Presumably finding arbitrarily many basic G*‘s will be hard. Two ideas for dealing with this: 1. Even if you only have finitely many and they’re all known, you could select one at random each time there’s a switch. 2. Each time there’s a switch, select a somehow-random linear (or some other sort, if you like) combination of your basic G*’s. (That would make guessing it in the first place quite hard, actually...)
Presumably finding arbitrarily many basic G*‘s will be hard. Two ideas for dealing with this: 1. Even if you only have finitely many and they’re all known, you could select one at random each time there’s a switch. 2. Each time there’s a switch, select a somehow-random linear (or some other sort, if you like) combination of your basic G*’s. (That would make guessing it in the first place quite hard, actually...)