Essentially optimal sparse polynomial multiplication

We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of characteristic zero or larger than the degree. We mainly rely on sparse interpolation and on a new algorithm for verifying a sparse product that has also a quasi-linear time complexity. Using Kronecker substitution techniques we extend our result to the multivariate case.

[1]  Victor Shoup,et al.  A computational introduction to number theory and algebra , 2005 .

[2]  Andrew Arnold,et al.  Multivariate sparse interpolation using randomized Kronecker substitutions , 2014, ISSAC.

[3]  Joachim von zur Gathen,et al.  Modern Computer Algebra (3. ed.) , 2003 .

[4]  Daniel S. Roche.  Chunky and equal-spaced polynomial multiplication , 2010, J. Symb. Comput..

[5]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[6]  Harvey,et al.  Integer multiplication in time O(n log n) , 2021, Annals of Mathematics.

[7]  Daniel S. Roche.  What Can (and Can't) we Do with Sparse Polynomials? , 2018, ISSAC.

[8]  Andrew Arnold,et al.  Sparse Polynomial Interpolation and Testing , 2016 .

[9]  Michael B. Monagan,et al.  Parallel sparse polynomial multiplication using heaps , 2009, ISSAC '09.

[10]  Pascal Giorgi,et al.  A probabilistic algorithm for verifying polynomial middle product in linear time , 2018, Inf. Process. Lett..

[11]  Michael B. Monagan,et al.  Sparse polynomial division using a heap , 2011, J. Symb. Comput..

[12]  Joris van der Hoeven,et al.  Faster polynomial multiplication over finite fields using cyclotomic coefficient rings , 2019, J. Complex..

[13]  Joris van der Hoeven,et al.  On the complexity of multivariate blockwise polynomial multiplication , 2012, ISSAC.

[14]  David A. Plaisted,et al.  New NP-hard and NP-complete polynomial and integer divisibility problems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[15]  Qiao-Long Huang.  Sparse Polynomial Interpolation over Fields with Large or Zero Characteristic , 2019, ISSAC.

[16]  Joris van der Hoeven,et al.  On the bit-complexity of sparse polynomial and series multiplication , 2013, J. Symb. Comput..

[17]  Andrew Chi-Chih Yao,et al.  On the Evaluation of Powers , 1976, SIAM J. Comput..

[18]  Joris van der Hoeven,et al.  Structured FFT and TFT: symmetric and lattice polynomials , 2013, ISSAC '13.

[19]  Vasileios Nakos.  Nearly Optimal Sparse Polynomial Multiplication , 2020, IEEE Transactions on Information Theory.

[20]  Stephen C. Johnson,et al.  Sparse polynomial arithmetic , 1974, SIGS.

[21]  Andrew Arnold,et al.  Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication , 2015, ISSAC.

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

[23]  Xiao-Shan Gao,et al.  Revisit Sparse Polynomial Interpolation Based on Randomized Kronecker Substitution , 2017, CASC.

[24]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[25]  Richard Cole,et al.  Verifying candidate matches in sparse and wildcard matching , 2002, STOC '02.