Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6861183 | Journal of Symbolic Computation | 2018 | 26 Pages |
Abstract
This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial degree s, independent of the underlying field extension degree m. We show that our multiplication algorithm is faster than all known ones when sâ¤m. Using a result by Caruso and Le Borgne (2017), this immediately implies a sub-quadratic division algorithm for linearized polynomials for arbitrary polynomial degree s. Also, we propose algorithms with sub-quadratic complexity for the q-transform, multi-point evaluation, computing minimal subspace polynomials, and interpolation, whose implementations were at least quadratic before. Using the new fast algorithm for the q-transform, we show how matrix multiplication over a finite field can be implemented by multiplying linearized polynomials of degrees at most s=m if an elliptic normal basis of extension degree m exists, providing a lower bound on the cost of the latter problem. Finally, it is shown how the new fast operations on linearized polynomials lead to the first error and erasure decoding algorithm for Gabidulin codes with sub-quadratic complexity.
Keywords
Related Topics
Physical Sciences and Engineering
Computer Science
Artificial Intelligence
Authors
Sven Puchinger, Antonia Wachter-Zeh,