Elliptic Curve Fast Fourier Transform (ECFFT) Part I: Low-degree Extension in Time O(n log n) over all Finite Fields

Given disjoint sets S, S^′ ⊆ Fq of size n and a function f : S → Fq, where Fq is a finite field, the low-degree extension (LDE) of f to S^′ is the function f^′ : S^′ → F_q obtained by restricting the interpolating polynomial of f to S^′. LDE computation is a fundamental primitive of modern algebraic coding theory and cryptography. The best asymptotic running time for LDE with parameter n is O(n log n) arithmetic operations over F_q - when q and the sets S, S^′ are special. This running time is achieved via the Fast Fourier Transform (FFT), and requires F_q to contain a multiplicative subgroup of smooth order ≥ n (smoothness means being the product of small primes). Another variant uses an additive subgroup of smooth order ≥ n. Most finite fields do not contain such a subgroup, which raises the question of computing the LDE in time O(n · log n) over general finite fields, for some disjoint pair of sets S, S^′ of size n. The main result of this paper is a positive answer to this question, presenting O(n log n)-time LDE for special S, S^′ shown to exist over all fields, as long as q = Ω(n²). This result is achieved by introducing a new FFT-like transform, the Elliptic Curve Fast Fourier Transform (ECFFT), which gives an approach to fast algorithms (using preprocessing) 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_q suitably related to a well-chosen elliptic curve group over F_q (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 ±2 √q + 1] and thus can have subgroups of large, smooth order, which an FFT-like divide and conquer algorithm can exploit. Compare this with multiplicative subgroups over F_q whose order must divide q - 1. By analogy, our method extends the standard, multiplicative FFT in a similar way to how Lenstra's elliptic curve method [Len87] extended Pollard's p - 1 algorithm [Pol74] for factoring integers. Representing polynomials by their evaluation over (well-chosen) subsets of L, we use the ECFFT to compute the LDE in time O(n log n). We also give small arithmetic circuits for polynomial multiplication, division, degree-computation, interpolation, evaluation and Reed-Solomon encoding (also known as low-degree extension) with fixed evaluation points, matching the circuit size of classical FFT-based algorithms when the field size q is special. For the classical problems (in the standard representation) of low degree extension with chosen evaluation points, and evaluating elementary symmetric polynomials, this yields the asymptotically smallest known arithmetic circuits. The efficiency of the classical FFT follows from using the 2-to-1 squaring map to reduce the evaluation set of roots of unity of order 2^k to similar groups of size 2^k-i, i > 0. Our algorithms operate similarly, using isogenies of elliptic curves with kernel size 2 as 2-to-1 maps to reduce L of size 2^k to sets of size 2^k-i that are, like L, suitably related to elliptic curves, albeit different ones.