- Author:
- Todd Mateer,
- Adviser:
- Shuhong Gao
This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the “Fast Fourier Transform” as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the “Fast Fourier Transform” as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the “Fast Fourier Transform” that can also be understood by someone who has an understanding of the topic from the engineering perspective.
The manuscript will also introduce several new algorithms that solve the fast multipoint evaluation problem over certain finite fields and require fewer finite field operations than existing techniques. The document will also demonstrate that these new algorithms can be used to multiply polynomials with finite field coefficients with fewer operations than Schönhage's algorithm in most circumstances.
A third objective of this document is to provide a mathematical perspective of several algorithms which can be used to multiply polynomials of size which is not a power of two. Several improvements to these algorithms will also be discussed.
Finally, the document will describe several applications of the “Fast Fourier Transform” algorithms presented and will introduce improvements in several of these applications. In addition to polynomial multiplication, the applications of polynomial division with remainder, the greatest common divisor, decoding of Reed-Solomon error-correcting codes, and the computation of the coefficients of a discrete Fourier Series will be addressed.
Cited By
Larrieu R The Truncated Fourier Transform for Mixed Radices Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, (261-268)- Arnold A and Schost É (2015). A Truncated Fourier Transform middle product, ACM Communications in Computer Algebra, 48:3/4, (98-99), Online publication date: 5-Feb-2015.
- Meng L and Johnson J High performance implementation of the inverse TFT Proceedings of the 2015 International Workshop on Parallel Symbolic Computation, (87-94)
- Ben-Sasson E, Chiesa A, Genkin D and Tromer E Fast reductions from RAMs to delegatable succinct constraint satisfaction problems Proceedings of the 4th conference on Innovations in Theoretical Computer Science, (401-414)
- Setty S, Braun B, Vu V, Blumberg A, Parno B and Walfish M Resolving the conflict between generality and plausibility in verified computation Proceedings of the 8th ACM European Conference on Computer Systems, (71-84)
- Arnold A A new truncated fourier transform algorithm Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, (15-22)
- Ben-Sasson E, Chiesa A, Genkin D and Tromer E On the concrete efficiency of probabilistically-checkable proofs Proceedings of the forty-fifth annual ACM symposium on Theory of Computing, (585-594)
- Chen Y and Nguyen P Faster algorithms for approximate common divisors Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques, (502-519)
- Gao S and Mateer T (2010). Additive fast Fourier transforms over finite fields, IEEE Transactions on Information Theory, 56:12, (6265-6272), Online publication date: 1-Dec-2010.
- De Feo L and Schost É Fast arithmetics in artin-schreier towers over finite fields Proceedings of the 2009 international symposium on Symbolic and algebraic computation, (127-134)
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