Abstract
We describe a unified framework to search for optimal formulae evaluating bilinear or quadratic maps. This framework applies to polynomial multiplication and squaring, finite field arithmetic, matrix multiplication, etc. We then propose a new algorithm to solve problems in this unified framework. With an implementation of this algorithm, we prove the optimality of various published upper bounds, and find improved upper bounds.
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References
Albrecht, M.R.: The M4RIE library for dense linear algebra over small fields with even characteristic (2011) (preprint), http://arxiv.org/abs/1111.6900
Artin, M.: Algebra. Prentice-Hall, Inc. (1991)
Bläser, M.: On the complexity of the multiplication of matrices of small formats. Journal of Complexity 19, 43–60 (2003)
Bürgisser, P., Clausen, M., Shokrollahi, M.: Algebraic complexity theory, vol. 315. Springer (1997)
Cenk, M., Özbudak, F.: Improved polynomial multiplication formulas over \({\mathbb F}_2\) using Chinese remainder theorem. IEEE Trans. Comput. 58(4), 572–576 (2009)
Cenk, M., Özbudak, F.: Efficient Multiplication in \({\mathbb F}_{3^{\ell m}}\), m ≥ 1 and 5 ≤ ℓ ≤ 18. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 406–414. Springer, Heidelberg (2008)
Cenk, M., Özbudak, F.: On multiplication in finite fields. J. Complexity 26, 172–186 (2010)
Cenk, M., Özbudak, F.: Multiplication of polynomials modulo x n. Theoret. Comput. Sci. 412, 3451–3462 (2011)
Chung, J., Hasan, M.A.: Asymmetric squaring formulae. In: Kornerup, P., Muller, J.M. (eds.) Proc. ARITH 18, pp. 113–122 (2007)
Comon, P., Golub, G., Lim, L., Mourrain, B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. & Appl. 30(3), 1254–1279 (2008)
Courtois, N.T., Bard, G.V., Hulme, D.: A new general-purpose method to multiply 3 ×3 matrices using only 23 multiplications (2011) (preprint), http://arxiv.org/abs/1108.2830
Fan, H., Hasan, A.: Comments on five, six, and seven-term Karatsuba-like formulae. IEEE Trans. Comput. 56(5), 716–717 (2007)
Hanrot, G., Quercia, M., Zimmermann, P.: The middle product algorithm, I. Speeding up the division and square root of power series. Appl. Algebra Engrg. Comm. Comput. 14(6), 415–438 (2004)
Hastad, J.: Tensor rank is NP-complete. J. Algorithms 11(4), 644–654 (1990)
Karatsuba, A.A., Ofman, Y.: Multiplication of multi-digit numbers on automata. Doklady Akad. Nauk SSSR 145(2), 293–294 (1962) (in Russian); translation in Soviet Physics-Doklady 7, 595–596 (1963)
Montgomery, P.: Five, six, and seven-term Karatsuba-like formulae. IEEE Trans. Comput. 54(3), 362–369 (2005)
Mulders, T.: On short multiplications and divisions. Appl. Algebra Engrg. Comm. Comput. 11(1), 69–88 (2000)
Oseledets, I.: Optimal Karatsuba-like formulae for certain bilinear forms in GF(2). Linear Algebra and its Applications 429, 2052–2066 (2008)
Strassen, V.: Gaussian elimination is not optimal. Numerische Mathematik 13(4), 354–356 (1969)
Toom, A.: The complexity of a scheme of functional elements realizing the multiplication of integers. Soviet Mathematics Doklady 3, 714–716 (1963)
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Barbulescu, R., Detrey, J., Estibals, N., Zimmermann, P. (2012). Finding Optimal Formulae for Bilinear Maps. In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_12
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DOI: https://doi.org/10.1007/978-3-642-31662-3_12
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