Algebras Having Linear Multiplicative Complexities

Fiduccia, Charles M.; Zalcstein, Yechezkel

doi:10.1145/322003.322014

Cited by 66 publications

(19 citation statements)

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“…Estimating the complexity only in terms of multiplication operations, we should take into account the possibility of multiplying two polynomials of degree r at the cost of only 2r -f 1 multiplications plus additional 2r + 1 multiplications by a constant which depends only on r [4]. In the case of block polynomials, we need (2r + l)p 3 multiplication operations plus (2r+ l)p 2 multiplications by a constant.…”

Section: Consequentlymentioning

confidence: 99%

Fast algorithms for block Toeplitz matrices

Тыртышников¹

1986

Russian Journal of Numerical Analysis and Mathematical Modelling

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This paper presents the solution of linear systems with block Toeplitz matrices by constructing block versions of the Fade approximation. This approach yields a fast method of solving systems with low Toeplitz rank matrices. Also discussed are the results on the description of inverse matrices for some special classes of matrices and new possibilities of realizing the method of Toeplitz embedding'.

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Section: Consequentlymentioning

confidence: 99%

Fast algorithms for block Toeplitz matrices

Тыртышников¹

1986

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…Among the problems reducible to this are fast multiplication of multiple-precision numbers, greatest common divisors in polynomial rings, Hankel matrix multiplication, computation of Padd approximations, and computation of finite Fourier transformations. Significant progress in this problem, due to Winograd (1), Fiduccia and Zalcstein (2), and Adler and Strassen (3), has mainly concentrated on minimal multiplicative complexities (m.m.c.) of polynomial multiplication over fields.…”

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confidence: 99%

“…schemes over A = Z or A = Z [1/2]. Schonhage and Strassen (5), Winograd (1), and Nussbaumer (6) and others constructed fast algorithms with divisions by 2 only by considering polynomial multiplications modulo cyclotomic divisors of X-_ 1. The best upper bound on m.m.c.…”

mentioning

confidence: 99%

“…Then (up to the choice of the basis of X over k), the multiplication of elements of X over the field of scalars k is determined by the multiplication of polynomials mod p(t). We are led to the traditional, in the field of algebraic complexities, problem of multiplication of two polynomials x(t)y(t) mod p(t) (1)(2)(3) The rank of an n X n matrix M is equal to n -no, where no is a number of zero eigenvalues of M-i.e., for the characteristic polynomial Xm(X) = deg(X-In + M) = Xn + . .…”

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confidence: 99%

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Algebraic complexities and algebraic curves over finite fields

Chudnovsky¹,

Chudnovsky²

1987

Proc. Natl. Acad. Sci. U.S.A.

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We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, [169][170][171][172][173][174][175][176][177][178][179][180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields.The algebraic complexity problem that is richest in underlying structure is the problem of fast polynomial multiplication. Among the problems reducible to this are fast multiplication of multiple-precision numbers, greatest common divisors in polynomial rings, Hankel (and, in particular, over any finite field) that one can achieve using variations of the fast Fourier transformation method is 0(n log n). The corresponding scheme for A = Z is far from simple. To study the optimal Zalgorithm one has to study their reductions mod p and m.m.c. algorithms of polynomial multiplication over finite fields, particularly over F2. Over finite fields the m.m.c. algorithms of polynomial multiplication are not, in general, given by the Toom-Cook scheme. For example, /.k(m, n) 2 m + n -1 for an arbitrary field k of scalars, but the inequality becomes equality only when the field k has at least m + n -2 elements (4). Better lower bounds on m.m.c. can be deduced using the theory of error-correcting linear codes. In refs. 7 and 8 it was proved that Fu2(n, n) > 3.52 n for large n [e.g., gz(n, n) > 3.52 n], using the upper bound (9). In this paper we explore various connections between m.m.c. algorithms for polynomial multiplication and multiplication in finite extensions of fields and optimal linear codes.First, we improve on lower bounds on m.m.c. over finite fields. Among bilinear algorithms that we study are m.m.c. algorithms for multiplication in commutative k-algebras without zero divisors. Then we use the connection between the theory of linear codes and algebraic curves over finite fields [Goppa codes (10)]. We present all the relevant information from the theory of algebraic curves over finite fields. Our algorithms of polynomial multiplication can be interpreted as interpolation methods on algebraic curves. As a corollary of our results, we prove that Ak(m, n) = 0(m + n) for an arbitrary finite field k. Moreover, for the multiplicative complexity ptk(N) of multiplication in a finite extension X over k we obtain a bound /uk(N) -2(1 + c/IkI"'2)[2K:kI comparing favorably with our lower bounds. Our results generalize to lower and upper bounds on m.m.c. of multiplication in other algebraic structures over finite fields including group algebras. Section

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“…Let T be the set of coefficients of R¡ • Sm and let Tp be the set of coefficients of R¡ • Sm mod P. It was shown in [3] that at least I + m + 1 multiplications are needed to compute T (multiplication by a fixed element g E G is not counted), and using the algorithm of [4] one can actually obtain an algorithm for computing T using I + m + 1 multiplications. Clearly, we can obtain Tp from T using only additions and multiplications by elements g E G; thus, the number of multiplications, which are counted, needed to compute Tp is at most I + m + 1.…”

mentioning

confidence: 99%