# fft.h – Schoenhage-Strassen FFT

## Split/combine FFT coefficients

mp_size_t fft_split_limbs(mp_limb_t **poly, mp_srcptr limbs, mp_size_t total_limbs, mp_size_t coeff_limbs, mp_size_t output_limbs)

  Split an integer `(limbs, total_limbs)` into coefficients of length `coeff_limbs` limbs and store as the coefficients of `poly` which are assumed to have space for `output_limbs + 1` limbs per coefficient. The coefficients of the polynomial do not need to be zeroed before calling this function, however the number of coefficients written is returned by the function and any coefficients beyond this point are not touched.

mp_size_t fft_split_bits(mp_limb_t **poly, mp_srcptr limbs, mp_size_t total_limbs, flint_bitcnt_t bits, mp_size_t output_limbs)

  Split an integer `(limbs, total_limbs)` into coefficients of the given number of `bits` and store as the coefficients of `poly` which are assumed to have space for `output_limbs + 1` limbs per coefficient. The coefficients of the polynomial do not need to be zeroed before calling this function, however the number of coefficients written is returned by the function and any coefficients beyond this point are not touched.

void fft_combine_limbs(mp_limb_t *res, mp_limb_t **poly, slong length, mp_size_t coeff_limbs, mp_size_t output_limbs, mp_size_t total_limbs)

  Evaluate the polynomial `poly` of the given `length` at `B^coeff_limbs`, where `B = 2^FLINT_BITS`, and add the result to the integer `(res, total_limbs)` throwing away any bits that exceed the given number of limbs. The polynomial coefficients are assumed to have at least `output_limbs` limbs each, however any additional limbs are ignored.

  If the integer is initially zero the result will just be the evaluation of the polynomial.

void fft_combine_bits(mp_limb_t *res, mp_limb_t **poly, slong length, flint_bitcnt_t bits, mp_size_t output_limbs, mp_size_t total_limbs)

  Evaluate the polynomial `poly` of the given `length` at `2^bits` and add the result to the integer `(res, total_limbs)` throwing away any bits that exceed the given number of limbs. The polynomial coefficients are assumed to have at least `output_limbs` limbs each, however any additional limbs are ignored. If the integer is initially zero the result will just be the evaluation of the polynomial.

## Test helper functions

void fermat_to_mpz(mpz_t m, mp_limb_t *i, mp_size_t limbs)

  Convert the Fermat number `(i, limbs)` modulo `B^limbs + 1` to an `mpz_t m`. Assumes `m` has been initialised. This function is used only in test code.

## Arithmetic modulo a generalised Fermat number

void mpn_negmod_2expp1(mp_limb_t *z, const mp_limb_t *a, mp_size_t limbs)

  Set `z` to the negation of the Fermat number \(a\) modulo `B^limbs + 1`. The input `a` is expected to be fully reduced, and the output is fully reduced. Aliasing is permitted.

void mpn_addmod_2expp1_1(mp_limb_t *r, mp_size_t limbs, mp_limb_signed_t c)

  Adds the signed limb `c` to the generalised Fermat number `r` modulo `B^limbs + 1`. The compiler should be able to inline this for the case that there is no overflow from the first limb.

void mpn_normmod_2expp1(mp_limb_t *t, mp_size_t limbs)

  Given `t` a signed integer of `limbs + 1` limbs in two’s complement format, reduce `t` to the corresponding value modulo the generalised Fermat number `B^limbs + 1`, where `B = 2^FLINT_BITS`.

void mpn_mul_2expmod_2expp1(mp_limb_t *t, mp_limb_t *i1, mp_size_t limbs, flint_bitcnt_t d)

  Given `i1` a signed integer of `limbs + 1` limbs in two’s complement format reduced modulo `B^limbs + 1` up to some overflow, compute `t = i1*2^d` modulo \(p\). The result will not necessarily be fully reduced. The number of bits `d` must be nonnegative and less than `FLINT_BITS`. Aliasing is permitted.

void mpn_div_2expmod_2expp1(mp_limb_t *t, mp_limb_t *i1, mp_size_t limbs, flint_bitcnt_t d)

  Given `i1` a signed integer of `limbs + 1` limbs in two’s complement format reduced modulo `B^limbs + 1` up to some overflow, compute `t = i1/2^d` modulo \(p\). The result will not necessarily be fully reduced. The number of bits `d` must be nonnegative and less than `FLINT_BITS`. Aliasing is permitted.

## Generic butterflies

void fft_adjust(mp_limb_t *r, mp_limb_t *i1, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w)

  Set `r` to `i1` times \(z^i\) modulo `B^limbs + 1` where \(z\) corresponds to multiplication by \(2^w\). This can be thought of as part of a butterfly operation. We require \(0 \leq i < n\) where \(nw =\) `limbs*FLINT_BITS`. Aliasing is not supported.

void fft_adjust_sqrt2(mp_limb_t *r, mp_limb_t *i1, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w, mp_limb_t *temp)

  Set `r` to `i1` times \(z^i\) modulo `B^limbs + 1` where \(z\) corresponds to multiplication by \(\sqrt{2}^w\). This can be thought of as part of a butterfly operation. We require \(0 \leq i < 2\cdot n\) and odd where \(nw =\) `limbs*FLINT_BITS`.

void butterfly_lshB(mp_limb_t *t, mp_limb_t *u, mp_limb_t *i1, mp_limb_t *i2, mp_size_t limbs, mp_size_t x, mp_size_t y)

  We are given two integers `i1` and `i2` modulo `B^limbs + 1` which are not necessarily normalised. We compute `t = (i1 + i2)*B^x` and `u = (i1 - i2)*B^y` modulo \(p\). Aliasing between inputs and outputs is not permitted. We require `x` and `y` to be less than `limbs` and nonnegative.

void butterfly_rshB(mp_limb_t *t, mp_limb_t *u, mp_limb_t *i1, mp_limb_t *i2, mp_size_t limbs, mp_size_t x, mp_size_t y)

  We are given two integers `i1` and `i2` modulo `B^limbs + 1` which are not necessarily normalised. We compute `t = (i1 + i2)/B^x` and `u = (i1 - i2)/B^y` modulo \(p\). Aliasing between inputs and outputs is not permitted. We require `x` and `y` to be less than `limbs` and nonnegative.

## Radix 2 transforms

void fft_butterfly(mp_limb_t *s, mp_limb_t *t, mp_limb_t *i1, mp_limb_t *i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w)

  Set `s = i1 + i2`, `t = z1^i*(i1 - i2)` modulo `B^limbs + 1` where `z1 = exp(Pi*I/n)` corresponds to multiplication by \(2^w\). Requires \(0 \leq i < n\) where \(nw =\) `limbs*FLINT_BITS`.

void ifft_butterfly(mp_limb_t *s, mp_limb_t *t, mp_limb_t *i1, mp_limb_t *i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w)

  Set `s = i1 + z1^i*i2`, `t = i1 - z1^i*i2` modulo `B^limbs + 1` where `z1 = exp(-Pi*I/n)` corresponds to division by \(2^w\). Requires \(0 \leq i < 2n\) where \(nw =\) `limbs*FLINT_BITS`.

void fft_radix2(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2)

  The radix 2 DIF FFT works as follows:

  Input: `[i0, i1, ..., i(m-1)]`, for \(m = 2n\) a power of \(2\).

  Output: `[r0, r1, ..., r(m-1)]` `= FFT[i0, i1, ..., i(m-1)]`.

  Algorithm:

  \(\bullet\) Recursively compute `[r0, r2, r4, ...., r(m-2)]`
  `= FFT[i0+i(m/2), i1+i(m/2+1), ..., i(m/2-1)+i(m-1)]`
  \(\bullet\) Let `[t0, t1, ..., t(m/2-1)]`
  `= [i0-i(m/2), i1-i(m/2+1), ..., i(m/2-1)-i(m-1)]`
  \(\bullet\) Let `[u0, u1, ..., u(m/2-1)]`
  `= [z1^0*t0, z1^1*t1, ..., z1^(m/2-1)*t(m/2-1)]`
  where `z1 = exp(2*Pi*I/m)` corresponds to multiplication by \(2^w\).
  \(\bullet\) Recursively compute `[r1, r3, ..., r(m-1)]`
  `= FFT[u0, u1, ..., u(m/2-1)]`

  The parameters are as follows:

  \(\bullet\) `2*n` is the length of the input and output arrays

  \(\bullet\) \(w\) is such that \(2^w\) is an \(2n\)-th root of unity in the ring \(\mathbf{Z}/p\mathbf{Z}\) that we are working in, i.e. \(p = 2^{wn} + 1\) (here \(n\) is divisible by

    `GMP_LIMB_BITS`)

  \(\bullet\) `ii` is the array of inputs (each input is an

    array of limbs of length `wn/GMP_LIMB_BITS + 1` (the extra limbs being a “carry limb”). Outputs are written in-place.

  We require \(nw\) to be at least 64 and the two temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void fft_truncate(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t trunc)

  As for `fft_radix2` except that only the first `trunc` coefficients of the output are computed and the input is regarded as having (implied) zero coefficients from coefficient `trunc` onwards. The coefficients must exist as the algorithm needs to use this extra space, but their value is irrelevant. The value of `trunc` must be divisible by 2.

void fft_truncate1(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t trunc)

  As for `fft_radix2` except that only the first `trunc` coefficients of the output are computed. The transform still needs all \(2n\) input coefficients to be specified.

void ifft_radix2(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2)

  The radix 2 DIF IFFT works as follows:

  Input: `[i0, i1, ..., i(m-1)]`, for \(m = 2n\) a power of \(2\).

  Output: `[r0, r1, ..., r(m-1)]`

    `= IFFT[i0, i1, ..., i(m-1)]`.

  Algorithm:

  \(\bullet\) Recursively compute `[s0, s1, ...., s(m/2-1)]`

    `= IFFT[i0, i2, ..., i(m-2)]`

  \(\bullet\) Recursively compute `[t(m/2), t(m/2+1), ..., t(m-1)]`

    `= IFFT[i1, i3, ..., i(m-1)]`

  \(\bullet\) Let `[r0, r1, ..., r(m/2-1)]`

    `= [s0+z1^0*t0, s1+z1^1*t1, ..., s(m/2-1)+z1^(m/2-1)*t(m/2-1)]` where `z1 = exp(-2*Pi*I/m)` corresponds to division by \(2^w\).

  \(\bullet\) Let `[r(m/2), r(m/2+1), ..., r(m-1)]`

    `= [s0-z1^0*t0, s1-z1^1*t1, ..., s(m/2-1)-z1^(m/2-1)*t(m/2-1)]`

  The parameters are as follows:

  \(\bullet\) `2*n` is the length of the input and output

    arrays

  \(\bullet\) \(w\) is such that \(2^w\) is an \(2n\)-th root of unity in the ring \(\mathbf{Z}/p\mathbf{Z}\) that we are working in, i.e. \(p = 2^{wn} + 1\) (here \(n\) is divisible by

    `GMP_LIMB_BITS`)

  \(\bullet\) `ii` is the array of inputs (each input is an array of limbs of length `wn/GMP_LIMB_BITS + 1` (the extra limbs being a “carry limb”). Outputs are written in-place.

  We require \(nw\) to be at least 64 and the two temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void ifft_truncate(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t trunc)

  As for `ifft_radix2` except that the output is assumed to have zeros from coefficient trunc onwards and only the first trunc coefficients of the input are specified. The remaining coefficients need to exist as the extra space is needed, but their value is irrelevant. The value of `trunc` must be divisible by 2.

  Although the implementation does not require it, we assume for simplicity that `trunc` is greater than \(n\). The algorithm begins by computing the inverse transform of the first \(n\) coefficients of the input array. The unspecified coefficients of the second half of the array are then written: coefficient `trunc + i` is computed as a twist of coefficient `i` by a root of unity. The values of these coefficients are then equal to what they would have been if the inverse transform of the right hand side of the input array had been computed with full data from the start. The function `ifft_truncate1` is then called on the entire right half of the input array with this auxiliary data filled in. Finally a single layer of the IFFT is completed on all the coefficients up to `trunc` being careful to note that this involves doubling the coefficients from `trunc - n` up to `n`.

void ifft_truncate1(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t trunc)

  Computes the first `trunc` coefficients of the radix 2 inverse transform assuming the first `trunc` coefficients are given and that the remaining coefficients have been set to the value they would have if an inverse transform had already been applied with full data.

  The algorithm is the same as for `ifft_truncate` except that the coefficients from `trunc` onwards after the inverse transform are not inferred to be zero but the supplied values.

void fft_butterfly_sqrt2(mp_limb_t *s, mp_limb_t *t, mp_limb_t *i1, mp_limb_t *i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w, mp_limb_t *temp)

  Let \(w = 2k + 1\), \(i = 2j + 1\). Set `s = i1 + i2`, `t = z1^i*(i1 - i2)` modulo `B^limbs + 1` where `z1^2 = exp(Pi*I/n)` corresponds to multiplication by \(2^w\). Requires \(0 \leq i < 2n\) where \(nw =\) `limbs*FLINT_BITS`.

  Here `z1` corresponds to multiplication by \(2^k\) then multiplication by `(2^(3nw/4) - 2^(nw/4))`. We see `z1^i` corresponds to multiplication by `(2^(3nw/4) - 2^(nw/4))*2^(j+ik)`.

  We first multiply by `2^(j + ik + wn/4)` then multiply by an additional `2^(nw/2)` and subtract.

void ifft_butterfly_sqrt2(mp_limb_t *s, mp_limb_t *t, mp_limb_t *i1, mp_limb_t *i2, mp_size_t i, mp_size_t limbs, flint_bitcnt_t w, mp_limb_t *temp)

  Let \(w = 2k + 1\), \(i = 2j + 1\). Set `s = i1 + z1^i*i2`, `t = i1 - z1^i*i2` modulo `B^limbs + 1` where `z1^2 = exp(-Pi*I/n)` corresponds to division by \(2^w\). Requires \(0 \leq i < 2n\) where \(nw =\) `limbs*FLINT_BITS`.

  Here `z1` corresponds to division by \(2^k\) then division by `(2^(3nw/4) - 2^(nw/4))`. We see `z1^i` corresponds to division by `(2^(3nw/4) - 2^(nw/4))*2^(j+ik)` which is the same as division by `2^(j+ik + 1)` then multiplication by `(2^(3nw/4) - 2^(nw/4))`.

  Of course, division by `2^(j+ik + 1)` is the same as multiplication by `2^(2*wn - j - ik - 1)`. The exponent is positive as \(i \leq 2\cdot n\), \(j < n\), \(k < w/2\).

  We first multiply by `2^(2*wn - j - ik - 1 + wn/4)` then multiply by an additional `2^(nw/2)` and subtract.

void fft_truncate_sqrt2(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t trunc)

  As per `fft_truncate` except that the transform is twice the usual length, i.e. length \(4n\) rather than \(2n\). This is achieved by making use of twiddles by powers of a square root of 2, not powers of 2 in the first layer of the transform.

  We require \(nw\) to be at least 64 and the three temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void ifft_truncate_sqrt2(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t trunc)

  As per `ifft_truncate` except that the transform is twice the usual length, i.e. length \(4n\) instead of \(2n\). This is achieved by making use of twiddles by powers of a square root of 2, not powers of 2 in the final layer of the transform.

  We require \(nw\) to be at least 64 and the three temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

## Matrix Fourier Transforms

void fft_butterfly_twiddle(mp_limb_t *u, mp_limb_t *v, mp_limb_t *s, mp_limb_t *t, mp_size_t limbs, flint_bitcnt_t b1, flint_bitcnt_t b2)

  Set `u = 2^b1*(s + t)`, `v = 2^b2*(s - t)` modulo `B^limbs + 1`. This is used to compute `u = 2^(ws*tw1)*(s + t)`, `v = 2^(w+ws*tw2)*(s - t)` in the matrix Fourier algorithm, i.e. effectively computing an ordinary butterfly with additional twiddles by `z1^rc` for row \(r\) and column \(c\) of the matrix of coefficients. Aliasing is not allowed.

void ifft_butterfly_twiddle(mp_limb_t *u, mp_limb_t *v, mp_limb_t *s, mp_limb_t *t, mp_size_t limbs, flint_bitcnt_t b1, flint_bitcnt_t b2)

  Set `u = s/2^b1 + t/2^b1)`, `v = s/2^b1 - t/2^b1` modulo `B^limbs + 1`. This is used to compute `u = 2^(-ws*tw1)*s + 2^(-ws*tw2)*t)`, `v = 2^(-ws*tw1)*s + 2^(-ws*tw2)*t)` in the matrix Fourier algorithm, i.e. effectively computing an ordinary butterfly with additional twiddles by `z1^(-rc)` for row \(r\) and column \(c\) of the matrix of coefficients. Aliasing is not allowed.

void fft_radix2_twiddle(mp_limb_t **ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs)

  As for `fft_radix2` except that the coefficients are spaced by `is` in the array `ii` and an additional twist by `z^c*i` is applied to each coefficient where \(i\) starts at \(r\) and increases by `rs` as one moves from one coefficient to the next. Here `z` corresponds to multiplication by `2^ws`.

void ifft_radix2_twiddle(mp_limb_t **ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs)

  As for `ifft_radix2` except that the coefficients are spaced by `is` in the array `ii` and an additional twist by `z^(-c*i)` is applied to each coefficient where \(i\) starts at \(r\) and increases by `rs` as one moves from one coefficient to the next. Here `z` corresponds to multiplication by `2^ws`.

void fft_truncate1_twiddle(mp_limb_t **ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs, mp_size_t trunc)

  As per `fft_radix2_twiddle` except that the transform is truncated as per `fft_truncate1`.

void ifft_truncate1_twiddle(mp_limb_t **ii, mp_size_t is, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_size_t ws, mp_size_t r, mp_size_t c, mp_size_t rs, mp_size_t trunc)

  As per `ifft_radix2_twiddle` except that the transform is truncated as per `ifft_truncate1`.

void fft_mfa_truncate_sqrt2(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t n1, mp_size_t trunc)

  This is as per the `fft_truncate_sqrt2` function except that the matrix Fourier algorithm is used for the left and right FFTs. The total transform length is \(4n\) where `n = 2^depth` so that the left and right transforms are both length \(2n\). We require `trunc > 2*n` and that `trunc` is divisible by `2*n1` (explained below). The coefficients are produced in an order different from `fft_truncate_sqrt2`.

  The matrix Fourier algorithm, which is applied to each transform of length \(2n\), works as follows. We set `n1` to a power of 2 about the square root of \(n\). The data is then thought of as a set of `n2` rows each with `n1` columns (so that `n1*n2 = 2n`).

  The length \(2n\) transform is then computed using a whole pile of short transforms. These comprise `n1` column transforms of length `n2` followed by some twiddles by roots of unity (namely `z^rc` where \(r\) is the row and \(c\) the column within the data) followed by `n2` row transforms of length `n1`. Along the way the data needs to be rearranged due to the fact that the short transforms output the data in binary reversed order compared with what is needed.

  The matrix Fourier algorithm provides better cache locality by decomposing the long length \(2n\) transforms into many transforms of about the square root of the original length.

  For better cache locality the sqrt2 layer of the full length \(4n\) transform is folded in with the column FFTs performed as part of the first matrix Fourier algorithm on the left half of the data.

  The second half of the data requires a truncated version of the matrix Fourier algorithm. This is achieved by truncating to an exact multiple of the row length so that the row transforms are full length. Moreover, the column transforms will then be truncated transforms and their truncated length needs to be a multiple of 2. This explains the condition on `trunc` given above.

  To improve performance, the extra twiddles by roots of unity are combined with the butterflies performed at the last layer of the column transforms.

  We require \(nw\) to be at least 64 and the three temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void ifft_mfa_truncate_sqrt2(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t n1, mp_size_t trunc)

  This is as per the `ifft_truncate_sqrt2` function except that the matrix Fourier algorithm is used for the left and right IFFTs. The total transform length is \(4n\) where `n = 2^depth` so that the left and right transforms are both length \(2n\). We require `trunc > 2*n` and that `trunc` is divisible by `2*n1`.

  We set `n1` to a power of 2 about the square root of \(n\).

  As per the matrix fourier FFT the sqrt2 layer is folded into the final column IFFTs for better cache locality and the extra twiddles that occur in the matrix Fourier algorithm are combined with the butterflied performed at the first layer of the final column transforms.

  We require \(nw\) to be at least 64 and the three temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void fft_mfa_truncate_sqrt2_outer(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t n1, mp_size_t trunc)

  Just the outer layers of `fft_mfa_truncate_sqrt2`.

void fft_mfa_truncate_sqrt2_inner(mp_limb_t **ii, mp_limb_t **jj, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t n1, mp_size_t trunc, mp_limb_t **tt)

  The inner layers of `fft_mfa_truncate_sqrt2` and `ifft_mfa_truncate_sqrt2` combined with pointwise mults.

void ifft_mfa_truncate_sqrt2_outer(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp, mp_size_t n1, mp_size_t trunc)

  The outer layers of `ifft_mfa_truncate_sqrt2` combined with normalisation.

## Negacyclic multiplication

void fft_negacyclic(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp)

  As per `fft_radix2` except that it performs a sqrt2 negacyclic transform of length \(2n\). This is the same as the radix 2 transform except that the \(i\)-th coefficient of the input is first multiplied by \(\sqrt{2}^{iw}\).

  We require \(nw\) to be at least 64 and the two temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void ifft_negacyclic(mp_limb_t **ii, mp_size_t n, flint_bitcnt_t w, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **temp)

  As per `ifft_radix2` except that it performs a sqrt2 negacyclic inverse transform of length \(2n\). This is the same as the radix 2 inverse transform except that the \(i\)-th coefficient of the output is finally divided by \(\sqrt{2}^{iw}\).

  We require \(nw\) to be at least 64 and the two temporary space pointers to point to blocks of size `n*w + FLINT_BITS` bits.

void fft_naive_convolution_1(mp_limb_t *r, mp_limb_t *ii, mp_limb_t *jj, mp_size_t m)

  Performs a naive negacyclic convolution of `ii` with `jj`, both of length \(m\), and sets \(r\) to the result. This is essentially multiplication of polynomials modulo \(x^m + 1\).

void _fft_mulmod_2expp1(mp_limb_t *r1, mp_limb_t *i1, mp_limb_t *i2, mp_size_t r_limbs, flint_bitcnt_t depth, flint_bitcnt_t w)

  Multiply `i1` by `i2` modulo `B^r_limbs + 1` where `r_limbs = nw/FLINT_BITS` with `n = 2^depth`. Uses the negacyclic FFT convolution CRT’d with a 1 limb naive convolution. We require that `depth` and `w` have been selected as per the wrapper `fft_mulmod_2expp1` below.

slong fft_adjust_limbs(mp_size_t limbs)

  Given a number of limbs, returns a new number of limbs (no more than the next power of 2) which will work with the Nussbaumer code. It is only necessary to make this adjustment if `limbs > FFT_MULMOD_2EXPP1_CUTOFF`.

void fft_mulmod_2expp1(mp_limb_t *r, mp_limb_t *i1, mp_limb_t *i2, mp_size_t n, mp_size_t w, mp_limb_t *tt)

  As per `_fft_mulmod_2expp1` but with a tuned cutoff below which more classical methods are used for the convolution. The temporary space is required to fit `n*w + FLINT_BITS` bits. There are no restrictions on \(n\), but if `limbs = n*w/FLINT_BITS` then if `limbs` exceeds `FFT_MULMOD_2EXPP1_CUTOFF` the function `fft_adjust_limbs` must be called to increase the number of limbs to an appropriate value.

## Integer multiplication

void mul_truncate_sqrt2(mp_ptr r1, mp_srcptr i1, mp_size_t n1, mp_srcptr i2, mp_size_t n2, flint_bitcnt_t depth, flint_bitcnt_t w)

  Integer multiplication using the radix 2 truncated sqrt2 transforms.

  Set `(r1, n1 + n2)` to the product of `(i1, n1)` by `(i2, n2)`. This is achieved through an FFT convolution of length at most `2^(depth + 2)` with coefficients of size \(nw\) bits where `n = 2^depth`. We require `depth >= 6`. The input data is broken into chunks of data not exceeding `(nw - (depth + 1))/2` bits. If breaking the first integer into chunks of this size results in `j1` coefficients and breaking the second integer results in `j2` chunks then `j1 + j2 - 1 <= 2^(depth + 2)`.

  If `n = 2^depth` then we require \(nw\) to be at least 64.

void mul_mfa_truncate_sqrt2(mp_ptr r1, mp_srcptr i1, mp_size_t n1, mp_srcptr i2, mp_size_t n2, flint_bitcnt_t depth, flint_bitcnt_t w)

  As for `mul_truncate_sqrt2` except that the cache friendly matrix Fourier algorithm is used.

  If `n = 2^depth` then we require \(nw\) to be at least 64. Here we also require \(w\) to be \(2^i\) for some \(i \geq 0\).

void flint_mpn_mul_fft_main(mp_ptr r1, mp_srcptr i1, mp_size_t n1, mp_srcptr i2, mp_size_t n2)

  The main integer multiplication routine. Sets `(r1, n1 + n2)` to `(i1, n1)` times `(i2, n2)`. We require `n1 >= n2 > 0`.

## Convolution

void fft_convolution(mp_limb_t **ii, mp_limb_t **jj, slong depth, slong limbs, slong trunc, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **s1, mp_limb_t **tt)

  Perform an FFT convolution of `ii` with `jj`, both of length `4*n` where `n = 2^depth`. Assume that all but the first `trunc` coefficients of the output (placed in `ii`) are zero. Each coefficient is taken modulo `B^limbs + 1`. The temporary spaces `t1`, `t2` and `s1` must have `limbs + 1` limbs of space and `tt` must have `2*(limbs + 1)` of free space.

## FFT Precaching

void fft_precache(mp_limb_t **jj, slong depth, slong limbs, slong trunc, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **s1)

  Precompute the FFT of `jj` for use with precache functions. The parameters are as for `fft_convolution`.

void fft_convolution_precache(mp_limb_t **ii, mp_limb_t **jj, slong depth, slong limbs, slong trunc, mp_limb_t **t1, mp_limb_t **t2, mp_limb_t **s1, mp_limb_t **tt)

  As per `fft_convolution` except that it is assumed `fft_precache` has been called on `jj` with the same parameters. This will then run faster than if `fft_convolution` had been run with the original `jj`.