In recent years, the discovery of new algorithms for dealing with polynomial equations,
coupled with their implementation on fast inexpensive computers, has sparked a minor …

Cryptographic ciphers come in two flavours: symmetric (AES, etc.) and asymmetric (RSA,
etc.). The symmetric ciphers are further divided into block ciphers and stream ciphers. Block …

Systems of polynomial equations are central to mathematics and its application to science
and engineering. Their solution sets, called algebraic sets, are studied in algebraic …

Many new ciphers target a concise algebraic description for efficient evaluation in a proof
system or a multi-party computation. This new target for optimization introduces algebraic …

The Gröbner walk is an algorithm for conversion between Gröbner bases for different term
orders. It is based on the polyhedral geometry of the Gröbner fan and involves tracking a line …

Definition 2.3 (Multisectioning). Let f(x) be a func- Page 1 We see that there are numerous
examples of rational poly-exponential functions that are of interest in mathematics. Two …

The notion of the compatibility between a term order in a polynomial ring R and a module
term order in R l is crucial to ensure the termination of a signature-based algorithm for …

We propose a method for computing the radical of an arbitrary ideal in the polynomial ring in
n variables over a perfect field of characteristic p> 0. In our method Buchberger's algorithm is …

The theory of Groebner Bases originated in the work of Buchberger and is now considered
to be one of the most important and useful areas of symbolic computation. A great deal of …

Buchberger's algorithm is the classical algorithm for computing a Gröbner basis, and
optimized implementations are crucial for many computer algebra systems. In this thesis we …