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Journal ArticleDOI

Some Graph-Based Encryption Schemes

19 Feb 2021-Journal of Mathematics (Hindawi Limited)-Vol. 2021, pp 1-8
TL;DR: This paper proposes some new encryption algorithms for secure transmission of messages using some special corona graphs and bipartite graph along with some algebraic properties to lead to more secure communication of secret messages.
Abstract: In today’s technological world, confidentiality is an important issue to deal with, and it is carried out through different proficiencies. Cryptography is a scientific technique of securing a communication from unauthenticated approach. There exist many encryption algorithms in cryptography for data security. The need of new nonstandard encryption algorithms has been raised to prevent the communication from traditional attacks. This paper proposes some new encryption algorithms for secure transmission of messages using some special corona graphs and bipartite graph along with some algebraic properties. These proposed encryption schemes will lead to more secure communication of secret messages.

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Citations
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Journal ArticleDOI
18 Oct 2021
TL;DR: In this paper, it was shown that the number of connected vertex labeled graphs of order seven with m vertices and t (t is the number edges that connect different pair of vertices) = ct C (m−1) t−1, with c6=6727, c7=30160, c8=30765, c9=21000, c10=28364, c11=26880, c12=26460, c13=20790, c14=10290, c15= 8022, c16=
Abstract: Given n vertices and m edges, m ≥ 1, and for every vertex is given a label, there are lots of graphs that can be obtained. The graphs obtained may be simple or not simple, connected or disconnected. A graph G(V,E) is called simple if G(V,E) not containing loops nor paralel edges. An edge which has the same end vertex is called a loop, and paralel edges are two or more edges which connect the same set of vertices. Let N(G7,m,t) as the number of connected vertex labeled graphs of order seven with m vertices and t (t is the number edges that connect different pair of vertices). The result shows that N(G7,m,t) = ct C (m−1) t−1, with c6=6727, c7=30160 , c8=30765, c9=21000, c10=28364, c11=26880, c12=26460, c13=20790, c14=10290, c15= 8022, c16=2940, c17=4417, c18=2835, c19=210, c20= 21, c21=1.
Journal ArticleDOI
TL;DR: In this paper , the authors proposed a cryptosystem using the Turan graph which has a complex graph structure, which is a unique multipartite complete graph with more edges than other multi-partite complete graphs.
Abstract: Encryption and decryption are the two processes in cryptography to conceal and convey important information to an authorized person without third-party interruption in a network. Cryptography is a branch of computer science in which the system has to be updated every second. It mainly depends on mathematical concepts like number theory and algebra. Recently, graph theory concepts are employed in cryptography to make it stronger. The usage of complex graphs in cryptosystems makes it difficult to hack. In this paper, we proposed a cryptosystem using the Turan graph which has a complex graph structure. The advantage of using a Turan graph is that it is a unique multipartite complete graph with more edges than other multipartite complete graphs. This adds robustness to the cryptosystem. The novelty of this paper is the decomposition of the Turan graph into paths and stars and applying edge labeling to them to encrypt and decrypt a sentence of k words. The algorithms for encryption and decryption are also proposed in this paper.
References
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Journal ArticleDOI
TL;DR: A theory of secrecy systems is developed on a theoretical level and is intended to complement the treatment found in standard works on cryptography.
Abstract: THE problems of cryptography and secrecy systems furnish an interesting application of communication theory.1 In this paper a theory of secrecy systems is developed. The approach is on a theoretical level and is intended to complement the treatment found in standard works on cryptography.2 There, a detailed study is made of the many standard types of codes and ciphers, and of the ways of breaking them. We will be more concerned with the general mathematical structure and properties of secrecy systems.

8,777 citations


"Some Graph-Based Encryption Schemes..." refers background in this paper

  • ...Modern cryptography was established by Shannon in 1949 [1]....

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Journal ArticleDOI
TL;DR: In this paper, a hash function is constructed from one of Pizers Ramanujan graphs, (the set of supersingular elliptic curves over with l-isogenies, l a prime different from p).
Abstract: We propose constructing provable collision resistant hash functions from expander graphs in which finding cycles is hard. As examples, we investigate two specific families of optimal expander graphs for provable collision resistant hash function constructions: the families of Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak and Pizer respectively. When the hash function is constructed from one of Pizers Ramanujan graphs, (the set of supersingular elliptic curves over with l-isogenies, l a prime different from p), then collision resistance follows from hardness of computing isogenies between supersingular elliptic curves. For the LPS graphs, the underlying hard problem is a representation problem in group theory. Constructing our hash functions from optimal expander graphs implies that the outputs closely approximate the uniform distribution. This property is useful for arguing that the output is indistinguishable from random sequences of bits. We estimate the cost per bit to compute these hash functions, and we implement our hash function for several members of the Pizer and LPS graph families and give actual timings.

283 citations

Journal ArticleDOI
TL;DR: A novel group theoretic and graphical method is proposed to construct S-box with optimal features to fulfill the requirement of robustness against linear and differential cryptanalyses.
Abstract: The success of AES encryption standard created challenges for the cryptographers to construct strong substitution-boxes using different underlying approaches. It is because they are solely responsible to decide the robustness of cryptosystem against linear and differential cryptanalyses. With an aim to fulfill the mentioned requirement of robustness, a novel group theoretic and graphical method is proposed to construct S-box with optimal features. Firstly, a strong S-box is generated with the help of orbits of coset graphs and the action of proposed powerful permutation of symmetric group S 256 . In addition, a specific group is designed the action of whose pairs of permutations has the ability to generate as many as 462422016 strong S-boxes. Few of such proposed S-boxes are reported and assessed against standard performance parameters to validate the effectiveness of proposed findings. The features of proposed S-boxes are compared with most of the recent S-boxes to validate the superior performance. Moreover, they are also applied for image encryption to demonstrate their suitability for multimedia security applications.

49 citations

Journal ArticleDOI
TL;DR: This paper uses coset diagram for the action of on projective line over the finite field to construct proposed S-box and applies a bijective map on each element of the matrix to evolve proposedS-box.
Abstract: The substitution box is a basic tool to convert the plaintext into an enciphered format. In this paper, we use coset diagram for the action of on projective line over the finite field to construct proposed S-box. The vertices of the cost diagram are elements of which can be represented by powers of , where is the root of irreducible polynomial over . Let denote the elements of which are of the form of even powers of . In the first step, we construct a matrix with the elements of in a specific order, determined by the coset diagram. Next, we consider defined by to destroy the structure of . In the last step, we apply a bijective map on each element of the matrix to evolve proposed S-box. The ability of the proposed S-box is examined by different available algebraic and statistical analyses. The results are then compared with the familiar S-boxes. We get encouraging statistics of the proposed box after comparison.

41 citations

Journal Article
TL;DR: The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs, and it is shown that they can be used for the implementation of secure and very fast symmetric encryption algorithms.
Abstract: We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large girth and their cryptographical applications. It contains new explicit algebraic constructions of in finite families of such graphs. We show that they can be used for the implementation of secure and very fast symmetric encryption algorithms. The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs.

35 citations