Universal Randomized Guessing Subject to Distortion

In this paper, we consider the problem of guessing a sequence subject to a distortion constraint. Specifically, we assume the following game between Alice and Bob: Alice has a sequence <inline-formula> <tex-math notation="LaTeX">${x}$ </tex-math></inline-formula> of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Bob wishes to guess <inline-formula> <tex-math notation="LaTeX">${x}$ </tex-math></inline-formula>, yet he is satisfied with finding any sequence <inline-formula> <tex-math notation="LaTeX">$\hat {x}$ </tex-math></inline-formula> which is within a given distortion <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>. Thus, he successively submits queries to Alice, until receiving an affirmative answer, stating that his guess was within the required distortion. Finding guessing strategies which minimize the number of guesses (the guesswork), and analyzing its properties (e.g., its <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula>–th moment) has several applications in information security, source and channel coding. Guessing subject to a distortion constraint is especially useful when considering contemporary biometrically–secured systems, where the “password” which protects the data is not a single, fixed vector but rather a ball of feature vectors centered at some <inline-formula> <tex-math notation="LaTeX">${x}$ </tex-math></inline-formula>, and any feature vector within the ball results in acceptance. We formally define the guessing problem under distortion in four different setups: memoryless sources, guessing through a noisy channel, sources with memory and individual sequences. We suggest a randomized guessing strategy which is asymptotically optimal for all setups and is five–fold universal, as it is independent of the source statistics, the channel, the moment to be optimized, the distortion measure and the distortion level.