??? ?? ? NTT(??? DFT) ???

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NTT? ??? ???? ?????. ??(?? ?? ?? ?? ??). ??? ??? ?? ? ??? ???? ????. .. 12000?? ??.

? ??? ??? ????.

1. ???? NTT ??? ????? ??? ???? ?? ???(???)?? ??? ???? ?? ????. ??? ?? ??? ????? ?????.


2. ?? ?? ??? ?? ? ?? ??? ????

??? NTT? C ?? ?????(???? ?? ?????? ?? ?? 100% C?? ???? ?????? ???? ???. ?? ??? ??? ????? ?? ?????!!! ?? ??? ????? ??? ?? ????.

???? ???

1. ?? ?? ?? ????: NTT ???? ?? W? iW? ????(?? ???? ? ??)? ?? ???? ???? ???? ?????. ??? ?? ??? ??? ??? ?? ???? ?? ??? ?????.
2. ?? ???: NTT ?????? ??? ????? ?? ??? ??? ????? ????. ??? ?? ?? ?? ?? ?? ??? ??? ? ????.
3. ??? ?? ???: ?? ?? ? ???? ??? ???? ??? ?? ??(??, ??, ?? ? ??)? ????? ?????. . ??? ?? ???? ?? ? ???? ???? ? ??? ??? ? ????.

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??? ?? ??? ????? ?? ??? ???? C?? ???? NTT ??? ????.

class NTT {
public:
    NTT() {
        // Initialize constants
        p = 0xc0000001;
        W = modpow(2, 0x30000000 / n);
        iW = modpow(2, p - 1 - 0x30000000 / n);
        rN = modpow(n, p - 2);
        NN = n >> 1;

        // Precompute W and iW powers
        WW = new uint32_t[n];
        iWW = new uint32_t[n];
        WW[0] = 1;
        iWW[0] = 1;
        for (uint32_t i = 1; i < n; i++) {
            WW[i] = modmul(WW[i - 1], W);
            iWW[i] = modmul(iWW[i - 1], iW);
        }
    }

    void NTT(uint32_t *dst, uint32_t *src, uint32_t n) {
        if (n > 0) {
            // Reorder even, odd elements
            for (uint32_t i = 0, j = 0; i < NN; i++, j += 2) {
                dst[i] = src[j];
            }
            for (j = 1; i < n; i++, j += 2) {
                dst[i] = src[j];
            }

            // Recursive NTT
            NTT(src, dst, NN);  // Even
            NTT(src + NN, dst + NN, NN);  // Odd

            // Restore results
            for (uint32_t i = 0, j = NN; i < NN; i++, j++) {
                uint32_t a0 = src[i];
                uint32_t a1 = modmul(src[j], WW[i]);
                dst[i] = modadd(a0, a1);
                dst[j] = modsub(a0, a1);
            }
        }
    }

private:
    uint32_t p, n, NN, W, iW, rN;
    uint32_t *WW, *iWW;

    // Modular arithmetic operations
    inline uint32_t modadd(uint32_t a, uint32_t b) {
        uint32_t d = a + b;
        if (d >= p) d -= p;
        return d;
    }

    inline uint32_t modsub(uint32_t a, uint32_t b) {
        uint32_t d = a - b;
        if (d > a) d += p;
        return d;
    }

    inline uint32_t modmul(uint32_t a, uint32_t b) {
        uint32_t m = (uint64_t)a * b;
        return m - (p * (m / p));
    }

    inline uint32_t modpow(uint32_t a, uint32_t b) {
        if (b == 0) return 1;
        uint32_t t = modpow(a, b / 2);
        t = modmul(t, t);
        if (b &amp; 1) t = modmul(t, a);
        return t;
    }
};

???

• ???? C? ?? ?? ?? ? ??? ????? ???? ?? ??.
• ?????? ???? ??? ?? ??? ???? ?? ??? ???? ?????.
• NTT ???? ???? ?????. ?? ??? ?? SIMD ???? ?????.

? ??? ?? ?? ? ?(?: 12000?? ??)? ?? ? ?? ??? ?? NTT(?? ??) ? ??? ??? ?????? ??? ?? ????? ?? ?????. ??? ??? PHP ??? ????? ?? ?? ??? ?????!