Examples: 1.Our October salary was given in advance, based on the assumption that the work would continue. 2. We continued working under the assumption that we would be paid on time. 3. With the assumption that work will continue past October and we will still be employed, we were paid an advanced salary. These all have similar meanings, but they are used differently in sentences. As an American, I hear "under the assumption" most often.

Generally 'on the assumption' but 'with the assumption' can work too. 'On the assumption' implies you did something because you assumed something. 'With the assumption' seems slightly less causal/logically connected to me - you did the thing while also believing that thing; it's another factor. 'I booked the restaurant on the assumption that you would be back in time.' 'I booked the restaurant because I know you like pasta, with the assumption that you would be back in time.'