, polynomials in…
The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a…
For various $2\leq n,m \leq 6$, we propose some new algorithms for multiplying an $n\times m$ matrix with an $m \times 6$ matrix over a possibly noncommutative coefficient ring.
We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…
Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this…
We study the complexity of polynomial multiplication over arbitrary fields. We present a unified approach that generalizes all known asymptotically fastest algorithms for this problem. In particular, the well-known algorithm for…
Quadratic permutation polynomial interleavers over integer rings have recently received attention in practical turbo coding systems from deep space applications to mobile communications. In this correspondence, a necessary and sufficient…
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified…
The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been…
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…
In this work, a rationalized algorithm for calculating the quotient of two quaternions is presented which reduces the number of underlying real multiplications. Hardware for fast multiplication is much more expensive than hardware for fast…
We show that the product of an nx3 matrix and a 3x3 matrix over a commutative ring can be computed using 6n+3 multiplications. For two 3x3 matrices this gives us an algorithm using 21 multiplications. This is an improvement with respect to…