Deterministic poly--log--time construction of a triple $(d,w,p)$ with $4p^{2}w\equiv-1\pmod d$ and $2pw\mid(P+d)$

For an arbitrary odd integer $P\ge 9$ such that $P = 4p + 1$, we give a deterministicalgorithm that, in $\bigO((\log P)^5)$ bit operations, finds integers$d,w,p$ such that\[ d\equiv3\pmod4,\qquad 4p^{2}w\equiv-1\pmod d,\qquad 2pw\mid P+d.\]The construction relies only on unconditional analytic tools(Bombieri--Vinogradov large sieve and the larger sieve of Greaves)and never requires a full factorisation of any number involved.