Browsing by Author "Wijesiri, G.S."

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An application of graph theory in asymmetric key cryptography
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Fernando, K.K.N.; Wijesiri, G.S.
The development of digital communication in many facets of our everyday lives significantly impacts the evolving world we live in today. The rapid growth and evolution of digital communication have become the backbone of how we interact with other people. Therefore, it is imperative to protect information and data from unauthorised activities, such as accessing, using, exposing, damaging, modifying, copying, or deleting them. Safeguarding data from these invalid operations is crucial to ensure its integrity, confidentiality, and availability. Cryptography plays a pivotal role in ensuring the security and privacy of information in various contexts, such as online banking, e-commerce transactions, and communication between governments, military organisations, and businesses. Various types of mathematical techniques are available for application in modern cryptology. The application of graph theory is widely utilised in the field of cryptography due to its straightforward representation in computers as a matrix. In this study, we propose a novel asymmetric key cryptography scheme for secure message transmission using graph theory and matrices. The proposed scheme consists of four algorithms. The key generation algorithm on the receiver side is based on the properties of matrices, which enables us to establish the relationship between private key and public key through matrix operations. On the sender-side graph generation algorithm, a graph theory approach is applied to encrypt the original message, and the message is converted into a splitting graph and its minimum spanning tree. Then, the sender-side encryption algorithm is used to generate a complex final ciphertext using the receiver’s public key. The decryption algorithm follows the same process in reverse order, employing the receiver’s private key. This system will provide better security while storing data in the financial retail industry and sharing passwords in transactions.
The chromatic number of prime graph of a noncommutative ring Mn×n(Z2)
(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Kolombage, K.A.D.D.B.V.; Wijesiri, G.S.
Graph theory is a significant area of Mathematics as its outstanding applications in many fields such as biochemistry, electrical engineering, computer science and operational research. Besides Graph theory, Ring theory is an abstract area in Mathematics. A ring consists of a set equipped with two binary operations that generalize the arithmetic operations of addition (+) and multiplication(∗). Theorems obtained as a result of abstract study of rings can be applied to solve problems arising in number theory, geometry and many other fields. The study of rings with the help of graphs began when a graph of a commutative ring was defined by I. Beck in 1988. Then a new bridge was formed between graph theory and the algebraic concept “ring” noted as prime graph of a ring 􀀌, denoted by 􀀍􀀎(􀀌) by B. Satyanarayana, K. Shyam Prasad, and D.Nagaraju in 2010. Later on with the help of existing concepts, K. Patra and S. Kalita investigated the chromatic number of prime graph, 􀀏􀀍􀀎(ℤ􀀐) of ring ℤ􀀐 for different values of 􀀑. Prime graph of a ring 􀀌 is a graph whose vertices are all elements of the ring and any two vertices 􀀒, 􀀔 of the vertex set are adjacent if and only if 􀀒 ∗ 􀀔 = 0 or 􀀔 ∗ 􀀒 = 0 and 􀀒 ≠ 􀀔 In this paper, we investigate the chromatic number of prime graph of some noncommutative rings 􀀘􀀐×􀀐(ℤ􀀙) for different values of n. The chromatic number of prime graph of some commutative rings are formed on the recognition of the conjecture that chromatic number, 􀀏(􀀌) and clique number are the same. But for non-commutative rings this is not always the case. Hence, in order to find the chromatic number of prime graph of a non-commutative ring, 􀀘􀀐×􀀐(ℤ􀀙), we have looked into MATLAB for a tactical solution.
The chromatic number of prime graph of a noncommutative ring Mn×n(Z2)
(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Kolombage, K.A.D.D.B.V.; Wijesiri, G.S.
Graph theory is a significant area of Mathematics as its outstanding applications in many fields such as biochemistry, electrical engineering, computer science and operational research. Besides Graph theory, Ring theory is an abstract area in Mathematics. A ring consists of a set equipped with two binary operations that generalize the arithmetic operations of addition (+) and multiplication(∗). Theorems obtained as a result of abstract study of rings can be applied to solve problems arising in number theory, geometry and many other fields. The study of rings with the help of graphs began when a graph of a commutative ring was defined by I. Beck in 1988. Then a new bridge was formed between graph theory and the algebraic concept “ring” noted as prime graph of a ring ��, denoted by ��(��) by B. Satyanarayana, K. Shyam Prasad, and D.Nagaraju in 2010. Later on with the help of existing concepts, K. Patra and S. Kalita investigated the chromatic number of prime graph, ��(ℤ��) of ring ℤ�� for different values of ��. Prime graph of a ring �� is a graph whose vertices are all elements of the ring and any two vertices ��, �� of the vertex set are adjacent if and only if �� ∗ �� = 0 or �� ∗ �� = 0 and �� ≠ �� In this paper, we investigate the chromatic number of prime graph of some noncommutative rings ��×��(ℤ��) for different values of n. The chromatic number of prime graph of some commutative rings are formed on the recognition of the conjecture that chromatic number, ��(��) and clique number are the same. But for non-commutative rings this is not always the case. Hence, in order to find the chromatic number of prime graph of a non-commutative ring, ��×��(ℤ��), we have looked into MATLAB for a tactical solution.
A masking method for resisting key attacks on AES
(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Perera, P.S.L.; Wijesiri, G.S.
Cryptography is the science of securing data. It makes us easy to store sensitive data and transmit across insecure networks. Modern cryptography is applied in smart cards, computer passwords, and electronic commerce. Data Encryption Standard (DES) was a popular symmetric-key (64-bit) block cipher in USA from 1970’s. It was broken in 1999 by making the key length insecure with the improved computer speed. Later AES (Advanced Encryption Standard) was introduced to replace DES. Both key expansion and encryption algorithms of AES depend on S-boxes and calculations are done on the finite field GF (28). AES uses substitution-permutation and 128 block size while DES uses a balanced fiestel structure and 64 block size. DES has already broken but AES is still in use and is not completely broken. Today, key attacks for AES are steadily in progress. Since AES has theoretically broken by some key attacks, it is already at a risk of practical breakthrough. Algebraic attacks and related- key attacks are two of the main key attacks. The objective of the algebraic attack is recovering the key by solving system of multivariate polynomial equations over a Galois field. Related-key attacks use multiple pairs of plaintexts and encrypt each of them using two keys. As a result, the encryption key can be recovered. In this paper, we are going to combine AES algorithm with DNA cryptography which is a branch of Nano-cryptography to enhance the security of AES. We propose an improved algorithm to provide security against key attacks. In the algorithm we use masking techniques and DNA cryptosystem concepts based on the Vigenere cipher. DNA cryptography concepts help to avoid the weaknesses of Vigenere cipher and provide both computational and biological security.
Maximal embedding genus of 3-edge connected harary graphs
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Withanaarachchi, W.A.K.D.H.; Almeida, S.V.A.; Wijesiri, G.S.
One of the most prominent problems of topological graph theory is to determine the type of surface a nonplanar graph can be embedded. Almost complete results have been obtained for 4-edge connected graphs. The methods that were used to obtain specific results (finding the maximum and minimum genus embedding) for 4-edge connected graphs do not generalise for 3-edge connected graphs. Graph embedding is an important representational technique that aims to maintain the structure of a graph while learning low-dimensional representations of its vertices. The aim of this research project was to study the embedding of 3-edge connected Harary graphs H3,n. Specifically to complete the problem of maximal embeddings of 3-edge connected Harary graphs. The result is proved using Jungerman’s study, which showed that for any graph, is upper-embeddable if and only if it has a spanning tree T such that has at most one component with an odd number of edges. More specifically, a spanning tree for each graph was observed by dividing all 3-edge connected Harary graphs into two groups: odd number of vertices and even number of vertices. The pattern of a set of deleting edges and corresponding spanning trees was generalised in both cases. It was proved that H3,n is upper-embeddable, and the maximum genus of H3,n is given by for each n, by analysing the odd components of the complement of the corresponding spanning trees.
Scalar and Multi-Scalar Addition Chain in Elliptic Curve Cryptography
(4th International Conference on Advances in Computing and Technology (ICACT ‒ 2019), Faculty of Computing and Technology, University of Kelaniya, Sri Lanka, 2019) Nisansala, W.V.A.; Wijesiri, G.S.
Cryptography is a mathematical based technology that ensure the security of communications in the presence of malicious adversaries. Nowadays, cryptography deals with designing of algorithms, protocols and systems to secure transfer of information. The Elliptic Curve Cryptography (ECC) is a main branch of the public key cryptography (asymmetric cryptosystem) which was introduced by Neal Koblitz and Victor Miller in 1985. Higher speed, the efficiency of using power, bandwidth and less storage are some advantages of ECC. The strength of ECC is based on the inability of determining the scalar k of the scalar multiplication kP, where P is a point of an elliptic curve in finite field and it is known as the Elliptic Curve Discrete Logarithm Problem (ECDLP). Hence, the scalar multiplication is the central operation of ECC. Since most of the efficient and secure exponentiation methods (i.e. double-and-add, triple-and-add methods) depend on the secret scalar or exponent, an attacker may reveal the secret information through the side channel analysis (side channel attack). Simple Power Analysis (SPA) is a type of side channel attack that an attacker retrieves secret key by observing the power consumption traces. One way to overcome this problem is the use of doubling free addition chain since it results a fixed sequence of operations, and an attacker cannot detect any information through SPA. Therefore, we have implemented a new methodology that is more secure and reasonably eﬃcient, a doubling free simultaneous addition chain involving Lucas pattern to compute the scalar and multiscalar multiplication.
Thetanulls of cyclic curves of genus 4
(University of Kelaniya, 2013) Wijesiri, G.S.
Let be an irreducible smooth projective cyclic curve of genus defined over the complex field . These are by definition compact Riemann surfaces of genus (unless we allow singular points) admitting an automorphism such that and generates a normal subgroup of the automorphism group of When the curve is hyperelliptic, then the curve has extra automorphisms, in particular is not the hyperelliptic involution. The condition implies to having an equation for the curve, where is an affine coordinate on and has order . The branch points of together with the signature of the cover provide algebraic coordinates for the curve in moduli. Choosing a symplectic homology basis for a given curve of genus such that the intersection products and where is the Kronecker delta and a basis for the space of holomorphic 1- forms such that we can define the period matrix of It can be shown that is an element of the Siegel upper-half space . For any and any the Riemann’s theta function is defined as Any point where is the Jacobian of the curve can be written uniquely as , where For any the theta function with rational characteristics is defined as When the entries of column vectors are from the set , then the corresponding theta functions with rational characteristics are known as theta characteristics. A scalar obtained by evaluating a theta characteristics at is called a thetanull. The problem of expressing branch points in terms of transcendentals (period matrix, thetanulls, etc.,) is classical. This is an old problem that goes back to Riemann, Jacobi, Picard and Rosenhein. We do not aim here at a complete account of the classical or contemporary work on these problems. We determine the curves of genus 4 in terms of thetanulls and further study relations among the classical thetanulls of cyclic curves (of genus 4) with an automorphisms. In our work we use formulas for small genus curves introduced by Rosenhein, Thomae’s formulas for hyperelliptic curves, some recent results of Hurwitz space theory, and symbolic manipulation. Inverting the period map has an application in fast genus two curves arithmetic incryptography. We determine similar formulas for genus 4 hyperelliptic curves as the one used in cryptography using genus 2 algebraic curves.