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  "body": "\n## Statement\n\nLet $p$ be a prime, let $\\omega = e^{-2\\pi i/p}$, and let $g$ be a [[primitive root modulo $p$|0F33KX9Q6F]].\nFor an integer $j$ write $g^{j}$ and $g^{-j}$ for the residues in $\\set{1, \\ldots, p-1}$ of these powers modulo $p$.\nLet $x \\in \\C^p$ and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \\sum_{n=0}^{p-1} x_n \\omega^{nk}$.\nDefine sequences of period $p - 1$ by $a_m = x_{g^m}$ and $c_m = \\omega^{g^{-m}}$.\nThen $X_0 = \\sum_{n=0}^{p-1} x_n$, and for $0 \\le q \\le p-2$,\n$$X_{g^{-q}} = x_0 + \\sum_{m=0}^{p-2} a_m\\, c_{q-m}.$$\nSo a DFT of prime length $p$ reduces to a cyclic [[convolution|2Z29JN3VES]] of length $p - 1$ and $O(p)$ further additions.\n\n## Remarks\n\nSince $g$ generates the multiplicative group modulo $p$, the maps $m \\mapsto g^m$ and $q \\mapsto g^{-q}$ are bijections from $\\set{0, \\ldots, p-2}$ onto $\\set{1, \\ldots, p-1}$.\nFor $n = g^m$ and $k = g^{-q}$ one has $nk \\equiv g^{-(q-m)} \\pmod p$, and $\\omega^{nk}$ depends only on $nk$ modulo $p$; this gives the formula.\nThe cyclic convolution of length $p-1$ can be computed by DFTs of length $p - 1$, which is composite for $p \u003e 3$.\nThese can use the [[Cooley–Tukey algorithm|9N8D9E5TB2]], possibly after zero padding to a power of $2$.\nExample: $p = 5$ and $g = 2$, so $(g^0, g^1, g^2, g^3) = (1, 2, 4, 3)$ and $(g^0, g^{-1}, g^{-2}, g^{-3}) = (1, 3, 4, 2)$.\nThe hypothesis that $p$ is prime gives a primitive root, and makes the nonzero indices a group under multiplication.\nFor composite lengths the nonzero indices do not form such a group.\n\n## Explanations\n\n### L1 human\n\nFast Fourier transforms usually split the length into factors, which fails when the length is prime.\nRader noticed that for a prime length the nonzero indices can be listed as powers of one number $g$.\nIn that order, the transform becomes a convolution, and convolutions can be done fast with other FFTs.\n\n### L2 human\n\nThe algorithm uses the structure of $(\\Z/p\\Z)^\\times$, which is cyclic of order $p - 1$.\nPermuting inputs by $n = g^m$ and outputs by $k = g^{-q}$ turns the $(p-1) \\times (p-1)$ block of the DFT matrix into a circulant matrix.\nCirculant matrices are diagonalised by the DFT, which is why the convolution can be done in $O(p \\log p)$ operations.\nWinograd's short-length FFT modules are built on the same reduction.\nBluestein's algorithm is an alternative that works for every length and does not need a primitive root.\n",
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