Elements In Group Rings Research Articles

Efficient quantum algorithms to break group ring cryptosystems

The security of widely-used public-key cryptographic protocols like RSA, Diffie–Hellman key exchange and the Digital Signature Algorithm (DSA) is under threat due to the emergence of quantum computers. Shor’s groundbreaking quantum algorithm poses a significant risk by efficiently factoring large integers into their prime factors, compromising RSA security. Additionally, it solves the Discrete Logarithm Problem, impacting certain Diffie–Hellman-based cryptosystems and digital signatures. Given this, it is imperative to enhance our current cryptographic tools for the post-quantum era, aiming to make it impractical, even with quantum algorithms, to breach the security of new cryptosystems. Prominent alternatives include elliptic curve and lattice-based cryptography, with exploration into other algebraic systems featuring difficult problems to ensure security. This paper establishes that systems based on the difficulty of inverting group ring elements are not quantum-resistant.

DNA CODES FROM REVERSIBLE GROUP CODES BY A VIRUS OPTIMISATION ALGORITHM

In this paper, we employ group rings and some known results on group codes to study reversible group DNA codes. We define and study reversible cyclic DNA codes from a group ring point of view and we also introduce the notion for self-reciprocal group ring elements. Moreover, we search for reversible group DNA codes with the use of a virus optimisation algorithm. We obtain many good DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement and the fixed GC-content constraints.

On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms

Under some assumptions, we prove the rank-part of the Mazur–Tate refined conjecture of BSD type. More concretely, we prove that the rank of the Selmer group of an elliptic modular form is less than or equal to the order of zeros of Mazur–Tate elements, or modular elements, which are elements in certain group rings constructed from special values of the associated L-function. Our main result is regarded as a generalization of our previous work on elliptic curves.

Aspects of Free Actions Based on Dependent Elements in Group Rings

This paper contains two directions of work. The first one gives material related to free action (an inner derivation) mappings on a group ring R[G] which is a construction involving a group G and a ring R and the dependent elements related to those mappings in R[G]. The other direction deals with a generalization of the definition of dependent elements and free actions. We concentrate our study on dependent elements, free action mappings and those which satisfy T(x)γ=δx,x∈R[G] and some fixed γ,δ∈R[G]. In the first part we work with one dependent element. In other words, there exists an element γ∈R[G] such that T(x)γ=γx,x∈R[G]. In second one, we characterize the two elements γ,δ∈R[G] which have the property T(x)γ=δx,x∈R[G] and some fixed γ,δ∈R[G], when T is assumed to have additional properties like generalized a derivation mappings.

Group LCD and group reversible LCD codes

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the n×n matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element v, one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element v, one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.

On partial augmentations of elements in integral group rings

Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.

On the probability of zero divisor elements in group rings

Let R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4.

Constructions for self-dual codes induced from group rings

In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.

Closed manifolds with transcendentalL2-Betti numbers:

In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an L2-Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L2-Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below). In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the L2-dimension of the kernel is the given number. We follow the method pioneered by Austin in "Rational group ring elements with kernels having irrational dimension" arXiv:0909.2360) but refine it to get very explicit calculations which make the above statements possible.

Group ring elements with large spectral density

Given \({\delta }>0\) we construct a group \(G\) and a group ring element \(S\in \mathbb Z[G]\) such that the spectral measure \(\mu \) of \(S\) fulfils \(\mu ((0,{\varepsilon })) > \frac{C}{|\log ({\varepsilon })|^{1+{\delta }}}\) for small \({\varepsilon }\). In particular the Novikov-Shubin invariant of any such \(S\) is \(0\). The constructed examples show that the best known upper bounds on \(\mu ((0,{\varepsilon }))\) are not far from being optimal.

A Brumer–Stark conjecture for non-abelian Galois extensions

Let K/k be an abelian extension of number fields. The Brumer–Stark conjecture predicts that a group ring element constructed from special values of L-functions associated to K/k annihilates the ideal class group of K. Moreover it specifies that the generators obtained have special properties. The aim of this article is to state and study a generalization of this conjecture to non-abelian Galois extensions that is, in spirit, very similar to the original conjecture.

Rational group ring elements with kernels having irrational dimension

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.

Involutions and Anticommutativity in Group Rings

AbstractLet g denote an involution on a group G. For any (commutative, associative) ring R (with 1), * extends linearly to an involution of the group ring RG. An element α ∊ RG is symmetric if α* = α and skew-symmetric if α* = -α. The skew-symmetric elements are closed under the Lie bracket, [αβ] = αβ - βα. In this paper, we investigate when this set is also closed under the ring product in RG. The symmetric elements are closed under the Jordan product, α˚α = αβ +βα. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ∗-algebra.

Characterizations of Bent and Almost Bent Function on $${\mathbb{Z}_p^2}$$

Bent and almost-bent functions on $${\mathbb{Z}_p^2}$$ are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on $${\mathbb{Z}_p^2}$$ are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on $${\mathbb{Z}_p^2}$$ in two classes of $${\mathcal{M}}$$ ’s and $${\mathcal{PS}}$$ ’s, and show that the graph set corresponding to a bent function on $${\mathbb{Z}_p^2}$$ can be written as the sum of a graph set of $${\mathcal{M}}$$ ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of $${\mathcal{M}}$$ ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.

Algebraic constructions of LDPC codes with no short cycles

An algebraic group ring method for constructing codes with no short cycles in the check matrix is derived. It is shown that the matrix of a group ring element has no short cycles if and only if the collection of group differences of this element has no repeats. When the method is applied to elements in the group ring with small support this gives a general method for constructing and analysing low density parity check (LDPC) codes with no short cycles from group rings. Examples of LDPC codes with no short cycles are constructed from group ring elements and these are simulated and compared with known LDPC codes, including those adopted for wireless standards.

Antisymmetric elements in group rings with an orientation morphism

Let R be a commutative ring, G a group and RG its group ring. Let φ σ : RG → RG denote the involution defined by φ σ (Σ r g g ) = Σ r g σ( g ) g –1 , where σ : G → {±1} is a group homomorphism (called an orientation morphism). An element x in RG is said to be antisymmetric if φ σ ( x ) = – x . We give a full characterization of the groups G and its orientations for which the antisymmetric elements of RG commute.

Mahler measure under variations of the base group

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the free abelian group or the infinite dihedral group, and others.

Codes from zero-divisors and units in group rings

A new construction method for codes using encodings from group rings is described and presented. They consist primarily of two types: zero-divisor and unit derived codes. Previous codes from group rings focused on ideals; for example, cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. The authors use the result that a group ring is isomorphic to a certain well-defined ring of matrices and thus, every group ring element has an associated matrix. This allows matrix algebra to be used as needed in the study and production of codes, enabling the creation of standard generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties, such as being LDPC or self-dual, may be expressed as properties within the group ring, thus enabling the construction of codes with these properties. The methods are general enabling the construction of codes with many types of group rings. There is no restriction on the ring and thus codes over the integers, over matrix rings or even over group rings themselves are possible and fruitful.

Lie properties of symmetric elements in group rings II

Let F be a field of characteristic different from 2, and G a group with involution ∗ . Write ( F G ) + for the set of elements in the group ring F G that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and ( F G ) + is Lie nilpotent (resp. Lie n -Engel), then F G is Lie nilpotent (resp. Lie m -Engel, for some m ). Here, we classify the groups containing 2-elements such that ( F G ) + is Lie nilpotent or Lie n -Engel.