A321 Captain reported encountering significant wake turbulence from an A380 in cruise flight on a trans-Atlantic route.

Narrative

While at cruise flight at FL340 we encountered wake turbulence from an A380. That aircraft was at FL350 approximately 20nm in front of us. The winds were about 215/80 which gave us a quartering tailwind from our left. Just prior to encountering the wake turbulence we had been discussing our offset we would initiate approaching the LOMSI fix to mitigate flying into any possible wake.Suddenly; the aircraft rolled right approximately 30 degrees; the autopilot kicked off; the airplane climbed 500 ft. - 600 ft. and the speed dropped just below green dot. We called ATC to advise the situation and immediately requested a lower altitude and a deviation to the right to get off the wake.Once stabilized I made a call to the cabin to check on the status of the crew members and customers. No injuries were reported. By this time we were entering the NAT and requested and received a 5 mile offset until the A380 moved farther along as it was much faster.No limitations were exceeded and I would classify the turbulence as moderate.