Explicit Triple $(d,b',c')$ for Any Prime $P$ and an Efficient Algorithm for Its Construction

Let $P=4p+1$ be a prime number.
We show the existence of an integer triple $(d,b',c')$, such that
\[
     d \equiv 3 \pmod{4}, \qquad
     4b'c' | (P + d), \qquad
     b' + c' \equiv 0 \pmod{d}.
\] 
and also:
\[
\gcd(b', c') = 1, \qquad
b' < c'.
\]
For $P \ge 10^{10}$, explicit bounds hold:
\[
d \le \frac{P}{2\ln^{2}P}, \quad
b' \le \frac{P^{3/2}}{\ln^{2}P}, \quad
c' \le \frac{\sqrt P}{2}.
\]
For $P < 10^{10}$, the triples computed by the algorithm are contained in the file \texttt{smallP\_triples.txt} (sample attached).
An algorithm with average complexity
$O(P^{1/2 + o(1)})$ is proposed, supported by numerical experiments and discussions on cryptographic applications (attacks on quasi-Blum moduli).