There exist very few concepts in computational algebra which are as central to theory and applications as Grobner bases. This thesis describes theory, algorithms and applications for the special case of Boolean polynomials. These parts form the mathematical foundations of the PolyBoRi framework (developed by the author together with Alexander Dreyer). The PolyBoRi framework has applications spread over a large number of domains ranging from formal verification, computational biology to cryptanalysis and many more. It is emerged to a worldwide audience by the Sage computational algebra system.

Abstract: "We studied several important properties of Boolean polynomial rings in [SaSa 90]. Especially we saw ideal plays a central role for solving a Boolean constraint. This paper gives an algorithm which produces a rewriting system for a given finitely generated ideal in the ring of Boolean polynomials. The rewriting system reduces all Boolean polynomials that are equivalent under the ideal to the same normal form."

Abstract: "Any constraint over sets can be represented in terms of a Boolean polynomial ring whenever the family of sets that we consider forms a Boolean ring. In this paper we give a complete solution method for such constraints using Boolean Gröbner bases. A Boolean Gröbner base is a modification of a standard Gröbner base which we developed to solve constraints of general Boolean polynomial rings."

Comprehensive account of theory and applications of Gröbner bases, co-edited by the subject's inventor.

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Gröbner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations. This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will find that all chapters fit nicely within a unified notation.

Abstract: "Detecting redundant S-polynomials raise efficiency of Buchberger's algorithm to construct Gröbner bases as was first pointed out in [Buchberger 79]. One of the most practical criteria for it is given in terms of a homogeneous basis of a module of syzygies. This criterion is also applicable in suitable forms even when coefficient domains are not fields. We show we can apply this criterion in construction of Boolean Gröbner bases introduced in [Sakai 92]. We examine its efficiency with some experimental results of our implementation."