flint_mpn_mulmid_complex, flint_mpn_sqrmid_complex

The two's complement window of a signed part is linear, so the real windows of the schoolbook and Karatsuba schemes combine modulo $B^{\mathit{hi} - \mathit{lo}}$, and it is what a Kronecker unpacking of signed fields reads. To extract the polynomial coefficients $[n_{lo}, n_{hi})$ of $b$-bit fields, take $\mathit{hi} = \lceil n_{hi} b / \mathtt{FLINT\_BITS} \rceil$ and $\mathit{lo}$ a few guard limbs below bit $n_{lo} b - 1$; the unpacked fields are exact provided that $|e| B^{\mathit{lo}}$ is below the distance from the low part $\sum_{k < n_{lo}} r_k 2^{k b}$ to $\pm 2^{n_{lo} b - 1}$, which a bound $|r_k| \le 2^{b - 1} - 2^{b - s}$ makes at least $2^{n_{lo} b - s}$ (see _fmpzi_poly_mulmid_KS, which uses three guard limbs).

The schoolbook and Karatsuba variants combine real windows from flint_mpn_mulmid. The transformed variant splits the integers into chunks and multiplies them as chunk sequences on fft_small transforms (four forward and two inverse transforms; two and two for the square), reconstructing only the coefficients that can reach limbs $[\mathit{lo}, \mathit{hi})$. When a power of two $2^d$ admits it (operands and the coefficients below the top of the window fitting in $2^d$, and the coefficients beyond $2^d$ aliasing below the first one reaching the window) the transforms are cyclic of length $2^d$. For the shape of the Newton middle product ($2n$ by $n$ limbs, window $[n, 2n)$) this gives a transform of about $2n$ chunks instead of $3n$ when $2n$ chunks lie a little below a power of two.

@misc{flint-doc-mpn-extras-flint-mpn-mulmid-complex,
  author       = {{The FLINT team}},
  title        = {{flint\_mpn\_mulmid\_complex, flint\_mpn\_sqrmid\_complex (mpn\_extras.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/mpn_extras/flint_mpn_mulmid_complex.html},
  note         = {Generated 2026-11-02}
}