Universal Randomized Guessing With Application to Asynchronous Decentralized Brute–Force Attacks

Consider the problem of guessing the realization of a random vector <inline-formula> <tex-math notation="LaTeX">${X}$ </tex-math></inline-formula> by repeatedly submitting queries (guesses) of the form “Is <inline-formula> <tex-math notation="LaTeX">${X}$ </tex-math></inline-formula> equal to <inline-formula> <tex-math notation="LaTeX">${x}$ </tex-math></inline-formula>?” until an affirmative answer is obtained. In this setup, a key figure of merit is the number of queries required until the right vector is identified, a number that is termed the <italic>guesswork</italic>. Typically, one wishes to devise a guessing strategy which minimizes a certain guesswork moment. In this work, we study a universal, decentralized scenario where the guesser does not know the distribution of <inline-formula> <tex-math notation="LaTeX">${X}$ </tex-math></inline-formula>, and is not allowed to use a strategy which prepares a list of words to be guessed in advance, or even remember which words were already used. Such a scenario is useful, for example, if bots within a Botnet carry out a brute–force attack in order to guess a password or decrypt a message, yet cannot coordinate the guesses between them or even know how many bots actually participate in the attack. We devise universal decentralized guessing strategies, first, for memoryless sources, and then generalize them for finite–state sources. In each case, we derive the guessing exponent, and then prove its asymptotic optimality by deriving a compatible converse bound. The strategies are based on randomized guessing using a universal distribution. We also extend the results to guessing with side information. Finally, for all above scenarios, we design efficient algorithms in order to sample from the universal distributions, resulting in strategies which do not depend on the source distribution, are efficient to implement, and can be used asynchronously by multiple agents.