Swastik Kopparty: Fast algorithms for polynomials over all finite fields via the Elliptic Curve Fast Fourier Transform (ECFFT)

Tuesday, October 5, 2021 - 4:00pm to 5:00pm
Email jmtaft@mit.edu for link
Swastik Kopparty, University of Toronto

For smooth finite fields Fq (i.e., when q−1 factors into small primes) the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division, interpolation and multi-point evaluation. However, the same operations over fields with no smooth order root of unity suffer from an asymptotic slowdown. The classical algorithm of Schonhage and Strassen incurred a multiplicative slowdown factor of log log n on top of the smooth case. Recent remarkable results of Harvey, van der Hoeven and Lecerf dramatically reduced this multiplicative overhead to exp(log∗(n)).

We introduce a new approach to fast algorithms for polynomial operations over all large finite fields. The key idea is to replace the group of roots of unity with a set of points L⊂F suitably related to a well-chosen elliptic curve group (the set L itself is not a group). The key advantage of this approach is that elliptic curve groups can be of any size in the Hasse-Weil interval [q+1±2sqrt(q)] and thus can have subgroups of large, smooth order, which an FFT-like divide and conquer algorithm (the ECFFT) can exploit. Compare this with multiplicative subgroups over Fq, whose order must divide q−1.

For polynomials represented by their evaluation over subsets of L, we show that multiplication, division, degree-computation, interpolation, evaluation and Reed-Solomon encoding (also known as low-degree extension) with fixed evaluation points can all be computed with arithmetic circuits of size similar to what is achievable with the classical FFTs when the field size is special. For several problems, this yields the asymptotically smallest known arithmetic circuits even in the standard monomial representation of polynomials.

Joint work with Eli Ben-Sasson, Dan Carmon and David Levit.