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:\[\gcd(b', c') = 1, \qquadb' < c'.\]For $P \ge 10^{10}$, explicit bounds hold:\[d \le \frac{P}{2\ln^{2}P}, \quadb' \le \frac{P^{3/2}}{\ln^{2}P}, \quadc' \le \frac{\sqrt P}{2}.\]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).