feat(NumberTheory): the number-theoretic transform#41924
feat(NumberTheory): the number-theoretic transform#41924junjihashimoto wants to merge 3 commits into
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Define the Cayley-Dickson double of a (possibly non-associative) star ring, and show that it preserves NonAssocRing and StarRing, so the construction iterates: quaternions -> octonions -> sedenions -> ... The doubling unit l = (0, 1) satisfies l * (l * x) = -x at every level of the tower, by a computation using only that star is an involution (no alternativity is developed). Left multiplication by l is therefore a complex structure on every Cayley-Dickson algebra.
The discrete Fourier transform of functions ZMod N -> M for M a module over a commutative domain containing a primitive N-th root of unity: orthogonality, the Fourier inversion formula, and the bilinear Parseval identity (the finite-field replacement for Plancherel's theorem). The module can be a hypercomplex algebra, giving hypercomplex NTTs.
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PR summary 93b9e11055Import changes for modified filesNo significant changes to the import graph Import changes for all files
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The number-theoretic transform is the discrete Fourier transform of functions
ZMod N → M, whereMis a module over a commutative domainRcontaining a primitiveN-th root of unityζ(e.g.R = ZMod pwithN ∣ p - 1):ntt hζ f k = ∑ j, ζ ^ (j * k) • f j.We prove the character orthogonality relation (
sum_zmodChar_mul, fromAddChar.sum_eq_ite), the Fourier inversion formulanttInv_ntt/ntt_nttInv(so the transform is bijective wheneverNis invertible inR), and the bilinear Parseval identitysum_bilin_ntt_ntt_neg— over a general coefficient ring there is no norm or conjugation, and this is the correct finite-field replacement for Plancherel's theorem.The module
Mis arbitrary; takingMto be a hypercomplex algebra overR(e.g. the octonions or sedenions overZMod pfrom #41919, whence the dependency) yields hypercomplex number-theoretic transforms, recorded as an example. This is the finite-field counterpart of the archimedean theory in #41921/#41922.The NTT is the transform underlying polynomial multiplication in lattice-based cryptography (e.g. ML-KEM/Kyber), so this also provides groundwork for formalizing those schemes.