Given a certain amount of known keystream from a keystream generator (KG), the most obvious way to determine the state of the generator is to search through all possible states, checking for a match between the resulting and observed keystream. In this paper, we draw attention to two attacks on stream cipher systems which, although their complexity grows exponentially with the size of the KG state, are more efficient than a simple-minded search through all possible KG states. Indeed, given sufficient storage and sufficient known keystream, each attack can almost halve the effective entropy of the state to be searched.