Perfectly Secure Steganography Using Minimum Entropy Coupling

Steganography is the practice of encoding secret information into innocuous content in such a manner that an adversarial third party would not realize that there is hidden meaning. While this problem has classically been studied in security literature, recent advances in generative models have led to a shared interest among security and machine learning researchers in developing scalable steganography techniques. In this work, we show that a steganography procedure is perfectly secure under Cachin (1998)'s information-theoretic model of steganography if and only if it is induced by a coupling. Furthermore, we show that, among perfectly secure procedures, a procedure maximizes information throughput if and only if it is induced by a minimum entropy coupling. These insights yield what are, to the best of our knowledge, the first steganography algorithms to achieve perfect security guarantees for arbitrary covertext distributions. To provide empirical validation, we compare a minimum entropy coupling-based approach to three modern baselines -- arithmetic coding, Meteor, and adaptive dynamic grouping -- using GPT-2, WaveRNN, and Image Transformer as communication channels. We find that the minimum entropy coupling-based approach achieves superior encoding efficiency, despite its stronger security constraints. In aggregate, these results suggest that it may be natural to view information-theoretic steganography through the lens of minimum entropy coupling

Infinite Divisibility of Information

We study an information analogue of infinitely divisible probability distributions, where the i.i.d. sum is replaced by the joint distribution of an i.i.d. sequence. A random variable $X$ is called informationally infinitely divisible if, for any $n\ge1$, there exists an i.i.d. sequence of random variables $Z_{1},\ldots,Z_{n}$ that contains the same information as $X$, i.e., there exists an injective function $f$ such that $X=f(Z_{1},\ldots,Z_{n})$. While there does not exist informationally infinitely divisible discrete random variable, we show that any discrete random variable $X$ has a bounded multiplicative gap to infinite divisibility, that is, if we remove the injectivity requirement on $f$, then there exists i.i.d. $Z_{1},\ldots,Z_{n}$ and $f$ satisfying $X=f(Z_{1},\ldots,Z_{n})$, and the entropy satisfies $H(X)/n\le H(Z_{1})\le1.59H(X)/n+2.43$. We also study a new class of discrete probability distributions, called spectral infinitely divisible distributions, where we can remove the multiplicative gap $1.59$. Furthermore, we study the case where $X=(Y_{1},\ldots,Y_{m})$ is itself an i.i.d. sequence, $m\ge2$, for which the multiplicative gap $1.59$ can be replaced by $1+5\sqrt{(\log m)/m}$. This means that as $m$ increases, $(Y_{1},\ldots,Y_{m})$ becomes closer to being spectral infinitely divisible in a uniform manner. This can be regarded as an information analogue of Kolmogorov's uniform theorem. Applications of our result include independent component analysis, distributed storage with a secrecy constraint, and distributed random number generation.Comment: 22 page

Algorithms and Data Structures for Coding, Indexing, and Mining of Sequential Data

In recent years, the production of sequential data has been rapidly increasing. This requires solving challenging problems about how to represent information, how to retrieve information, and how to extract knowledge, from sequential data. These questions belong to the areas of coding, indexing, and mining, respectively. In this thesis, we investigate problems from those three areas. Coding refers to the way in which information is represented. Coding aims at generating optimal codes, that are codes having a minimum expected length. Codes can be generated for different purposes, from data compression to error detection/correction. The Lempel-Ziv 77 parsing produces an asymptotically optimal code in terms of compression. We study algorithms to efficiently decompress strings from the Lempel-Ziv 77 parsing, using memory proportional to the size of the parsing itself. We provide the first implementation of an algorithm by Bille et al., the only work we are aware of on this problem. We present a practical evaluation of this approach and several optimizations which improve the performance on all datasets we tested. Through the Ulam-R{'e}nyi game, it is possible to provide optimal adaptive error-correcting codes. The game consists of discovering an unknown $m$-bit number by asking membership questions the answers to which can be erroneous. Questions are formulated knowing the answers to all previous ones. We want to find an optimal strategy, i.e., a strategy that can identify any $m$-bit number using the theoretical minimum number of questions. We studied the case where questions are a union of up to a fixed number of intervals, and up to three answers can be erroneous. We first show that for any sufficiently large $m$, there exists a strategy to identify an initially unknown $m$-bit number which uses at most four intervals per question. We further refine our main tool to turn the above asymptotic result into a complete characterization of those instances of the Ulam-R{'e}nyi game that admit optimal strategies. Indexing refers to the way in which information is retrieved. An index for texts permits finding all occurrences of any substring, without traversing the whole text. Many applications require to look for approximate substrings. One of these is the problem of jumbled pattern matching, where two strings match if one is a permutation of the other. We study combinatorial aspects of prefix normal words, a class of binary words introduced in this context. These words can be used as indices for the Indexed Binary Jumbled Pattern Matching problem. We present a new recursive generation algorithm for prefix normal words that is competitive with the previous one but allows to list all prefix normal words sharing the same prefix. This sheds lights on novel insights that may help solving the problem of counting the number of prefix normal words of a given length. We then introduce infinite prefix normal words, and we show that one of the operations used by the algorithm, when repeatedly applied to extend a word, produces an infinite prefix normal word. This motivates the seeking for other operations that produce infinite prefix normal words. We found that one of these operations establishes a connection between prefix normal words and Sturmian words. We also explored the relationship between prefix normal words and Abelian complexity, as well as between prefix normal words and lexicographic order. Mining refers to the way in which information is converted into knowledge. The process of knowledge discovery covers several processing steps, including knowledge extraction. We analyze the problem of mining assertions for an embedded system from its simulation traces. This problem can be modeled as a pattern discovery problem on colored strings. We present two problems of pattern discovery on colored strings: patterns for one color only, or for all colors at the same time. We present two suffix tree-based algorithms. The first algorithm solves both the one color problem and the all colors problem. We then, introduce modifications which improve performance of the algorithm both on synthetic and on real data. We implemented and evaluated the proposed approaches, highlighting time trade-offs that can be obtained. A different way of knowledge extraction is based on the information-theoretic perspective of Pearl's model of causality. It has been postulated that the true causality direction between two phenomena A and B is related to the problem of finding the minimum entropy joint distribution between A and B. This problem is known to be NP-hard, and greedy algorithms have recently been proposed. We provide a novel analysis of one of the proposed heuristic showing that this algorithm guarantees an additive approximation of 1 bit. We then, provide a general criterion for guaranteeing an additive approximation factor of 1. This criterion may be of independent interest in other contexts where couplings are used