You want to get one of these (double sides, so 4 abrasive surfaces)
Start with the pad numbered 1, and buff all over the crystal (working around the cyclops just requires a little caution, but would not be impossible) You want to buff the crystal till you can't really see any of the original scratches, and the entire crystal will appear 'frosted' and you won't really be able to see through it. Then move on to pad 2, and repeat the process. Same with pads 3 and 4, and by that time the crystal should be near pristine again. If you want to really take the polish to the next level, put a smear of toothpaste on the crystal (no where near as much as you would use when cleaning your teeth) and rub in tight, circular motions with either your thumb, or a bit of kitchen roll. Do that for maybe 4-5 minutes, then rinse under the cold tap, and it should be good as new Of course, if it's not a plastic crystal, then the above will not work