Zero Divisors Research Articles

The QARMA Block Cipher Family. Almost MDS Matrices Over Rings With Zero Divisors, Nearly Symmetric Even-Mansour Constructions With Non-Involutory Central Rounds, and Search Heuristics for Low-Latency S-Boxes

This paper introduces QARMA, a new family of lightweight tweakable block ciphers targeted at applications such as memory encryption, the generation of very short tags for hardware-assisted prevention of software exploitation, and the construction of keyed hash functions. QARMA is inspired by reflection ciphers such as PRINCE, to which it adds a tweaking input, and MANTIS. However, QARMA differs from previous reflector constructions in that it is a three-round Even-Mansour scheme instead of a FX-construction, and its middle permutation is non-involutory and keyed. We introduce and analyse a family of Almost MDS matrices defined over a ring with zero divisors that allows us to encode rotations in its operation while maintaining the minimal latency associated to {0, 1}-matrices. The purpose of all these design choices is to harden the cipher against various classes of attacks. We also describe new S-Box search heuristics aimed at minimising the critical path. QARMA exists in 64- and 128-bit block sizes, where block and tweak size are equal, and keys are twice as long as the blocks. We argue that QARMA provides sufficient security margins within the constraints determined by the mentioned applications, while still achieving best-in-class latency. Implementation results on a state-of-the art manufacturing process are reported. Finally, we propose a technique to extend the length of the tweak by using, for instance, a universal hash function, which can also be used to strengthen the security of QARMA.

The QARMA Block Cipher Family. Almost MDS Matrices Over Rings With Zero Divisors, Nearly Symmetric Even-Mansour Constructions With Non-Involutory Central Rounds, and Search Heuristics for Low-Latency S-Boxes

This paper introduces QARMA, a new family of lightweight tweakable block ciphers targeted at applications such as memory encryption, the generation of very short tags for hardware-assisted prevention of software exploitation, and the construction of keyed hash functions. QARMA is inspired by reflection ciphers such as PRINCE, to which it adds a tweaking input, and MANTIS. However, QARMA differs from previous reflector constructions in that it is a three-round Even-Mansour scheme instead of a FX-construction, and its middle permutation is non-involutory and keyed . We introduce and analyse a family of Almost MDS matrices defined over a ring with zero divisors that allows us to encode rotations in its operation while maintaining the minimal latency associated to {0, 1}-matrices. The purpose of all these design choices is to harden the cipher against various classes of attacks. We also describe new S-Box search heuristics aimed at minimising the critical path. QARMA exists in 64- and 128-bit block sizes, where block and tweak size are equal, and keys are twice as long as the blocks. We argue that QARMA provides sufficient security margins within the constraints determined by the mentioned applications, while still achieving best-in-class latency. Implementation results on a state-of-the art manufacturing process are reported. Finally, we propose a technique to extend the length of the tweak by using, for instance, a universal hash function, which can also be used to strengthen the security of QARMA.

Corrigendum: Covering of Line Graph of Zero Divisor Graph over Ring Zn

Corrigendum: Covering of Line Graph of Zero Divisor Graph over Ring Zn

Some noncommutative rings constructed on the base of polynomials x2 + tx + 1 and their zero divisors

Some noncommutative rings constructed on the base of polynomials x2 + tx + 1 and their zero divisors

Finiteness of 2-reflective lattices of signature (2,n)

A modular form for an even lattice L of signature (2,n) is said to be 2-reflective if its zero divisor is set-theoretically contained in the Heegner divisor defined by the (-2)-vectors in L. We prove that there are only finitely many even lattices with n>6 which admit 2-reflective modular forms. In particular, there is no such lattice in n>25 except the even unimodular lattice of signature (2,26). This proves a conjecture of Gritsenko and Nikulin in the range n>6.

The Zero Divisor Graph of 2 × 2 Matrices Over a Field

A zero divisor graph, Γ(R), is formed from a ring R by having each element of Z(R) \\ {0} to be a vertex in the graph and having two vertices u and v adjacent if the corresponding elements from the ring are nonequal and have product equal to zero. In this paper, the structure of the zero-divisor graph of 2 × 2 matrices over a field, Γ(M2(F)), are completely determined.

Projective zero divisor graphs of partially ordered sets

The zero divisor graph of a partially ordered set (poset, briefly) \((P, \le )\) with the least element 0, which is denoted by \(G^*(P)\), is an undirected graph with vertex set \(P^*= P{\setminus } \{0\}\) and, for two distinct vertices x and y, x is adjacent to y in \(G^*(P)\) if and only if \(\{x,y\}^l=\{0\}\), where, for a subset S of P, \(S^l\) is the set of all elements \(x\in P\) with \(x\le s\), for all \(s\in S\). In this paper we completely characterize all posets P with projective zero divisor graphs \(G^*(P)\).

Note on Sus̆kevic̆’s problem on zero divisors

ABSTRACTWe solve an old Sus̆kevic̆’s problem on right zero divisors in the ring T(∞,K) of infinite ℕ×ℕ upper triangular matrices over a field K. In the solution we use a notion of strong linear independence.

Algebras of cubic matrices

We consider algebras of -cubic matrices (with ). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size m. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff . We introduce a notion of accompanying algebra (which is -dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed.

Solvability conditions of problem of synthesizing proportional-integral control of a linear MIMO system

The solvability conditions and just the solution of the problem of the regular and irregular proportional-integral (PI) control are found in accordance with the properties of invariant zeros of a multi-input multioutput (MIMO) system. It is proved that the problem of synthesizing the control of the MIMO system is solvable if and only if the pair of matrices (A, B) that describes a control plant is controllable and the matrix BL?ACR? (where BL? is the left zero divisor of the matrix B and CR? is the right zero divisor of the output matrix C) has a complete row rank.

On the Zero Divisor Graphs of a Class of Commutative Completely Primary Finite Rings

Let R be a Completely Primary Finite Ring with a unique maximal ideal Z(R)), satisfying ((Z(R))n−1 ̸= (0) and (Z(R))n = (0): The structures of the units some classes of such rings have been determined. In this paper, we investigate the structures of the zero divisors of R:

The kernel of Ribet's isogeny for genus three Shimura curves

There are exactly nine reduced discriminants D of indenite quater- nion algebras over Q for which the Shimura curve XD attached to D has genus 3. We present equations for these nine curves and, moreover, for each D we de- termine a subgroup c(D) of cuspidal divisors of degree zero of Jac(X0(D)) new such that the abelian variety Jac(X0(D)) new =c(D) is the jacobian of the curve XD.

Functional equations for double series of Euler type with coefficients

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors.

Exact pairs of homogeneous zero divisors

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that if S is a compressed level algebra, then S does not contain any homogeneous zero divisors.

On the extended zero divisor graph of commutative rings

In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors $x$ and $y$ of a commutative ring $R$ are adjacent whenever there exist two nonnegative integers $n$ and $m$ such that $x^ny^m=0$ with $x^n\neq 0$ and $y^m\neq 0$. This yields an extension of the classical zero divisor graph $\Gamma(R)$ of $R$, which will be denoted by $\overline{\Gamma}(R)$. First we distinguish when $\overline{\Gamma}(R)$ and $\Gamma(R)$ coincide. Various examples in this context are given. We show that if $\overline{\Gamma}(R) \not=\Gamma(R)$, then $\overline{\Gamma}(R)$ must contain a cycle. We also show that if $\overline{\Gamma}(R) \not=\Gamma(R)$ and $\overline{\Gamma}(R)$ is complemented, then the total quotient ring of $R$ is zero-dimensional. Among other things, the diameter and girth of $\overline{\Gamma}(R)$ are also studied.

Local nilpotency of the McCrimmon radical of a Jordan system

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwiches of the corresponding Tits–Kantor–Koecher Lie algebras, we prove local nilpotency of the McCrimmon radical of a Jordan system (algebra, triple system, or pair) over an arbitrary ring of scalars. As an application, we show that simple Jordan systems are always nondegenerate.

The Zero Divisor Graph of the Ring Zqp

In this paper we construct a star zero divisor graph from the zero divisor graph of the ring Zqp. The star zero divisor graph is obtained by removing some vertices from the zero divisor graph Γ(Zqp), in different ways , but the best way to get star zero divisor graph SΓ(Zqp) is by removing an odd number of zero divisors. Finally the crossing number, the girth and the diameter of this graph is also determined.

Book Thickness of Planar Zero Divisor Graphs

Let $R$ be a finite commutative ring with identity. We form the zero divisor graph of $R$ by taking the nonzero zero divisors as the vertices and connecting two vertices, $x$ and $y$, by an edge if and only if $xy=0$. We establish that if the zero divisor graph of a finite commutative ring with identity is planar, then the graph has a planar supergraph with a Hamiltonian cycle. We also determine the book thickness of all planar zero divisor graphs.

NÚMEROS PERPLEXOS: UMA ABORDAGEM PARA O ENSINO MÉDIO

http://dx.doi.org/10.5902/2179460X14316In the search for disseminating the subject among mathematics students, in this article, adapted of a dissertation of Professional Master’s degree in Mathematics - does PROFMAT come the group of the perplexed numbers (P), done take by analogy with the complex numbers, will the addition operations and multiplication of the perplexed numbers be defined, in elapsing of the presentation it is done necessary we admit coming results of the hyperbolic functions in order to we present the perplexed numbers in his/her exponential form. It will be made the discussion concerning the geometry of the perplexed plan, including the description of the divisors of zero of this group.. The polar form of a Perplexed number will be studied placing them in the 4 quadrants of the perplexed plan. Finally a comparative picture will be presented between the perplexed ones and the compounds.

Zero Divisors in Skew Power Series Rings

We study the rigid property of rings in the concrete, via power series rings, matrix rings, and Insertion-of-Factors-Property. In the procedure, we concentrate our attention on the skew power-serieswise Armendariz property. We next apply the McCoy condition to skew power series rings, which induces a generalization of skew power-serieswise Armendariz. The relationship is investigated among the properties above and related ring properties, and several known results are obtained as consequences of our results.