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Adaptability of additive FFT techniques to the p-adic setting

Determine whether additive FFT techniques, which crucially rely on positive characteristic and evaluate over finite additive subgroups, admit meaningful adaptations to the p-adic field ℚ_p or its finite-degree extensions, and, if so, construct and analyze such p-adic additive FFT algorithms.

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Background

Additive FFTs in positive characteristic evaluate polynomials over additive subgroups and form a separate family of fast transforms from multiplicative FFTs. They have seen substantial development over finite fields, notably in characteristic two.

Because ℚ_p has characteristic zero, it is unclear whether the structural properties enabling additive FFTs in positive characteristic carry over in any useful way. Clarifying this and, if possible, developing an analogue for p-adic fields would broaden the algorithmic toolbox for p-adic polynomial transforms.

References

On the other hand, additive FFTs crucially rely on the characteristic being positive, and it is unclear whether such techniques can be adapted in any meaningful way to the p-adic setting.

— Cooley-Tukey FFT over $\mathbb{Q}_p$ via Unramified Cyclotomic Extension (2505.02509 - Kondo, 5 May 2025) in Section 1, Subsection “Relation to existing algebraic FFTs,” bullet “The additive FFT”