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Watermarking embeds a secret message into a cover message. In media watermarking the secret is usually a copyright notice and the cover a digital image. Watermarking an object discourages intellectual property theft, or when such theft has occurred, allows us to prove ownership. The Software Watermarking problem can be described as follows. Embed a structure W into a program P such that: W can be reliably located and extracted from P even after P has been subjected to code transformations such as translation, optimization and obfuscation; W is stealthy; W has a high data rate; embedding W into P does not adversely affect the performance of P ; and W has a mathematical property that allows us to argue that its presence in P is the result of deliberate actions. In the first part of the paper we construct an informal taxonomy of software watermarking techniques. In the second part we formalize these results. Finally, we propose a new software watermarking technique in which a dynamic gr...

This report is a contributed chapter to the Handbook of Theoretical Computer Science (North-Holland, 1990). Its aim is to describe the main mathematical methods and applications in the average-case analysis of algorithms and data structures. It comprises two parts: First, we present basic combinatorial enumerations based on symbolic methods and asymptotic methods with emphasis on complex analysis techniques (such as singularity analysis, saddle point, Mellin transforms). Next, we show how to apply these general methods to the analysis of sorting, searching, tree data structures, hashing, and dynamic algorithms. The emphasis is on algorithms for which exact "analytic models" can be derived.

A survey is presented of some methods and results on counting words that satisfy various restrictions on subwords (i.e., blocks of consecutive symbols). Various applications to comma-free codes, games, pattern matching, and other subjects are indicated. The emphasis is on the unified treatment of those topics through the use of generating functions. 1.

Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.

This paper develops a unified enumerative and asymptotic theory of directed 2-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1-dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.

Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.

An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.

this paper, we consider the problem of enumerating the fully commutative elements of these groups. The main result (Theorem 2.6) is that for six of the seven infinite families (we omit the trivial dihedral family I 2 (m)), the generating function for the number of fully commutative elements can be expressed in terms of three simpler generating functions for certain formal languages over an alphabet with at most four letters. The languages in question vary from family to family, but have a uniform description. The resulting generating function one obtains for each family is algebraic, although in some cases quite complicated. (See (3.7) and (3.11).) In a general Coxeter group, the fully commutative elements index a basis for a natural quotient of the corresponding Iwahori-Hecke algebra [G]. (See also [F1] for the simplylaced case.) For An , this quotient is the Temperley-Lieb algebra. Recently, Fan [F2] has shown that for types A, B, D, E and (in a sketched proof) F , this quotient is generically semisimple, and gives recurrences for the dimensions of the irreducible representations. (For type H, the question of semisimplicity remains open.) This provides another possible approach to computing the number of fully commutative elements in these cases; namely, as the sum of the squares of the dimensions of these representations. Interestingly, Fan also shows that the sum of these dimensions is the number of fully commutative involutions

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class -- an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.