Abstract: Although Strassen method is not the
asymptotically faster matrix multiplication known, it is the most
widely used for large matrices on finite fields. After his first
paper, some variant have been proposed, with different additive
complexity, here we describe a new one.
The new variant is as
good as those already known for a simple matrix multiplication, but
can save operations either when more than two matrices are to be
multiplied or for squaring. Moreover it can be proved optimal for this
tasks.
The biggest gain is shown for nth-power computation,
in this scenario the additive complexity can be halved, with respect
to original Strassen's.
Published: Preprint N.622 Centro Interdipartimentale "Vito Volterra" - Università di Roma "Tor Vergata" (2008)
Download: DVI-paper (25 kB), PDF-paper (247 kB), DjVu-paper (71 kB), software.
Abstract:
Toom-Cook strategy is a well-known method for building algorithms to
efficiently multiply dense univariate polynomials. Efficiency of the
algorithm depends on the choice of interpolation points and on the
exact sequence of operations for evaluation and interpolation. If
carefully tuned, it gives the fastest algorithm for a wide range of
inputs. This work smoothly extends the Toom strategy to polynomial
rings, with a focus on
GF(2)[x]. Moreover a method is
proposed to find the faster Toom multiplication algorithm for any
given splitting order. New results found with it, for polynomials in
characteristic 2, are presented.
A new extension for
multivariate polynomials is also introduced; through a new definition
of density leading Toom strategy to be efficient.
Published: In C.Carlet and B.Sunar, editors, WAIFI'07 proceedings, volume 4547 of LNCS, pages 116-133. Springer, Madrid, España, June 21-22, 2007.
Also known as: Searching Optimal Toom-Cook Algorithms for Polynomials in Characteristic 2 and 0 (working title).
Download: DVI-paper (38 kB), PDF-paper (320 kB), DjVu-paper (132 kB), PDF-slides (380K), TeX-slides (7K), BibTex-entry, software.
Abstract: The use of Toom-Cook sub-quadratic
polynomial multiplication was recently shown to be possible also when
the coefficient field does not have elements enough, particularly for
𝔽2[x]. This paper focus on how Toom's
strategies can be adapted to polynomials on non-binary small
fields. In particular we describe the Toom-3 algorithm for
𝔽p[x] with p∈{3, 5, 7}, and
some other unbalanced algorithms.
Algorithms are described with full details, not only the asymptotic
complexity is given, but the exact sequence of operations for possibly
optimal implementations.
Algorithms given here for
𝔽p[x] perfectly
works, as well, for
𝔽pn[x],
and may be useful for computations inside
𝔽pn for large enough
n.
Moreover some connections with FFT-based multiplication are found,
slightly improving the worst cases of FFT.
Published: Preprint Centro Interdipartimentale "Vito Volterra" - Università di Roma "Tor Vergata" (2007)
Download: DVI-paper (32 kB), PDF-paper (307 kB), DjVu-paper (104 kB), BibTex-entry, software.
Abstract: We improve our proposal of a new variant of the McEliece cryptosystem based on QC-LDPC codes. The original McEliece cryptosystem, based on Goppa codes, is still unbroken up to now, but has two major drawbacks: long key and low transmission rate. Our variant is based on QC-LDPC codes and is able to overcome such drawbacks, while avoiding the known attacks. Recently, however, a new attack has been discovered that can recover the private key with limited complexity. We show that such attack can be avoided by changing the form of some constituent matrices, without altering the remaining system parameters. We also propose another variant that exhibits an overall increased security level. We analyze the complexity of the encryption and decryption stages by adopting efficient algorithms for processing large circulant matrices. The Toom-Cook algorithm and the short Winograd convolution are considered, that give a significant speed-up in the cryptosystem operations.
Published: in Proceedings of the Sixth Conference SCN ,volume 5229 of LNCS, pages 246-262. Springer, Amalfi, Italy, September 10-12, 2008.
Download: DVI-paper (34 kB), PDF-paper (251 kB), DjVu-paper (111 kB), PDF-slides (523K), BibTex-entry, software.
Ask for a copy: waiting for expiration of publisher restrictions, ask for a copy of the paper to one of the authors: Marco Bodrato <>.
Abstract: Karatsuba and Toom-Cook are well-known methods used to multiply efficiently two long integers. There have been different proposal about the interpolating values used to determine the matrix to be inverted and the sequence of operations to invert it. A definitive word about which is the optimal matrix (values) and the (number of) basic operations to invert it seems still not to have been said. In this paper we present some particular examples of useful matrices and a method to generate automatically, by means of optimised exhaustive searches on a graph, the best sequence of basic operations to invert them.
Published: Preprint N.605 Centro Interdipartimentale "Vito Volterra" - Università di Roma "Tor Vergata" (2006)
Published (full revised version): Integer and Polynomial Multiplication: Towards Optimal Toom-Cook Matrices in Proceedings of the ISSAC 2007 conference, Ontario, Canada, July 29-August 1, 2007, ACM press.
Download: PDF-preprint (263 kB), DjVu-preprint (104 kB), PDF-paper (287 kB), PDF-slides (475kB), BibTex-entry, software.
Abstract: In Gröbner bases computation, as in other algorithms in commutative algebra, a general open question is how to guide the calculations coping with numerical coefficients and/or not exact input data. It often happens that, due to error accumulation and/or insufficient working precision, the obtained result is not one expects from a theoretical derivation. The resulting basis may have more or less polynomials, a different number of solution, roots with different multiplicity, another Hilbert function, and so on. Augmenting precision we may overcome algorithmic errors, but one does not know in advance how much this precision should be, and a trial-and-error approach is often the only way to follow. Coping with initial errors is an even more difficult task. In this experimental work we propose the combined use of syzygies and interval arithmetic to decide what to do at each critical point of the algorithm.
Published: Proceedings workshop CASC 2006. V.G. Ganzha, E.W. Mayr, E.V. Vorozhtsov ed. Springer-Verlag LNCS 4194. Chişinau, Moldova 2006, pp. 64-76, ISBN: 3-540-45182-X
Download: PDF-paper (250 kB), DjVu-paper (87 kB), PDF-slides (538K), BibTex-entry,
software.
Abstract: In Gröbner bases computation a general open question is how to guide calculations coping with numerical coefficients and/or not exact input data. It may happen that, due to error accumulation or insufficient working precision, the result is not one theoretically expects. The basis may have more or less polynomials, a different number of solutions, a zero set with wrong multiplicity, and so on. Augmenting precision we may overcome algorithmic errors, but we don't know in advance how much it should be, and a trial-and-error approach is often the only way. Coping with initial errors is an even more difficult task. In this work the combined use of syzygies and interval arithmetic is proposed as a technique to decide at each critical point of the algorithm what to do.
Published: (reduced version) Proceedings of the 6th SYNASC Symposium. D. Petcu, D. Zaharie, V. Negru, T. Jebelean ed. MIRTON, Timişoara, Romania 2004, pp. 77 - 89, ISBN 973-661-441-7
Published: (full version) Analele Universitatii din Timisoara, Vol. XLII, Fasc.special 2, T. Jebelean, V. Negru, A. Popovici ed., Timişoara, Romania 2004, pp. 13 - 30, ISSN 1224-970X
Download: PDF-paper, BibTex-entry, software.
Marco Bodrato -
27 marzo 2009