A321 pilot reported an airspeed autoflight malfunction during cruise in the indicated airspeed fluctuations. The flight descended to a lower altitude and diverted to an alternate airport.

Narrative

Cruising at 36;000 ft we hit a light to moderate turbulence for just a few seconds; we were cruising at .77 or .78 and we had 15 knots or more on either side off our speed; after the bumps I noticed the speed bleeding off and at 5 kts below target speed I said hey look at this ; it's acting up again; this was the third time ; once on take off and the other in cruise. I watched the speed come down and failed the speed up thinking it would make the power kick in but the speed was not coming back and at green dot. I then put 34;000 in the window and open D down to 34;000 and the speed wasn't increasing went through 35;000 and the speed then went up fast past the bug speed by more then 15 knots into an over speed situation. We contacted Denver center and let them know we had a problem and needed to descend to 34;000Three times we watched the speed do abnormal things and the whole flight it had trouble maintaining the target speed always going plus 5 to 8 knots over the bug speed. Airplane just couldn't manage speed very well. On take off the speed went up to almost a flap over speed before trust reduction alt. Maybe a bad fmgc computer I don't know.