Multiplicative complexity of polynomial multiplication over finite fields
TL;DR: It is shown that if q/2 ≪ n ≤ q + 1, the tight bound Mq(n) = 3n + 1 - ⌊q/2⌋ is established.
Abstract: Let Mq(n) denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q-element field by means of bilinear algorithms. It is shown that Mq(n) n 3n - o(n). In particular, if q/2
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TL;DR: This paper surveys bit-parallel multipliers for finite field GF according to quadratic and subquadratic arithmetic complexities of the underlying algorithms, various bases used for representing the field elements, and design approaches that rely on polynomial and matrix operations.
37 citations
01 Feb 1989
TL;DR: It is shown that the complexity of one direct sum is the sum of the complexityof the summands and that every minimal quadratic algorithm for computing the direct sums is a direct-sum algorithm.
Abstract: We consider the quadratic complexity of certain sets of quadratic forms. We study a classes of direct sums of quadratic forms. For these classes of problems we show that the complexity of one direct sum is the sum of the complexity of the summands and that every minimal quadratic algorithm for computing the direct sums is a direct-sum algorithm.
22 citations
TL;DR: The number of non-isomorphic functional graphs of affine-linear transformations from ( F q ) n to itself is studied, and upper and lower bounds on this quantity as n→∞ are proved.
13 citations
Cites methods from "Multiplicative complexity of polyno..."
...As we only require a weaker big-O estimate, we present a simplified version of the proof in [10] which also works for q = 2....
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TL;DR: The authors prove the lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight-line algorithms.
Abstract: The authors prove the $2.5 n - o(n)$ lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight-line algorithms.
12 citations
TL;DR: Improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most (n-1) modulo x^n over R are given.
11 citations
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Book•
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher:
With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter.
0201000296B04062001
9,262 citations
Book•
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3,617 citations
2,541 citations
TL;DR: In this article, it was shown that the use of divisions does not decrease the number of (*,/)-operations for multiplication of general matrices, and that multiplication of orthogonal matrices does not increase the computational complexity.
Abstract: The extent to whieh the use of divisions may speed up the evaluation of polynomials is estimated from above. In particular it is shown that for multiplying general matrices the use of divisions does not decrease the number of (*,/)-operations. The computational complexity of the multiplication of matrices from a linear algebraic group G is estimated from below by a simple algebraic invariant of the Liering of G. In particular, the multiplication of orthogonal matrices is treated.
394 citations
"Multiplicative complexity of polyno..." refers background in this paper
...The following corollary is a partial case of the direct sum conjecture conjectured by Strassen in [16]....
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...It is known from [16] that if a set of bilinear forms over an infinite field can be computed in t multiplications/divisions, then it can be computed in t multiplications by a bilinear algorithm whose total number of operations differs from that of the original one by a factor of a small constant....
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