The geometric-quadratic index (GQ) is a recently introduced degree-based topological descriptor, and Kumar et al. observed that it is potentially a very good molecular descriptor. In this thesis, we characterize the extremal (chemical) graphs and trees with respect to (GQ) for a given order and size. Then, the n-vertex trees, unicyclic and bicyclic graphs with the first, second, third, fourth, fifth, and sixth maximum values of this geometric-quadratic index are determined.  

  For a graph G, the first Zagreb index is defined as the sum of the squares of the vertex degrees, while the second Zagreb index is the sum of the products of the degrees of adjacent vertices. The aim of this paper is to completely characterize n-vertex trees with given k ? 1 vertices that have a fixed maximum degree ? ? 3 with respect to the maximal and minimal Zagreb indices. Furthermore, our results provide detailed insights into the structure of extremal trees and are equally applicable to the class of chemical trees.

We introduce the notion of coretractable modules. A module $M$ is said to be coretractable if for every nonzero factor module $M/N$ (where $N \\leq M$), there exists a nonzero $R$-homomorphism $f: M/N \\to M$. We prove that all right (left) modules over a ring are coretractable if and only if the ring is Morita equivalent to a finite product of local right and left perfect rings.  

In this thesis, we focus on the Ekeland’s variational principle and some of its applications to equilibrium and quasi-equilibrium problems and existential results for equilibrium and Minty equilibrium problems. First, we state the Ekeland’s variational principle and then, using it, we present equilibrium forms of the Ekeland’s variational principle for cyclically antimonotone and cyclically monotone functions, and using these equilibrium forms, we present two existence theorems for equilibrium and Minty equilibrium problems, respectively. We also state an existence theorem for the quasiequilibrium problem using the Ekeland’s variational principle. Next, we study the existence of solutions for equilibrium and Minty equilibrium problems and then the relationship between them. It should be noted that with different assumptions such as the Minty lemma, cyclically monotonicity and cyclically antimonotonicity, we will study this relationship in topological spaces, topological vector spaces, and Banach spaces.   

We study a multi-group SAIRS-type epidemic model with vaccination. the role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. This is a natural extension of the homogeneous mixing SAIRS model with vaccination studied in Ottaviano et. al(2021) to a network of communities. We provide a global stability analysis for the model. We determine the value of the basic reproduction number R0 and prove that the disease-free equilibrium is globally asymptotically stable if R0<1. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when R0=1. Moreover, if R0>1, the disease-free equilibrium is unstable and a unique equilibrium exists. First, we investigate the local asymptotic stability of the endemic equilibrium and subsequently its global stability, for two variations of the original model. Last, we provide numerical simulations to compare the epidemic spreading on different networks topologies. 

   Today, the emergence of generative models based on machine learning has provided a significant reduction in the bit rate of speech codecs. However, in real conditions and in the presence of destructive factors such as noise and distortion, the above processes face serious problems, which is caused by the sensitivity of the maximum likelihood criterion to outliers, as well as the inefficiency of modeling the sum of independent signals with Autoregressive model is used. In this thesis, a method based on predicting variance regularization is introduced to reduce sensitivity to outliers and thus increase system performance. In addition, it is shown that noise reduction to remove unwanted signals can increase the performance significantly. Also, extensive objective evaluations will be presented, which show that the proposed system based on the generative model provides a new coding performance mode for real-time speech signals at 3 Kbit/s.

  در

   تعامل شكار-شكارچي يا منبع-مصرف كننده، اساسي ترين و مهم ترين فرآيند در پويايي جمعيت است. بسياري از گونه   ها، مانند گياهان تك باره و جانوران يك بار زا كه پس از زادآوري مي ميرند، داراي نسل هاي ناهمپوشان گسسته هستند و تولد آنها در فصول توليد مثل به طور منظم اتفاق مي افتد. فعل و انفعالات آنها با معادلات تفاضلي توصيف و يا به صورت نگاشت هاي زمان-گسسته فرموله مي شوند. در اين پايان نامه، انشعابات را در يك مدل شكار-شكارچي گسسته با تابع پاسخ غير يكنوا كه توسط تابع ساده شده هالينگ IV توصيف شده است، مطالعه مي كنيم. همچنين ثابت مي كنيم كه مدل فوق انشعاب هاي مختلفي از هم بعد 1 را نشان مي دهد، كه شامل انشعابات فولد، انشعاب ترا بحراني، انشعاب فليپ و انشعاب نيمارك-ساكر مي باشد، زيرا مقادير پارامترها متفاوت است. علاوه بر اين، وجود انشعاب بوگدانوف -تاكنز از هم بعد 2 را مشخص و عبارات تقريبي منحني هاي انشعاب را محاسبه مي كنيم شبيه سازي هاي عددي نيز دهد براي نشان دادن تحليل نظري ارائه شده اند اين نتايج نشان مي دهد كه انشعاب بوگدانوف-تاكنز از هم بعد 2 در تكيني تبهگن در هر سه نسخه زمان-پيوسته، زمان-گسسته و زمان-تاخيري از مدل شكار-شكارچي با تابع پاسخ غير يكنوا برقرار است.

  مدولهاي درون منظم موضوع بسياري از مقالات در طول شصت سال گذشته بوده كه فوچز اين سوال را مطرح كرد كه كدام گروه آبلي درون منظم هستند. گلاز و ويكلس در [19[ و رنگسومي در [30 [به اين سوال براي طبقات بزرگي از گروههاي آبلي پاسخ دادند. اما مسئله همچنان باز است. وير ? در [3? [جز اولين كساني بود كه روي مدول هاي درون منظم روي حلقه هاي دلخواه بررسي كرد، او بيشتر بر مدول تصويري تمركز داشت. لي و همكارانش، بعدها در [23 [تحقيقات كلي تري در مورد مدولهاي درون منظم انجام دادند. حلقه هاي منظم يكه يكطرفه و مدولهاي درون منظم يكه يكطرفه، براي اولين بار توسط ارليچ در [10 ،11 [مورد مطالعه قرار گرفت. لي و ژانگ در [38 [در مورد اين موضوع توضيح دادند. همچنين مدولهاي درون منظم قوي توسط لي و همكارانش در [23 ،38 [مورد بحث قرار گرفتند كه اين مدول ها را “مدولهاي درون منظم آبلي” ناميدند. [همچنين گلاز و ويكلس در [19 ،([نتايجي در مورد ايده آل درون منظم(براي مثال، گروه هاي آبلي ) ثابت كرد. در مطالب پيش رو هدف بررسي سه مورد خواهد بود. ابتدا چندين نتيجه كلي در مورد اشكال مختلف “درون منظم” را ثابت خواهيم كرد، كه بر تئوري توسعه يافته قبلي بسط داده شده است. سپس بسياري از نتايج شناخته شده در مورد “درون منظم” در گروههاي آبلي را به مدولها روي حلقه هاي جابه جايي با طيف نوتري گسترش خواهيم داد. در نهايت تعميم مفيدي از مدولهاي درون منظم (روي حلقه هاي جابهجايي) را تعريف مي كنيم كه آن را مدولهاي درون منظم ضعيف مي ناميم و بسياري از ويژگيهاي اين مدولها را بررسي خواهيم كرد

  ‎In this thesis‎, ‎we study the anticenter(Lie-center) of Leibniz algebras and give several concepte that are related to this notion‎. Also give some bounds on the dimension of hypercenter of a Liebniz algebra.‎‎ Furthermore, we study Lie-central extensions and we obtain a six-term exact sequence of Lie-homology groups associated to a Lie-central extension. This allow us to characterize Lie-stem ‎extensions,‎ stem-covers and Lie-capable Liebniz algebras.

  ‎Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements‎. ‎The Grassmann graph of $V$‎, ‎denoted by $\\Gamma_k(V)$‎, ‎is a simple graph whose vertex set is the set of all $k$-dimensional su  aces of $V$‎, ‎with <k<n‎ -‎1$ and two distinct vertices are adjacent if their intersection is $k-1$-dimentional su  ace of $V$‎. ‎Denote by $\\Gamma(n‎, ‎k)_q$ the restriction of $\\Gamma_k(V)$ to the set of all non-degenerate linear $[n‎, ‎k]_q$-codes‎. ‎In this thesis‎, ‎we show that if $n$ is sufficiently large then there exists pairs of codes whose distances in the graphs $\\Gamma_k(V)$ and $\\Gamma(n‎, ‎k)_q$ are distinct‎. ‎Also‎, ‎one 0px; margin-right: 0px; text-indent: 0px;">‎Among other results‎, ‎it is shown that the induced subgraph of $\\Gamma_k(V)$ on projective $[n‎, ‎k]_q$-codes is connected and its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph‎. ‎Then we study the graphs of simplex codes‎. ‎Finally‎, ‎we prove that binary simplex codes of dimension $3$ are precisely maximal singular su  aces of a non-degenerate quadratic form‎.

  Let $G$ be a graph with adjacency matrix $A(G)$‎. ‎The nullity $\\eta(G)$ of $G$ is the multiplicity of zero as an eigenvalue of $A(G)$‎, ‎which has received a lot of attention because of its chemical importance‎. ‎Here‎, ‎some upper bounds for $\\eta(G)$ are given‎. ‎For example‎, ‎it is shown that $\\eta(G) \\leq \\frac{(\\Delta‎- ‎2)n‎ + ‎2}{\\Delta‎- ‎1}$ and the equality holds if and only if $G \\cong C_n$ ($n \\equiv 0({\\rm mod} 4)$) or $G \\cong K_{\\Delta‎, ‎\\Delta}$‎. ‎The multiplicity of an arbitrary eigenvalue $\\lambda$ of $A(G)$ is denoted as $m(G‎, ‎\\lambda)$‎. ‎Let $\\theta(G) =|E(G)|-|V(G)|‎ +‎1$ be the cyclomatic number of $G$ and $p(G)$ denote the number of pendant vertices of $G$‎. ‎In this thesis‎, ‎it is proved that for a connected graph $G$‎, ‎$m(G‎, ‎\\lambda) \\leq 2\\theta(G)‎ +‎p(G)$ and the equality holds if and only if $G$ is a cycle $C_n$ and $\\lambda=2\\cos \\frac{2k\\pi}{n}$ with $k=1,2,\\ldots \\lceil\\frac{n}{2}\\rceil-1$‎.

In this thesis, we have examined the necessary and sufficient conditions for a group ring to be weakly nil-neat.  The first chapter contains the primitive concepts and definitions. In the seconds chapter, we define nil-clean group ring and weakly nil-clean group ring. The main purpose of the third chapter is to examine the weakly nil-clean group ring.

   Let T be a triangular algebra over a commutative ring R, _ be an automorphism of T and Z_(T ) be the _-center of T . Suppose that q : T _ T ??! T is an R-bilinear mapping and that Tq : T ??! T is a trace of q. Our aim is to describe the form of Tq satisfying the commuting condition [Tq; x]_ = 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain classical traingular algebras.

In this thesis a representation of continuous functions is presented and then the Reisz representation theorem by using equilibrium problem is investigated.

Encompassing many standard notions such as Dedekind finite and reversible rings we introduce and study a new property for subsets of a ring. We give many examples and characterize some rings such as 2-primal rings with the aid of this notion.In ‎addition‎, We study some properties relatedto zero divisors and reversibility in noncommutative rings .  

   Let nq(k‎, ‎d) be the minimum length n for which an [n‎, ‎k‎, ‎d] q-code exists‎. In coding theory‎, ‎there is a natural lower bound on nq(k‎, ‎d) ‎, ‎the Griesmer bound‎: nq(k‎, ‎d) ? gq(k‎, ‎d) =‎‎i=0k-1[dqi] ‎.‎ ‎In this thesis‎,   ‎a lot of new [n‎, ‎4‎, ‎d] 9-codes whose lengths are close to the Griesmer‎ bound were be constructed‎. ‎Also‎, ‎we prove the nonexistence of some linear codes attaining the Griesmer bound‎ using some geometric techniques through projective geometries to determine the‎ exact value of n9(4‎, ‎d) or to improve the known bound on n9(4‎, ‎d) for given values‎ of d‎. ‎Finally‎, ‎the updated table for n9

self-dual and cyclic codes over noncommutative rings

   ‎A ‎‎$‎‎k‎$‎‎-rose graph is a graph consisting of ‎‎$‎‎k‎$‎ cycles that all meet in one vertex. ‎In this thesis, it‎ is ‎shown‎ that except for two specific examples, these rose graphs are determined by the Laplacian spectrum. ‎Then‎ ‎it‎ is proved that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927–2016), ‎‎we‎ show the specific case of the latter result for the adjacency matrix by using ‎Sachs' ‎theorem and a new result on the number of matchings in the disjoint union of paths.‎‎ ‎T‎hen a new method to determine the degree sequence of cospectral mates of a graph ‎is‎ introduced. ‎Among‎ other results, we prove that ‎all‎‎ $‎2‎$‎-rose ‎graphs,‎ with one exception, are determined by their signless Laplacian ‎spectrum‎.‎

  ‎In addition to their applications in data communication and storage‎, ‎linear codes also have nice applications in combinatorics and cryptography‎. ‎As a special class of linear codes‎, ‎minimal linear codes have important applications in secret sharing and secure two-party computation‎. ‎Constructing minimal linear codes with new and desirable parameters has been an interesting research topic in coding theory and cryptography‎. ‎Ashikhmin and Barg showed that $\\frac{w_{\\min}}{w_{\\max}}>\\frac{q-1}{q}$ is a sufficient condition for a linear code over the finite field$F_q$ to be minimal‎, ‎where $q$ is a prime power‎, ‎$w_{\\min}$ and $w_{max}$ denote the minimum and maximum nonzero weights in the code‎, ‎respectively‎. ‎The first objective of this thesis is to present a sufficient and necessary condition for linear codes over finite fields to be minimal‎. ‎This condition enables us to‎ ‎obtain several infinite families of minimal linear codes with $\\frac{w_{\\min}}{w_{\\max}}\\leq \\frac{q-1}{q}$‎. ‎The second objective of this thesis is to construct infinite families of binary and ternary minimal linear codes‎, ‎violating Ashikhmin and Barg's condition‎. ‎The weight distributions of all these minimal linear codes are also determined‎.  

  A ring R is called left comorphic if, for each a ? R, there exists b ? R such that Ra = l(b)and r(a) = bR. Examples include (von Neumann) regular rings, and Z p n for a prime p and n ? 1.

Let G be a simple graph with Laplacian spectrum  ????_1???_2???????_(n-1)???_n=0. The Laplacian spread of  G  is defined as S_L (G)=?_1-?_(n-1). In this thesis،the newlower bounds on the Laplacian spread of graphs in terms of invariantparameters of graphs such as bandwidth، independence number and vertex connectivity، are obtained. Then we present a new sharp upper bound for  SL(G)and use it to prove the conjecture that ”S_L (G)?n-1” for  t-quasi-regulargraphs when t??(n-3+2/n)  Among other results، it is shown that this conjecture is true for some special graphs such as triangle-free graphs. Finally، we give several sharp lower bounds for  SL(G) as well.  

  ‎‎Let $G$ be an $n$-vertex graph with $m$ edges, maximum degree $\\Delta$‎, ‎average degree $\\overline{d}=\\frac{2m}{n}$‎ ‎and‎‎ ‎clique number $\\omega$ having‎‎Laplacian spectrum $\\mu_1 \\geq \\mu_2 \\geq\\cdots\\geq ‎\\mu_{n-‎1} \\geq \\mu_n = 0$‎. ‎Let $S_k(G)=\\sum_{i=1}^k\\mu_i$ for every \\leq k \\leq n$‎. ‎Also‎, ‎assume ‎that $\\sigma$‎‎ ‎is ‎the number of Laplacian eigenvalues greater than or equal to average degree‎ ‎$\\overline{d}$‎. ‎In this thesis‎, ‎a lower bound for $S_{\\omega‎ -‎1}$ and an upper bound for $S_\\sigma(G)$ in terms of $m,\\Delta,\\omega$ are obtained‎.‎As an application‎, ‎we obtain the stronger bounds for the Laplacian energy $LE =\\sum_{i=1}^n|\\mu_i-\\frac{2m}{n}|$‎, ‎which improve some well known earlier bounds‎.‎Among other results‎, ‎all connected threshold graphs are characterized‎. ‎A Nordhaus-Gaddum-type result for $\\sigma$ is proved‎, ‎too. ‎Finally‎, ‎some relations between $\\sigma$ with other graph invariants are presented‎.

  An element a of a ring R is nil-clean, if a = e + b, where e2 = e ? R and b is a nilpotentelement, andthering R iscallednil-cleanifeachofitselementsisnil-clean. In [22], it was proved that, for a commutative ring R and an abelian group G, the group ring R[G] is nil-clean, i? R is nil-clean and G is a 2-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally ?nite 2-group over a nil-clean ring is nil-clean, and that the hypercenter of the group G must be a 2-group if a group ring of G is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, i? the ring is a nil-clean ring and the group is a 2-group.Keywords:Nil-clean ring, nil-clean group ring

We obtain ‎ relation between an exact   g -frame and a   g -Riesz basis under some conditions. We alsoprove that the stability of an exact   g -frame for a Hilbert space under perturbation is ‎di‎fferent to that of a   g -frame. These properties of exact   g -frames for Hilbert spaces are not similar to those of exact frames. then we mainly discuss the exact   K-g -frames in theHilbert spaces.  

In this thesis, two spectral upper bounds for the k-independencenumber of a graph which is the maximum size of a set of verticesat pairwise distance greater than k, are obtained. Also, we constructgraphs that attain equality for our first bound and show that our secondbound compares favorably to previous bounds on the k-independencenumber. Among other results, some lower bounds for the chromaticand fractional chromatic numbers of a graph in terms of its interiaare presented. Extremal graphs for this bound are investigated, too.Moreover, it is proved that this bounds are not lower bounds for thevector chromatic number of graph. Finally, some Nordhause-Gaddumtype results are proved.  

  In this thesis, the existence of weak solutions to a franctional system of p-Laplacianequations and a singular p-Laplacian equation with sing-changing functions, is investigatedvia the Nehari manifold method. To do this, corresponding to critical pointsof fibering maps, the Nehari manifold is divided into three sets. Moreove is provedthat the energy functional is coercive and bounded from below on the Nehari manifoldand local minima on the Nehari manifold are critical points of the energy functional.Then by showing the existence of two minima on the subsets of Nehari manifoldcorrespanding to maxima and minima of fibering maps, the existence of two criticalpoints of the energy functional will be proved.

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classi?cation the lines lying on the Grassmannian variety play a central role. Related codes,namely the af?ne Grassmann codes. In this paper we also classify and count the minimum weight codewords of the dual af?ne Grassmann codes. Combining the above classi?cation results, we are able to show that the dual of a Grassmann code is generated by its minimum weight codewords.  

One of the continuity conditions identified by Utumi on self-injective rings isthe $C_3$-condition, where a module $M$ is called a $C_3$-module if whenever $A$ and $B$ are directsummands of $M$ and $A \\cap B = 0$, then $A \\oplus B$ is a summand of $M$. In addition to injectiveand direct-injective modules, the 0px; TEXT-INDENT: 0px; -qt-block-indent: 0">indecomposable and regular modules. Indeed, every commutative ring is a $C_3$-ring. Inthis thesis provide a general and unified treatment of the above mentioned 0px; TEXT-INDENT: 0px; -qt-block-indent: 0">modules in terms of the $C_3$-condition, and establish new characterizations of several wellknown classes of rings.

  This‎ thesis is devoted to the study ‎of‎ properties and ‎‎‎irre‎d‎ucible‎ representations of ‎Le‎ibniz ‎algebras‎.‎ We use the liezation method to translate some fandamental theorems of Lie ‎‎algebras‎‎ ‎such‎ ‎as‎‎ the Engels theorem, the Levi decomposition, ... to Liebniz ‎algebras.‎ Also, we prove that any irreducible representation of ‎a‎ Leibniz algebra can be obtained from an irreducible representation of the ‎semi-‎simple Lie algebra from the Levi decomposition. As an application, we determine the irreducible representations of sl‎2.‎‎‎

  The (sign-less) laplacian spectrum of graphs is extensively studied, however, the value of the least sign-less laplacian eigenvalue of graphs and its lower and upper bounds is less considered. The main aim of this thesis is to investigate and optimize the least sign-less Laplacian eigenvalue of some classes of graphs. A connected graph is called unicyclic if it has the same number of vertices and edges. Here, a graph with the maximum least signless Laplacian eigenvalue among all connected unicyclic graphs with fixed order is determined. Also, some relations between the least sign-less Laplacian eigenvalue of a connected n-vertex graph G and its independence number (covering number) are studied. Amongotherresults, wedeterminethegraphswhichhavethe minimum least sign-less Laplacian eigenvalue among all nonbipartite graphs with given either independence number at least n?1 2 or covering number at most n+1 2 .

One of the most important subjects that matters to either government organization and non-government specially for Judicial, security and law enforcement systems is preventing forgery, Undoubtedly, in order to prevent, it is necessary to take this issue into the account. As the field of knowledge and technology progressing, the methods of committing crime are also evolving in various dimensions and levels. Therefore, obtaining effective results in the prevention of documents forgery requires the application of indicators and security components in documents as well as utilizing modern technologies in the prevention and protection of documents forgery. Establish bank checks in a digital and secure way, and preventing counterfeiting of information, faces two major challenges including error and security. In this thesis, by implementation of digital signature based on elliptical curves all check data is generated from this type of signature, tended to keep information safe and secure by appointing this signature as resistant watermarking to the blue component of the image and DTC middle frequencies domain. Even though, AES cryptography and logistic mapping are used to increase the security of cache information.Besides, the inclusion of fragile cryptographic information and reconstruction of this information to investigate the non-manipulation of the red component of the image by encryption and decryption elliptical curve in the spatial domain being concluded.‎ While data transferring, due to the noise and interference, the data can be subjected to be incorrect. In this thesis applying codes‎ BCH(31‎, ‎16)it is possible to correct up to three possible errors. ‎The production of digital signature is done applying the software of ‎SAGE‎ and the placement of the information cryptography and reconstruction of this information is implemented by MATLAB software.

  The dynamics of a general di?usive predator–prey system is considered. Existence and nonexistence of non-constant positive steady state solutions are shown to iden- tify the ranges of parameters of spatial pattern formation. Bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non-constant steady state solutions are studied. Keywords: Reactiondi?usion; Patternformation; Globalbifurcation; Predatorprey; StrongAllee e?ect; Spatiotemporal patterns

‎In this thesis‎, ‎we compare the abelian subalgebras and ideals of maximal‎‎dimension in some 0px; MARGIN: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎abelian subalgebras of solvable Lie algebras and study the 0px; MARGIN: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎an abelian subalgebra of codimension 2‎. ‎Also‎, ‎we prove that nilpotent Lie algebras with‎‎an abelian subalgebra of codimension 1,2 or 3 contain an abelian ideal with the same dimension‎.‎Furthermore‎, ‎we investigate‎ ‎the structure of Lie algebras with a core-free‎ ‎maximal subalgebra and use them to introduce the conceptes of crown and pre-crown for chief factors‎.

  DNA cryptography is a new branch of cryptography that utilizes DNA as an in- formational and computational carrier with the aid of molecular techniques. Most of the modern encryption algorithms have been broken fully or partially. The world of information security looks for new directions to protect the data and their transmi- tion. The DNA computing in the ?elds of cryptography has been identi?ed as a new hope to create some unbreakable algorithms. In this thesis two digital signatures are discussed and analaysed, the ?rst method uses DNA coding and XOR operation with a symmetric key and the second uses DNA coding, Polymerase Chain Reaction and RSA encryption.

  امنيت يكي از اركان موجودات زنده و احساس امنيت يكي از اساسيترين نيازهاي نوع بشر است. امروزه با گستري وسايل ارتباطي و حجم اطلاعات مبادله شده در شبكههاي رايانهاي و همچنين توسعه و پيشرفتهاي صنعت مخابرات ند رسانه اي، مفهوم مخابرات تصويري متحول شده است. امنيت رسانه هاي ديجيتال يكي از مسائل مهم و مطرا جامعه رمزنگاري در دنياي امروز است. با توجه به كاربرد روزافزون رايانه و گستري زيرساخت هاي ارتباطي مثل شبكههاي سيار و اينترنت، حفظ محرمانگي و تاييد صحت تصاوير روز به روز اهميت بيشتري مييابد. با توجه به كاربرد روزافزون اينترنت و افزايش حجم اطلاعات مبادله شده، حفظ امنيت و تاييد صحت مخابره شده كه ميتواند كاربردهايي در امور تجاري، نظامي و حتي پزشكي داشته باشند نيز روز به روز اهميت بيشتري ميابد. در جهان ديجيتال امروزي امنيت تصاوير ديجيتال بيش از پيش اهميت پيدا كرده است. در سالهاي اخير سرعت زيادي در رشد انتقال تصاوير ديجيتال از طريق كامپيوتر بويژه اينترنت صورت گرفته است.به عنوان مثال ما در اين پاياننامه از يك روي رمزنگاري تصوير صحبت ميكنيم كه در امور پزشكي كه مرتبط با عكسبرداري از بيماران ميباشد و همچنين صحت و امنيت اطلاعات بيمار از اهميّت بالايي برخوردار است، استفاده ميشود. اين روي از رمزنگاري مبتني بر ميدان گالواست و ما آن را به طور كامل پيادهسازي كرده و علاوه بر بيان نقاط ضعف آن، راهي براي بهبود كارايي اين روي را بيان ميكنيم