Nonadaptive Mastermind Algorithms for String and Vector Databases, with Case Studies

In this paper, we study sparsity-exploiting Mastermind algorithms for attacking the privacy of an entire database of character strings or vectors, such as DNA strings, movie ratings, or social network friendship data. Based on reductions to nonadaptive group testing, our methods are able to take advantage of minimal amounts of privacy leakage, such as contained in a single bit that indicates if two people in a medical database have any common genetic mutations, or if two people have any common friends in an online social network. We analyze our Mastermind attack algorithms using theoretical characterizations that provide sublinear bounds on the number of queries needed to clone the database, as well as experimental tests on genomic information, collaborative filtering data, and online social networks. By taking advantage of the generally sparse nature of these real-world databases and modulating a parameter that controls query sparsity, we demonstrate that relatively few nonadaptive queries are needed to recover a large majority of each database

New Constructions for Competitive and Minimal-Adaptive Group Testing

Group testing (GT) was originally proposed during the World War II in an attempt to minimize the \emph{cost} and \emph{waiting time} in performing identical blood tests of the soldiers for a low-prevalence disease. Formally, the GT problem asks to find $d\ll n$ \emph{defective} elements out of $n$ elements by querying subsets (pools) for the presence of defectives. By the information-theoretic lower bound, essentially $d\log_2 n$ queries are needed in the worst-case. An \emph{adaptive} strategy proceeds sequentially by performing one query at a time, and it can achieve the lower bound. In various applications, nothing is known about $d$ beforehand and a strategy for this scenario is called \emph{competitive}. Such strategies are usually adaptive and achieve query optimality within a constant factor called the \emph{competitive ratio}. In many applications, queries are time-consuming. Therefore, \emph{minimal-adaptive} strategies which run in a small number $s$ of stages of parallel queries are favorable. This work is mainly devoted to the design of minimal-adaptive strategies combined with other demands of both theoretical and practical interest. First we target unknown $d$ and show that actually competitive GT is possible in as few as $2$ stages only. The main ingredient is our randomized estimate of a previously unknown $d$ using nonadaptive queries. In addition, we have developed a systematic approach to obtain optimal competitive ratios for our strategies. When $d$ is a known upper bound, we propose randomized GT strategies which asymptotically achieve query optimality in just $2$, $3$ or $4$ stages depending upon the growth of $d$ versus $n$. Inspired by application settings, such as at American Red Cross, where in most cases GT is applied to small instances, \textit{e.g.}, $n=16$. We extended our study of query-optimal GT strategies to solve a given problem instance with fixed values $n$, $d$ and $s$. We also considered the situation when elements to test cannot be divided physically (electronic devices), thus the pools must be disjoint. For GT with \emph{disjoint} simultaneous pools, we show that $\Theta (sd(n/d)^{1/s})$ tests are sufficient, and also necessary for certain ranges of the parameters

Optimal Strategies for Static Black-Peg AB Game With Two and Three Pegs

The AB~Game is a game similar to the popular game Mastermind. We study a version of this game called Static Black-Peg AB~Game. It is played by two players, the codemaker and the codebreaker. The codemaker creates a so-called secret by placing a color from a set of $c$ colors on each of $p \le c$ pegs, subject to the condition that every color is used at most once. The codebreaker tries to determine the secret by asking questions, where all questions are given at once and each question is a possible secret. As an answer the codemaker reveals the number of correctly placed colors for each of the questions. After that, the codebreaker only has one more try to determine the secret and thus to win the game. For given $p$ and $c$, our goal is to find the smallest number $k$ of questions the codebreaker needs to win, regardless of the secret, and the corresponding list of questions, called a $(k+1)$-strategy. We present a $\lceil 4c/3 \rceil-1)$-strategy for $p=2$ for all $c \ge 2$, and a $\lfloor (3c-1)/2 \rfloor$-strategy for $p=3$ for all $c \ge 4$ and show the optimality of both strategies, i.e., we prove that no $(k+1)$-strategy for a smaller $k$ exists

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum