# Fast integer multiplication using \goodbreak generalized Fermat primes

@article{Covanov2019FastIM, title={Fast integer multiplication using \goodbreak generalized Fermat primes}, author={Svyatoslav Covanov and Emmanuel Thom{\'e}}, journal={Math. Comput.}, year={2019}, volume={88}, pages={1449-1477} }

For almost 35 years, Schonhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n · log n · log log n) for multiplying n-bit inputs. In 2007, Furer proved that there exists K > 1 and an algorithm performing this operation in O(n · log n · K log n). Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get K = 8, and conjecturally K = 4. Using an alternative algorithm, which…

## 4 Citations

Polynomial Multiplication over Finite Fields in Time \( O(n \log n \)

- Computer Science, MathematicsJ. ACM
- 2022

It is shown that polynomials of degree less than n over a finite field with q elements can be multiplied in time, uniformly in q, and how to multiply two n -bit integers in time is shown.

Brief Announcement: On the I/O Complexity of Sequential and Parallel Hybrid Integer Multiplication Algorithms

- Computer ScienceProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures
- 2022

This work presents an Ω((n/max(M,n0) (max(1,n_0/M))2M) lower bound for the I/O complexity of a class of "uniform, non-stationary" hybrid algorithms, where n0 denotes the threshold size of sub-problems which are computed using standard algorithms with algebraic complexity Ω(n2).

On the I/O complexity of hybrid algorithms for Integer Multiplication

- Computer Science, MathematicsArXiv
- 2019

The lower bound is derived for the more general class of "non-uniform, non-stationary" hybrid algorithms which allow recursive calls to have a different structure, even when they refer to the multiplication of integers of the same size and in the same recursive level including those where the value of $k$ is allowed to vary with the level of recursion.

Big Prime Field FFT on Multi-core Processors

- Computer Science, MathematicsISSAC
- 2019

A multi-threaded implementation of Fast Fourier Transforms over generalized Fermat prime fields is reported on, overcome the less fine control of hardware resources by successively using FFT in support of the multiplication in those fields.

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