  The objective of this paper concerns at first the motivation and the method
of Shor's algorithm including an excursion into quantum mechanics and quantum
computing introducing an algorithmic description of the method. The corner
stone of the Shor's algorithm is the modular exponentiation that is the most
computational component (in time and space). Second, a linear depth unit based
on phase estimation is introduced and a description of a generic version of a
modular multiplier based on phases is introduced to build block of a modular
exponentiation circuit. Our proposal includes numerical experiments achieved on
both the IBM simulator using the Qiskit library and on quantum physical
optimizers provided by IBM.
