Bijective Map Research Articles

Code Design for Non-Coherent Index Modulation

This letter explores the concept of non-coherent index modulation (IM) in a general setting from the standpoint of coding theory. In doing so, we establish a bijective mapping from the set of activation patterns of an IM system to the set of codewords of a binary block code. We show that a non-coherent IM system whose active indices are independently affected by additive white Gaussian noise and Rayleigh fading can achieve a diversity order of $t+1$ if its activation patterns follow a $t$ -error correcting code designed for a Z-channel. Activation codes employed for IM systems are either constant or variable-weight codes. Using a variable-weight code requires a slightly higher detection complexity, but it can result in a diversity order significantly higher than the maximum achievable diversity order of a constant-weight code.

Enhanced Huffman Coded OFDM With Index Modulation

In this paper, we propose an enhanced Huffman coded orthogonal frequency-division multiplexing with index modulation (EHC-OFDM-IM) scheme. The proposed scheme is capable of utilizing all legitimate subcarrier activation patterns (SAPs) and adapting the bijective mapping relation between SAPs and leaves on a given Huffman tree according to channel state information (CSI). As a result, a dynamic codebook update mechanism is obtained, which can provide more reliable transmissions. We take the average block error rate (BLER) as the performance evaluation metric and approximate it in closed form when the transmit power allocated to each subcarrier is independent of channel states. Also, we propose two CSI-based power allocation schemes with different requirements for computational complexity to further improve the error performance. Subsequently, we carry out numerical simulations to corroborate the error performance analysis and the proposed dynamic power allocation schemes. By studying the numerical results, we find that the depth of the Huffman tree has a significant impact on the error performance when the SAP-to-leaf mapping relation is optimized based on CSI. Meanwhile, through numerical results, we also discuss the trade-off between error performance and data transmission rate and investigate the impacts of imperfect CSI on the error performance of EHC-OFDM-IM.

Noise Free Fully Homomorphic Encryption Scheme Over Non-Associative Algebra

Among several approaches to privacy-preserving cryptographic schemes, we have concentrated on noise-free homomorphic encryption. It is a symmetric key encryption that supports homomorphic operations on encrypted data. We present a fully homomorphic encryption (FHE) scheme based on sedenion algebra over finite $Z_{n}$ rings. The innovation of the scheme is the compression of a 16-dimensional vector for the application of Frobenius automorphism. For sedenion, we have $p^{16}$ different possibilities that create a significant bijective mapping over the chosen 16-dimensional vector that adds permutation to our scheme. The security of this scheme is based on the assumption of the hardness of solving a multivariate quadratic equation system over finite $Z_{n}$ rings. The scheme results in $256n$ multivariate polynomial equations with $256+16n$ unknown variables for $n$ messages. For this reason, the proposed scheme serves as a security basis for potentially post-quantum cryptosystems. Moreover, after sedenion, no newly constructed algebra loses its properties. This scheme would therefore apply as a whole to the following algebras, such as 32-dimensional trigintadunion.

Multiplicative Mappings of Gamma Rings

Let Mi and Γi (i = 1, 2) be abelian groups such that Mi is a Γi-ring. An ordered pair (ϕ, φ) of mappings is called a multiplicative isomorphism of M1 onto M2 if they satisfy the following properties: (i) ϕ is a bijective mapping from M1 onto M2, (ii) φ is a bijective mapping from Γ1 onto Γ2 and (iii) ϕ(xγy) = ϕ(x)φ(γ)ϕ(y) for every x, y ∈ M1 and γ ∈ Γ1. We say that the ordered pair (ϕ, φ) of mappings is additive when ϕ(x + y) = ϕ(x) + ϕ(y), for all x, y ∈ M1. In this paper we establish conditions on M1 that assures that (ϕ, φ) is additive.

Computation of Even-Odd Harmonious Labeling of Certain Family of Acyclic Graphs

Let ( ) , GVE be a graph with p number of vertices and q number of edges. An injective function f :V →1, 3, 5, , 2 p −1 is called an even-odd harmonious labeling of the graph G if there exists an induced edge function : →0,2,,2( −1) * f E q such that i) * f is bijective function ii) ( = ) = ( ( ) + ( ))( 2 ) * f e uv f u f v mod q The graph obtained from this labeling is called even-odd harmonious graph.

A Diffie-Hellman Encryption Scheme over Elliptic Curvesusing Golden Matrices

In this paper, we proposed Diffie-Hellman encryption scheme based on golden matrices over the elliptic curves. This algorithm works with a bijective function defined as characters of ASCII from the elliptic curve points and the matrix developed the additional personal key, which was obtained from the golden matrices.

On maps preserving rank-one nilpotents

In this paper, we show that a linear, bijective map such that whenever XY = M, for M, N rank-one nilpotents, is of the form for an invertible and a non-zero . We extend this result to show in general that, given any M and N in the ring, , of infinite matrices over a field with finitely many non-zero entries, a linear, bijective map , such that whenever XY = M must satisfy for all and .

Isometries of absolute order unit spaces

We prove that a unital, bijective linear map between absolute order unit spaces is an isometry if and only if it is absolute value preserving. We deduce that, on (unital) JB-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that a unital, bijective $$*$$ -linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) $$\hbox {C}^*$$ -algebras such maps are precisely $$\hbox {C}^*$$ -algebra isomorphisms.

Probability representation of quantum mechanics and star product quantization

This paper presents a review of star-product formalism. This formalism provides a description for quantum states and observables by means of the functions called’ symbols of operators’. Those functions are obtained via bijective maps of the operators acting in Hilbert space. Examples of the Wigner-Weyl symbols (Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified for the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators required for construction of bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relationship between structure constants of associative star-product of operator symbols and quantizer-dequantizer operators is reviewed.

Exploration of Non-Volatile MTJ/CMOS Circuits for DPA-Resistant Embedded Hardware

Magnetic tunnel junction (MTJ/CMOS-based Logic-in-Memory (LiM) circuits have nearly zero leakage power dissipation, and they are very appropriate to design low-power hardware. However, the differences in power consumption between the switching of MTJ devices increase the vulnerability of differential power analysis (DPA)-based side-channel attacks. Furthermore, the MTJ/CMOS hybrid logic circuits that require frequent switching of MTJs are not very energy efficient due to the significant energy required to switch the MTJ devices. MTJ/CMOS circuits consume uniform power if there is no switching of MTJs. In this article, we have investigated the novel approach of building cryptographic hardware in MTJ/CMOS circuits using a lookup table (LUT)-based method where the data stored in MTJs are constant during the entire encryption/decryption operation. As a case study, we have designed a non-linear bijective function of the PRESENT-80 lightweight cryptographic algorithm called substitution box or S-box and one round of PRESENT-80 cryptographic hardware using MTJ/CMOS circuits. The designs are simulated using 45 nm CMOS technology with perpendicular anisotropy CoFeB/MgO MTJ model using Cadence Spectre simulator. From our simulations, we found that the PRESENT-80 S-box circuit and one round of PRESENT-80 cryptographic hardware implemented using MTJ/CMOS circuits save up to 26% and 29% of energy, respectively, compared to the conventional CMOS-based designs at 50 MHz. Furthermore, the security of the MTJ/CMOS circuits has been evaluated by performing a simulation-based DPA attack. From our simulations, we found that the PRESENT-80 cryptographic hardware implemented using MTJ/CMOS circuits is resistant against DPA attack. Low-energy and DPA-resistant property along with high density and low leakage make MTJ/CMOS circuits suitable to implement in low-energy and secure embedded cryptographic hardware.

On (a, d)-SEAT labeling of forests of subdivided stars

Graph ϒ = (V(ϒ), E(ϒ)) contain finite nodes V(ϒ) and finite edges E(ϒ). We also represent the order of the graph and size as μ = |V(ϒ)| and v = |V(ϒ)|. A graph is called (a, d)-edge magic total (EAT) labeling if there exists a bijective map ϕ from V(ϒ) ∪ E(ϒ) to the elements if the weight-set X = {ω(qr)|qr ∈ E(ϒ)} is arithmetic progression (A. P.) of positive integers which is started with a, d as common difference and ω(qr) = ϕ(q) + ϕ(qr) + ϕ(r). We say ω as the set of edge-weights. Further, if then the graph ϒ is said to be super (a, d)-edge antimagic total((a, d)-SEAT) labeling. In this article, we represent some new result on the (a, d)-SEAT labeling of some disjoint copies for subdivided star for the parameter .

Analisis Kemampuan Mahasiswa dalam Menyelesaikan Soal Pembuktian tentang Isomorfisma Grup

Group Isomorphism is one of the sub topic in the Algebra Structure. This sub topic required prerequisite material about the bijective function. If this prerequisite material is not mastered, it will be difficult to study Isomorphism Group material. There are still many students who found difficulties to solve probative questions about Group Isomorphism. The major problem are they were forgetting and not understanding the prerequisite material. Therefore, it should be a further research on the ability of students to prove the questions of proof and errors in preparing evidence about Group Isomorphism.

On maps preserving Lie products equal to e11 - e22

Bijective linear maps preserving Lie products equal to rank-one nilpotents were recently shown to have nonstandard descriptions. In this paper, we show that bijective linear maps preserving Lie products equal to , a rank-two matrix, are standard commutativity preserving maps.

An Interaction Between User and an Augmented Reality System using A Generalized Finite State Automata and A Universal Turing Machine

Most interactions between users and augmented reality system (ARS) are that user assigns a marker to ARS, and the ARS responds the marker. In this context, a marker is mapped to an ARS's response, or in general, an array of markers is mapped to an array of ARS's responses. This interaction is a constant or linear complexity interaction since there is only a bijective mapping between a set of markers and a set of ARS's responses. In this research, we propose the expansion of user - ARS complexity into the polynomial. It is an interaction in which not only one marker for a single response (or an array of markers for an array of ARS's responses), but the interaction by which user provides a string of markers as a word of markers (i.e., a combination of multiple markers as a word) for a single ARS's response. The set of strings of markers to the ARS provided by users built a regular language. So that, the complexity of the user-ARS interaction became polynomial. This interaction was implemented by stating the user's language by means of a generalization of finite state automata (gFSA) and placing a universal Turing machine (UTM) between user and ARS, where the UTM as an interpreter translating or mapping the user language to ARS. To summarize our research, overall we apply the idea of a formal language into the interaction between the user and ARS, thereby changing the complexity of the interaction to polynomial even expandable to nondeterministic polynomials.

The local antimagic on disjoint union of some family graphs

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Shake genus and slice genus

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0(K)$, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41 of [Kir97]. As corollaries we show that Rasmussen's $s$ invariant is not a $0$-trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from [4MKC16].

Differential Spectrum of Kasami Power Permutations Over Odd Characteristic Finite Fields

Functions with low differential uniformity have important applications in cryptography, coding theory, and sequence design. The differential spectrum of a cryptographic function is of great interest for estimating its resistance to some variants of differential cryptanalysis. Finding power permutations (i.e., monomial bijective mappings) over finite fields with low differential uniformity and determining their differential spectra have received a lot of attention over the past two decades. The objective of this paper is to study the differential properties of the well-known Kasami power permutations $x^{p^{2k}-p^{k}+1}$ over $ {\mathrm {GF}}(p^{n})$ , where $p$ is an odd prime and $k$ is an integer with $\gcd (n,k)=1$ . It turns out that this family of monomials is differentially $(p+1)$ -uniform. Our result in the case of $p=3$ gives an affirmative solution to a recent conjecture by Xu, Cao, and Xu. Most notably, the differential spectrum of this family of power permutations is completely determined.

Low Complexity NOMA System With Combined Constellations

In this letter, a non-orthogonal multiple access system based on combinatorial designs (C-NOMA) is studied. The sparse pattern of balanced incomplete block designs provide an inherent mapping for this system. A novel constellation design is proposed to avoid the indeterminate equations when multiuser data is linearly combined over the resources. It can form a bijection mapping with much lower complexity compared to other NOMA approaches that require the design of complex codebooks and decoders. This will allow a larger number of users. The simulation results show that our system obtains an equivalent and/or better performance compared to other NOMA systems.

Additivity of Quadratic Maps on JB Algebras

The problem of automatic additivity of Jordan type homomorphisms has received a great deal of attention in theory of operator algebras as well in ring theory. We study additivity of quadratic maps, that is the maps between Jordan Banach algebras that preserve the quadratic product (a, b) → aba. The main result shows that any continuous quadratic bijective map between unital Jordan Banach algebras that is linear on associative subalgebras is automatically additive. This contributes to the Borification program in quantum theory and Mackey-Gleason problem on linearity of quasi-linear maps.

Lie maps on alternative rings

Let $${\mathfrak {R}}\, $$ be an alternative ring containing a nontrivial idempotent and $${\mathfrak {R}}\, '$$ be another alternative ring. Suppose that a bijective mapping $$\varphi : {\mathfrak {R}}\, \rightarrow {\mathfrak {R}}\, '$$ is a Lie multiplicative mapping and $${\mathfrak {D}}\, $$ is a Lie triple derivable multiplicative mapping from $${\mathfrak {R}}\, $$ into $${\mathfrak {R}}\, $$. Under a mild condition on $${\mathfrak {R}}\, $$, we prove that $$\varphi $$ and $${\mathfrak {D}}\, $$ are almost additive, that is, $$\varphi (a + b) - \varphi (a) - \varphi (b) \in \mathcal {Z}({\mathfrak {R}}\, ')$$ and $${\mathfrak {D}}\, (a+b) - {\mathfrak {D}}\, (a) - {\mathfrak {D}}\, (b) \in \mathcal {Z}({\mathfrak {R}}\, )$$ for all $$a,b \in {\mathfrak {R}}\, $$, where $$\mathcal {Z}({\mathfrak {R}}\, ')$$ is the commutative centre of $${\mathfrak {R}}\, '$$ and $$\mathcal {Z}({\mathfrak {R}}\, )$$ is the commutative centre of $${\mathfrak {R}}\, $$. As applications, we show that every Lie multiplicative bijective mapping and Lie triple derivable multiplicative mapping on prime alternative rings are almost additive.