I think it is reasonable to push back and ask for more requirements. If they say that they don't exist yet, estimate it will take 6 months. When they accuse you

I think it is reasonable to push back and ask for more requirements. If they say that they don't exist yet, estimate it will take 6 months. When they accuse you of being unhelpful, restate that at present you don't have enough information to make any kind of reasonable ballpark estimate let alone something for them to put in their awful gantt chart. If that path won't work, then I'd tell them that I need two days to break it down into all the tasks that I think I'll need to do and then give them an estimate after you do that investigation.