This paper presents a generalized form of the fractional logistic map. Two general parameters a and b are added to the classical fractional logistic equation. The effect of such parameters on the map is studied explicitly, in combination with the fractional order parameter α, which offers an extra degree of freedom increasing the design flexibility and adding more controllability on the design. The vertical and the zooming map are two special maps that arise as a result of the added parameters. Moreover, different design problems are offered in this work, as a resultant of the control of all these parameters at hand. This shows that any application specific map can be designed, highlighting the flexibility and integrity of the design. The combination of the added extra parameters a and b in addition to the system parameter ρ and the initial condition X_0, as well as the fractional order parameter α makes the proposed generalized fractional logistic map the most favorable in constructing more efficient encryption keys.
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