@phdthesis{Ali2011, author = {Ali, Rashid}, title = {Weyl Gr{\"o}bner Basis Cryptosystems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-23195}, school = {Universit{\"a}t Passau}, year = {2011}, abstract = {In this thesis, we shall consider a certain class of algebraic cryptosystems called Gr{\"o}bner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gr{\"o}bner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of Gr{\"o}bner basis is, in general, EXPSPACE-hard. Cryptanalysis of these commutative Polly Cracker type cryptosystems is possible by using attacks that do not require the computation of Gr{\"o}bner basis for breaking the system, for example, the attacks based on linear algebra. To secure these (commutative) Gr{\"o}bner basis cryptosystems against various attacks, among others, Ackermann and Kreuzer introduced a general class of Gr{\"o}bner Basis Cryptosystems that are based on the difficulty of computing module Gr{\"o}bner bases over general non-commutative rings. The objective of this research is to describe a special class of such cryptosystems by introducing the Weyl Gr{\"o}bner Basis Cryptosystems. We divide this class of cryptosystems in two parts namely the (left) Weyl Gr{\"o}bner Basis Cryptosystems and Two-Sided Weyl Gr{\"o}bner Basis Cryptosystems. We suggest to use Gr{\"o}bner bases for left and two-sided ideals in Weyl algebras to construct specific instances of such cryptosystems. We analyse the resistance of these cryptosystems to the standard attacks and provide computational evidence that secure Weyl Gr{\"o}bner Basis Cryptosystems can be built using left (resp. two-sided) Gr{\"o}bner bases in Weyl algebras.}, subject = {Gr{\"o}bner-Basis}, language = {en} }