Shuffling Cards When You Are of Very Little Brain: Low Memory Generation of Permutations

Abstract

How can we generate a permutation of the numbers $1$ through $n$ so that it is hard to guess the next element given the history so far? The twist is that the generator of the permutation (the ``Dealer") has limited memory, while the ``Guesser" has unlimited memory. With unbounded memory (actually $n$ bits suffice), the Dealer can generate a truly random permutation where $\ln n$ is the expected number of correct guesses. Our main results establish tight bounds for the relationship between the guessing probability and the memory $m$ required to generate the permutation. We suggest a method for an $m$-bit Dealer that operates in constant time per turn, and any Guesser can pick correctly only $O(n/m+\log m)$ cards in expectation. The method is fully transparent, requiring no hidden information from the Dealer (i.e., it is "open book" or "whitebox"). We show that this bound is the best possible, even with secret memory. Specifically, for any $m$-bit Dealer, there is a (computationally powerful) guesser that achieves $Ω(n/m+\log m)$ correct guesses in expectation. We point out that the assumption that the Guesser is computationally powerful is necessary: under cryptographic assumptions, there exists a low-memory Dealer that can fool any computationally bounded guesser. We also give an $O(n)$ bit memory Dealer that generates perfectly random permutations and operates in constant time per turn.