scholarly journals On ideal lattices, Gröbner bases and generalized hash functions

2018 ◽  
Vol 17 (06) ◽  
pp. 1850112 ◽  
Author(s):  
Maria Francis ◽  
Ambedkar Dukkipati
Keyword(s):  
Gröbner Bases ◽  
Hash Functions ◽  
Monic Polynomial ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Worst Case ◽  
Ideal Lattices ◽  
Cyclic Lattices ◽  
Residue Class Ring ◽  
Class Ring

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, [Formula: see text] (here [Formula: see text] is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Gröbner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in [Formula: see text], and use its computational hardness to establish the collision resistance of the hash functions.

scholarly journals Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

2014 ◽  
Vol 24 (08) ◽  
pp. 1157-1182 ◽  
Author(s):  
Roberto La Scala
Keyword(s):  
Associative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Free Associative Algebra ◽  
Commutative Algebras ◽  
Skew Polynomial Rings ◽  
Experimental Implementation ◽  
Buchberger Algorithm ◽  
Skew Polynomial

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

scholarly journals Distributive laws between the operads Lie and Com

2020 ◽  
Vol 30 (08) ◽  
pp. 1565-1576
Author(s):  
Murray Bremner ◽  
Vladimir Dotsenko
Keyword(s):  
Lie Algebras ◽  
Computer Algebra ◽  
Gröbner Bases ◽  
Classification Problem ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Associative Algebras ◽  
Distributive Law ◽  
Free Modules

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.

scholarly journals Indispensable Hibi Relations and Gröbner Bases

2015 ◽  
Vol 22 (04) ◽  
pp. 567-580
Author(s):  
Ayesha Asloob Qureshi
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Lexicographic Order ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Ideal Lattices ◽  
The Ideal

In this paper we consider Hibi rings and Rees rings attached to a poset. We classify the ideal lattices of posets whose Hibi relations are indispensable and the ideal lattices of posets whose Hibi relations form a quadratic Gröbner basis with respect to the rank lexicographic order. Similar classifications are obtained for Rees rings of Hibi ideals.

scholarly journals Subalgebra Analogue to standard basis for ideal

2011 ◽  
Vol 48 (4) ◽  
pp. 458-474
Author(s):  
Junaid Khan
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Standard Basis ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis

A theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial orderings which are not necessarily well orderings [1]) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Analogue to Gröbner Basis for Ideals”, is treated in [6]. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them.

scholarly journals An extension of Buchberger’s criteria for Gröbner basis decision

2010 ◽  
Vol 13 ◽  
pp. 111-129
Author(s):  
John Perry
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Characterization Theorem ◽  
Polynomial Systems ◽  
Polynomial Ideal ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Case Scenario ◽  
Worst Case ◽  
New Criterion

AbstractTwo fundamental questions in the theory of Gröbner bases are decision (‘Is a basisGof a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known thatGis a Gröbner basis if and only if a certain set of polynomials (theS-polynomials) satisfy a certain property. In general there arem(m−1)/2 of these, wheremis the number of polynomials inG, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systemsGfor which the new criterion has dramatic impact, reducing the worst-case scenario fromm(m−1)/2 S-polynomials tom−1.

Border bases and kernels of homomorphisms and of derivations

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Janusz Zieliński
Keyword(s):  
Polynomial Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
The Ideal

AbstractBorder bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.

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