We demonstrate the physical principles for the construction of a nanometer sized magnetoresistance device based on the Aharonov-Bohm effect. The proposed device is made of a short single-walled carbon nanotube (SWCNT) placed on a substrate and coupled to a tip. We consider conductance due to motion of electrons along the circumference of the tube (as opposed to motion parallel to its axis). We find that the circumference conductance is sensitive to magnetic fields threading the SWCNT due to the Aharonov-Bohm effect, and show that by retracting the tip, so that its coupling to the SWCNT is reduced, very high sensitivity to the threading magnetic field develops. This is due to the formation of a narrow resonance through which the tunneling current flows. Using a bias potential the resonance can be shifted to low magnetic fields, allowing the control of conductance with magnetic fields of the order of 1 Tesla.