Distributed storage for data security

We study the secrecy of a distributed storage system for passwords. The encoder, Alice, observes a length-n password and describes it using two hints, which she then stores in different locations. The legitimate receiver, Bob, observes both hints. In one scenario we require that the number of guesses it takes Bob to guess the password approach 1 as n tends to infinity and in the other that the size of the list that Bob must form to guarantee that it contain the password approach 1. The eavesdropper, Eve, sees only one of the hints; Alice cannot control which. For each scenario we characterize the largest normalized (by n) exponent that we can guarantee for the number of guesses it takes Eve to guess the password.