Modular Composition Modulo Triangular Sets and Applications

We generalize Kedlaya and Umans’ modular composition algorithm to the multivariate case. As a main application, we give fast algorithms for many operations involving triangular sets (over a finite field), such as modular multiplication, inversion, or change of order. For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead exponential in the number of variables. As a further application, we show that, from the complexity viewpoint, Charlap, Coley, and Robbins’ approach to elliptic curve point counting can be competitive with the better known approach due to Elkies.

[1]  Victor Shoup,et al.  Fast construction of irreducible polynomials over finite fields , 1994, SODA '93.

[2]  Erich Kaltofen,et al.  Greatest common divisors of polynomials given by straight-line programs , 1988, JACM.

[3]  Joachim von zur Gathen,et al.  Computing Frobenius maps and factoring polynomials , 2005, computational complexity.

[4]  Bican Xia,et al.  A complete algorithm for automated discovering of a class of inequality-type theorems , 2001, Science in China Series F Information Sciences.

[5]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[6]  W. Cooper,et al.  SCIENCE IN CHINA. , 1940, Science.

[7]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[8]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

[9]  Philippe Flajolet,et al.  Fast computation of special resultants , 2006, J. Symb. Comput..

[10]  Éric Schost,et al.  Tellegen's principle into practice , 2003, ISSAC '03.

[11]  Joris van der Hoeven,et al.  Homotopy techniques for multiplication modulo triangular sets , 2009, J. Symb. Comput..

[12]  Éric Schost,et al.  Computing the eigenvalue in the schoof-elkies-atkin algorithm using abelian lifts , 2007, ISSAC '07.

[13]  Alexei Yu. Uteshev,et al.  On the Bézout Construction of the Resultant , 1999, J. Symb. Comput..

[14]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[15]  Christopher Umans,et al.  Fast Polynomial Factorization and Modular Composition , 2011, SIAM J. Comput..

[16]  M. M. Maza.  On Triangular Decompositions of Algebraic Varieties , 2000 .

[17]  H. T. Kung,et al.  Fast Algorithms for Manipulating Formal Power Series , 1978, JACM.

[18]  Christopher Umans,et al.  Fast polynomial factorization and modular composition in small characteristic , 2008, STOC.

[19]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[20]  Evelyne Hubert,et al.  Notes on Triangular Sets and Triangulation-Decomposition Algorithms I: Polynomial Systems , 2001, SNSC.

[21]  Erich Kaltofen,et al.  Improved Sparse Multivariate Polynomial Interpolation Algorithms , 1988, ISSAC.

[22]  A. J. Stothers.  On the complexity of matrix multiplication , 2010 .

[23]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[24]  Victor Shoup.  A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic , 1991, ISSAC '91.

[25]  Victor Shoup,et al.  New algorithms for finding irreducible polynomials over finite fields , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[26]  Giovanni Manzini,et al.  Inversion of two level circulant matrices over Zp , 2003 .

[27]  Marie-Françoise Roy,et al.  Zeros, multiplicities, and idempotents for zero-dimensional systems , 1996 .

[28]  Marc Moreno Maza,et al.  On the Theories of Triangular Sets , 1999, J. Symb. Comput..

[29]  Marc Moreno Maza,et al.  On the complexity of the D5 principle , 2005, SIGS.

[30]  J. E. Morais,et al.  Straight--Line Programs in Geometric Elimination Theory , 1996, alg-geom/9609005.

[31]  Marc Giusti,et al.  A Gröbner Free Alternative for Polynomial System Solving , 2001, J. Complex..

[32]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[33]  Éric Schost,et al.  Complexity results for triangular sets , 2003, J. Symb. Comput..

[34]  Victor Y. Pan.  Simple Multivariate Polynomial Multiplication , 1994, J. Symb. Comput..

[35]  Michael Kalkbrener,et al.  A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties , 1993, J. Symb. Comput..

[36]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[37]  R. Schoof,et al.  Elliptic curves over finite fi elds and the computation of the square roots modulo p , 1984 .

[38]  Victor Y. Pan,et al.  Fast Rectangular Matrix Multiplication and Applications , 1998, J. Complex..

[39]  Éric Schost,et al.  Change of order for bivariate triangular sets , 2006, ISSAC '06.

[40]  Marc Moreno Maza,et al.  Fast arithmetic for triangular sets: from theory to practice , 2007, ISSAC '07.

[41]  R. Lercier,et al.  On Elkies subgroups of $\ell $-torsion points in elliptic curves defined over a finite field , 2008 .

[42]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[43]  Daniel Reischert.  Asymptotically fast computation of subresultants , 1997, ISSAC.

[44]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[45]  Joris van der Hoeven,et al.  Homotopy methods for multiplication modulo triangular sets , 2009, ArXiv.

[46]  Patrizia M. Gianni,et al.  Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases , 1987, AAECC.

[47]  Erich Kaltofen,et al.  Challenges of Symbolic Computation: My Favorite Open Problems , 2000, J. Symb. Comput..

[48]  Joachim von zur Gathen,et al.  Functional Decomposition of Polynomials: The Tame Case , 1990, J. Symb. Comput..