Computing Gröbner bases in monoid and group rings

Madlener, Klaus; Reinert, Birgit

doi:10.1145/164081.164139

Cited by 20 publications

(14 citation statements)

References 8 publications

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“…In the following we collect the necessary results generalizing the theory of prefix rewriting and prefix Gröbner bases for one-sided ideals of K[M ] developed in [20], [21], [22], and [27]. A Gröbner basis theory for the two-sided case can be introduced in a similar way.…”

Section: Remark 42 the Assumption Thatmentioning

confidence: 99%

“…The generalized word problem (also called the submonoid membership problem) was discussed in [20]. It was shown that it leads to a subalgebra membership problem in K[M ].…”

Section: Proposition 52 (The Submodule Membership Problem)mentioning

confidence: 99%

“…In order to prevent linear algebra attacks (see next section) T. Rai suggested in his recent doctoral thesis [26] to construct Gröbner basis cryptosystems based on twosided ideals. The corresponding Gröbner basis theory was sketched in [20], [21], [22], [27] and Section 3.…”

Section: We Use the K[x]-module Fmentioning

confidence: 99%

“…Alas, for the monoid and group rings we are interested in such bases are usually not available. Thus we resort to another generalization of Mora's approach: Klaus Madlener, Birgit Reinert, and their co-workers (see [19], [20], [21], [22], and [27]) have successfully described a Gröbner basis theory for monoid and group rings based on a presentation of the monoid or group by a convergent term rewriting system.…”

Section: Introductionmentioning

confidence: 99%

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Gröbner Basis Cryptosystems

Ackermann

Kreuzer

2006

AAECC

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In the first sections we extend and generalize Gröbner basis theory to submodules of free right modules over monoid rings. Over free monoids, we adapt the known theory for right ideals and prove versions of Macaulay's basis theorem, the Buchberger criterion, and the Buchberger algorithm. Over monoids presented by a finitely generated convergent string rewriting system we generalize Madlener's Gröbner basis theory based on prefix reduction from right ideals to right modules. After showing how these Gröbner basis theories relate to classical group-theoretic problems, we use them as a basis for a new class of cryptosystems that are generalizations of the cryptosystems described in [2] and [8]. Well known cryptosystems such as RSA, El-Gamal, Polly Cracker, Polly Two and a braid group cryptosystem are shown to be special cases. We also discuss issues related to the security of these Gröbner basis cryptosystems.

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Section: Remark 42 the Assumption Thatmentioning

confidence: 99%

“…The generalized word problem (also called the submonoid membership problem) was discussed in [20]. It was shown that it leads to a subalgebra membership problem in K[M ].…”

Section: Proposition 52 (The Submodule Membership Problem)mentioning

confidence: 99%

Section: We Use the K[x]-module Fmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gröbner Basis Cryptosystems

Ackermann

Kreuzer

2006

AAECC

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“…This provides a link to the theory of term rewriting and the Zharkov/Blinkov method appears as an application of the pre x reduction/saturation technique of Madlener and Reinert (c.f. [MR93]) with a restricted saturation. The restricted saturation has its natural origin in the syzygy theory and heavily improves the termination behaviour in the particular case of Pommaret bases.…”

Section: Introductionmentioning

confidence: 99%

The Computation of Gröbner Bases Using an Alternative Algorithm

Apel¹

1998

Symbolic Rewriting Techniques

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A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings

Madlener

Reinert

1998

Journal of Symbolic Computation

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It is well-known that for the integral group ring of a polycyclic group several decision problems are decidable, in particular the ideal membership problem. In this paper we define an effective reduction relation for group rings over polycyclic groups. This reduction is based on left multiplication and hence corresponds to left ideals. Using this reduction we present a generalization of Buchberger's Gröbner basis method by giving an appropriate definition of "Gröbner bases" in this setting and by characterizing them using the concepts of saturation and s-polynomials. The approach is extended to two-sided ideals and a discussion on a Gröbner bases approach for right ideals is included.

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Computing Gröbner bases in monoid and group rings

Abstract: FollowingBuchberger's approach to computing a

Cited by 20 publications

References 8 publications

Gröbner Basis Cryptosystems

Gröbner Basis Cryptosystems

The Computation of Gröbner Bases Using an Alternative Algorithm

A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings

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