  Bogopolski, Martino and Ventura in [BMV10] introduced a general criteria to
construct groups extensions with unsolvable conjugacy problem using short exact
sequences. We prove that such extensions have always solvable word problem.
This makes the proposed construction a systematic way to obtain finitely
presented groups with solvable word problem and unsolvable conjugacy problem.
It is believed that such groups are important in cryptography. For this, and as
an example, we provide an explicit construction of an extension of Thompson
group F and we propose it as a base for a public key cryptography protocol.
