Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method

Journal of Complexity

Volumn 28, Issue 4, 2012, Pages 489-517

  Randriambololona, Hugues ^a  

^a INSTITUT TELECOM   (France)

Author keywords

Algebraic curves; Finite fields; Interpolation; Multiplication algorithm; Non special divisors; Tensor rank

Indexed keywords

INTERPOLATION;

ALGEBRAIC CURVES; FINITE FIELDS; MULTIPLICATION ALGORITHMS; NON-SPECIAL DIVISORS; TENSOR RANKS;

ALGEBRA;

EID: 84861646697     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2012.02.005     Document Type: Article

References (38)

1. S. Ballet On the tensor rank of the multiplication in the finite fields J. Number Theory 128 2008 1795 1806
2. S. Ballet A note on the tensor rank of the multiplication in certain finite fields J. Chaumine, J. Hirschfeld, R. Rolland, Algebraic Geometry and its Applications, Proceedings of the First SAGA Conference Papeete, France, 7-11 May 2007 Ser. Number Theory Appl. vol. 5 2008 World Sci. Publ. 332 342
3. Fq J. Number Theory 116 2006 293 310
4. S. Ballet, D. Le Brigand, and R. Rolland On an application of the definition field descent of a tower of function fields F. Rodier, S. Vladut, Proceedings of the Conference "Arithmetic, Geometry and Coding Theory" (AGCT 2005) Séminaires et Congrès vol. 21 2010 Société Mathématique de France 187 203
5. 2 J. Complexity 27 2011 230 245
6. Fq Acta Arith. 143 2010 377 392
7. S. Ballet, and R. Rolland Multiplication algorithm in a finite field and tensor rank of the multiplication J. Algebra 272 2004 173 185
8. N. Bourbaki Éléments de Mathématique, Algèbre Commutative, Chapitres 8 et 9 1983 Masson Reprint: Springer-Verlag, 2006
9. R.W. Brocket, and D. Dobkin On the optimal evaluation of a set of bilinear forms Linear Algebra Appl. 19 1978 624 628
10. P. Bürgisser, M. Clausen, and A. Shokrollahi Algebraic Complexity Theory Grundlehren der Math. Wissenschaften vol. 315 1997 Springer-Verlag
11. I. Cascudo, R. Cramer, and C. Xing The torsion-limit for algebraic functions fields and its application to arithmetic secret sharing Ph. Rogaway, Advances in Cryptology - CRYPTO 2011 Lecture Notes in Comp. Science vol. 6841 2011 Springer-Verlag 685 705
12. I. Cascudo Pueyo, On asymptotically Good strongly multiplicative linear secret sharing, Tesis Doctoral, Universidad de Oviedo, 2010.
13. I. Cascudo Pueyo, Personal communication.
14. M. Cenk, and F. Özbudak On multiplication in finite fields J. Complexity 26 2010 172 186
15. J. Chaumine Multiplication in small finite fields using elliptic curves J. Chaumine, J. Hirschfeld, R. Rolland, Algebraic Geometry and its Applications, Proceedings of the First SAGA Conference Papeete, France, 7-11 May 2007 Ser. Number Theory Appl. vol. 5 2008 World Sci. Publ. 343 350
16. H. Chen, and R. Cramer Algebraic geometric secret sharing schemes and secure multi-party computations over small fields C. Dwork, Advances in Cryptology - CRYPTO 2006 Lecture Notes in Comp. Science vol. 4117 2006 Springer-Verlag 521 536
17. D.V. Chudnovsky, and G.V. Chudnovsky Algebraic complexities and algebraic curves over finite fields J. Complexity 4 1988 285 316
18. R. Cramer, I. Damgård, and U. Maurer Efficient general secure multi-party computation from any linear secret-sharing scheme B. Prenel, Advances in Cryptology - EUROCRYPT 2000 Lecture Notes in Comp. Science vol. 1807 2000 Springer-Verlag 316 334
19. H.F. de Groote Lectures on the Complexity of Bilinear Problems Lecture Notes in Comp. Science vol. 245 1987 Springer-Verlag
20. C. Fiduccia, and Y. Zalcstein Algebras having linear multiplicative complexities J. Assoc. Comput. Mach. 24 1977 311 331
21. A. Garcia, and H. Stichtenoth A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound Invent. Math. 121 1995 211 222
22. R. Hartshorne Algebraic Geometry Graduate Texts in Mathematics vol. 52 1977 Springer-Verlag
23. A., and 145 1962 293 294 English translation: A. Karatsuba, Yu. Ofman, Multiplication of multi-digit numbers on automata, Soviet Physics Doklady 7 (1963) 595-596
24. J.M. Landsberg Geometry and the complexity of matrix multiplication Bull. Amer. Math. Soc. 45 2008 247 284
25. A. Lempel, and S. Winograd A new approach to error-correcting codes IEEE Trans. Inform. Theory 23 1977 503 508
26. H. Niederreiter, and C. Xing Low-discrepancy sequences and global function fields with many rational places Finite Fields Appl. 2 1996 241 273
27. H. Randriam, Hecke operators with odd determinant and binary frameproof codes beyond the probabilistic bound?, in: Proc. of ITW 2010 Dublin - IEEE Information Theory Workshop, Dublin, Ireland, 2010.
28. H. Randriambololona, (2, 1) -separating systems beyond the probabilistic bound, Israel J. Math. (in press). http://arXiv.org/abs/1010.5764.
29. H. Randriambololona, Diviseurs de la forme 2D-G sans sections et rang de la multiplication dans les corps finis, Preprint. http://arXiv.org/abs/1103. 4335.
30. M.A. Shokrollahi Optimal algorithms for multiplication in certain finite fields using elliptic curves SIAM J. Comput. 21 1992 1193 1198
31. I. Shparlinski, M. Tsfasman, and S. Vladut Curves with many points and multiplication in finite fields H. Stichtenoth, M.A. Tsfasman, Coding Theory and Algebraic Geometry Luminy, 1991 Lecture Notes in Math. vol. 1518 1992 Springer-Verlag 145 169
32. H. Stichtenoth Algebraic Function Fields and Codes Universitext 1993 Springer-Verlag
33. V. Strassen Gaussian elimination is not optimal Numer. Math. 13 1969 354 356
34. V. Strassen Vermeidung von Divisionen J. Reine Angew. Math. 264 1973 184 202
35. V. Strassen Rank and optimal computation of generic tensors Linear Algebra Appl. 52-53 1983 645 685
36. W. Waterhouse Abelian varieties over finite fields Ann. Sci. Éc. Norm. Supér. 2 1969 521 560
37. S. Winograd Some bilinear forms whose multiplicative complexity depends on the field of constants Math. Syst. Theory 10 1977 169 180
38. C. Xing Asymptotic bounds on frameproof codes IEEE Trans. Inform. Theory 48 2002 2991 2995