Testing numbers of the form $N=2kp_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}-1$ for primality

We suggest an algorithm to test numbers of the form $N=2kp_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}-1$ for primality, where $2k<p_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}$, $k$ is an odd positive integer, $\pi$ is a prime number, $i=1,\dots,n$, and $p_1p_2\cdots p_n=3\pmod4$. The algorithm makes use of the Lucas functions. The algorithm suggested is of complexity $\widehat O(\log^2N)$.