Search results for: elliptic curve cryptography (ECC)

Authors: Abdelhakim Chillali, Abdelhamid Tadmori, Muhammed Ziane

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In this article we will study the elliptic curve defined over the ring An and we define the mathematical operations of ECC, which provides a high security and advantage for wireless applications compared to other asymmetric key cryptosystem.

Keywords: elliptic curves, finite ring, cryptography, study

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Authors: K. Sanjayani, C. Saraswathy, S. Sreenivasan, S. Sudhahar, D. Suganya, K. S. Neelukumari, N. Vijayarangan

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Double Base Number System (DBNS) is an imminent system of representing a number using two bases namely 2 and 3, which has its application in Elliptic Curve Cryptography (ECC) and Digital Signature Algorithm (DSA).The previous binary method representation included only base 2. DBNS uses an approximation algorithm namely, Greedy Algorithm. By using this algorithm, the number of digits required to represent a larger number is less when compared to the standard binary method that uses base 2 algorithms. Hence, the computational speed is increased and time being reduced. The standard binary method uses binary digits 0 and 1 to represent a number whereas the DBNS method uses binary digit 1 alone to represent any number (canonical form). The greedy algorithm uses two ways to represent the number, one is by using only the positive summands and the other is by using both positive and negative summands. In this paper, arithmetic operations are used for elliptic curve cryptography. Elliptic curve discrete logarithm problem is the foundation for most of the day to day elliptic curve cryptography. This appears to be a momentous hard slog compared to digital logarithm problem. In elliptic curve digital signature algorithm, the key generation requires 160 bit of data by usage of standard binary representation. Whereas, the number of bits required generating the key can be reduced with the help of double base number representation. In this paper, a new technique is proposed to generate key during encryption and extraction of key in decryption.

Keywords: cryptography, double base number system, elliptic curve cryptography, elliptic curve digital signature algorithm

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Authors: Mohamad Khairi Ishak

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Conventional public key crypto systems such as RSA (Ron Rivest, Adi Shamir and Leonard Adleman), DSA (Digital Signature Algorithm), and Elgamal are no longer efficient to be implemented in the small, memory constrained devices. Elliptic Curve Cryptography (ECC), which allows smaller key length as compared to conventional public key crypto systems, has thus become a very attractive choice for many applications. This paper describes implementation of an elliptic curve cryptography (ECC) encryption engine on a FPGA. The system has been implemented in 2 different key sizes, which are 131 bits and 163 bits. Area and timing analysis are provided for both key sizes for comparison. The crypto system, which has been implemented on Altera’s EPF10K200SBC600-1, has a hardware size of 5945/9984 and 6913/9984 of logic cells for 131 bits implementation and 163 bits implementation respectively. The crypto system operates up to 43 MHz, and performs point multiplication operation in 11.3 ms for 131 bits implementation and 14.9 ms for 163 bits implementation. In terms of speed, our crypto system is about 8 times faster than the software implementation of the same system.

Keywords: elliptic curve cryptography, FPGA, key sizes, memory

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Authors: Zhifu Li, Lei Li, Wanting Zhou, Yuanhang He

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This paper proposed a lightweight certificate-less authentication and key exchange protocol (Light-CL-PKC) based on elliptic curve cryptography and the Chinese Remainder Theorem for smart home scenarios. Light-CL-PKC can efficiently reduce the computational cost of both sides of authentication by forgoing time-consuming bilinear pair operations and making full use of point-addition and point-multiplication operations on elliptic curves. The authentication and key exchange processes in this system are also completed in a a single round of communication between the two parties. The analysis result demonstrates that it can significantly minimize the communication overhead of more than 32.14% compared with the referenced protocols, while the runtime for both authentication and key exchange have also been significantly reduced.

Keywords: authentication, key exchange, certificateless public key cryptography, elliptic curve cryptography

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Authors: Tun Myat Aung, Ni Ni Hla

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This paper begins by describing basic properties of finite field and elliptic curve cryptography over prime field and binary field. Then we discuss the discrete logarithm problem for elliptic curves and its properties. We study the general common attacks on elliptic curve discrete logarithm problem such as the Baby Step, Giant Step method, Pollard’s rho method and Pohlig-Hellman method, and describe in detail experiments of these attacks over prime field and binary field. The paper finishes by describing expected running time of the attacks and suggesting strong elliptic curves that are not susceptible to these attacks.c

Keywords: discrete logarithm problem, general attacks, elliptic curve, prime field, binary field

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Authors: D. M. S. Bandara, Yunqi Lei, Ye Luo

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Fingerprints are suitable as long-term markers of human identity since they provide detailed and unique individual features which are difficult to alter and durable over life time. In this paper, we propose an algorithm to encrypt and decrypt fingerprint images by using a specially designed Elliptic Curve Cryptography (ECC) procedure based on block ciphers. In addition, to increase the confusing effect of fingerprint encryption, we also utilize a chaotic-behaved method called Arnold Cat Map (ACM) for a 2D scrambling of pixel locations in our method. Experimental results are carried out with various types of efficiency and security analyses. As a result, we demonstrate that the proposed fingerprint encryption/decryption algorithm is advantageous in several different aspects including efficiency, security and flexibility. In particular, using this algorithm, we achieve a margin of about 0.1% in the test of Number of Pixel Changing Rate (NPCR) values comparing to the-state-of-the-art performances.

Keywords: arnold cat map, biometric encryption, block cipher, elliptic curve cryptography, fingerprint encryption, Koblitz’s encoding

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Authors: A. Andreasyan, C. Connors

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The Elliptic Curve Digital Signature algorithm-based X509v3 certificates are becoming more popular due to their short public and private key sizes. Moreover, these certificates can be stored in Internet of Things (IoT) devices, with limited resources, using less memory and transmitted in network security protocols, such as Internet Key Exchange (IKE), Transport Layer Security (TLS) and Secure Shell (SSH) with less bandwidth. The proposed method gives another advantage, in that it increases the performance of the above-mentioned protocols in terms of key exchange by saving one scalar multiplication operation.

Keywords: cryptography, elliptic curve digital signature algorithm, key exchange, network security protocol

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Authors: Siti Noor Farwina Anwar, Hailiza Kamarulhaili

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In this paper, we present the idea of implementing the Integer Sub-Decomposition (ISD) method on elliptic curves with j-invariant 1728. The ISD method was proposed in 2013 to compute scalar multiplication in elliptic curves, which remains to be the most expensive operation in Elliptic Curve Cryptography (ECC). However, the original ISD method only works on integer number field and solve integer scalar multiplication. By extending the method into the complex quadratic field, we are able to solve complex multiplication and implement the ISD method on elliptic curves with j-invariant 1728. The curve with j-invariant 1728 has a unique discriminant of the imaginary quadratic field. This unique discriminant of quadratic field yields a unique efficiently computable endomorphism, which later able to speed up the computations on this curve. However, the ISD method needs three endomorphisms to be accomplished. Hence, we choose all three endomorphisms to be from the same imaginary quadratic field as the curve itself, where the first endomorphism is the unique endomorphism yield from the discriminant of the imaginary quadratic field.

Keywords: efficiently computable endomorphism, elliptic scalar multiplication, j-invariant 1728, quadratic field

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Authors: Sartaj Singh, Amar Singh, Ashok Sharma, Sandeep Kaur

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With upcoming threats in a digital world, we need to work continuously in the area of security in all aspects, from hardware to software as well as data modelling. The rise in social media activities and hunger for data by various entities leads to cybercrime and more attack on the privacy and security of persons. Cryptography has always been employed to avoid access to important data by using many processes. Symmetric key and asymmetric key cryptography have been used for keeping data secrets at rest as well in transmission mode. Various cryptosystems have evolved from time to time to make the data more secure. In this research article, we are studying various cryptosystems in asymmetric cryptography and their application with usefulness, and much emphasis is given to Elliptic curve cryptography involving algebraic mathematics.

Keywords: cryptography, symmetric key cryptography, asymmetric key cryptography

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Authors: H. Parveen Begam, M. A. Maluk Mohamed

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Grid computing is an environment that allows sharing and coordinated use of diverse resources in dynamic, heterogeneous and distributed environment using Virtual Organization (VO). Security is a critical issue due to the open nature of the wireless channels in the grid computing which requires three fundamental services: authentication, authorization, and encryption. The privacy and anonymity are considered as an important factor while communicating over publicly spanned network like web. To ensure a high level of security we explored an extension of onion routing, which has been used with dynamic token exchange along with protection of privacy and anonymity of individual identity. To improve the performance of encrypting the layers, the elliptic curve cryptography is used. Compared to traditional cryptosystems like RSA (Rivest-Shamir-Adelman), ECC (Elliptic Curve Cryptosystem) offers equivalent security with smaller key sizes which result in faster computations, lower power consumption, as well as memory and bandwidth savings. This paper presents the estimation of the performance improvements of onion routing using ECC as well as the comparison graph between performance level of RSA and ECC.

Keywords: grid computing, privacy, anonymity, onion routing, ECC, RSA

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Authors: Eun-Jun Yoon

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Deniable authentication protocol is a new security authentication mechanism which can enable a receiver to identify the true source of a given message, but not to prove the identity of the sender to a third party. In 2013, Kar proposed a secure ID-based deniable authentication protocol whose security is based on computational infeasibility of solving Elliptic Curve Diffie-Hellman Problem (ECDHP). Kar claimed that the proposed protocol achieves properties of deniable authentication, mutual authentication, and message confidentiality. However, this paper points out that Kar's protocol still suffers from sender spoofing attack and message modification attack unlike its claims.

Keywords: deniable authentication, elliptic curve cryptography, Diffie-Hellman problem, cryptanalysis

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Authors: Lee Feng Koo, Tze Jin Wong, Pang Hung Yiu, Nik Mohd Asri Nik Long

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Greater common divisor (GCD) attack is an attack that relies on the polynomial structure of the cryptosystem. This attack required two plaintexts differ from a fixed number and encrypted under same modulus. This paper reports a security reaction of Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field under GCD attack. Lucas Based El-Gamal Cryptosystem in the Elliptic Curve group over finite field was exposed mathematically to the GCD attack using GCD and Dickson polynomial. The result shows that the cryptanalyst is able to get the plaintext without decryption by using GCD attack. Thus, the study concluded that it is highly perilous when two plaintexts have a slight difference from a fixed number in the same Elliptic curve group over finite field.

Keywords: decryption, encryption, elliptic curve, greater common divisor

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Authors: Despoina Chochtoula, Aristidis Ilias, Yannis Stamatiou

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Modbus is a protocol that enables the communication among devices which are connected to the same network. This protocol is, often, deployed in connecting sensor and monitoring units to central supervisory servers in Supervisory Control and Data Acquisition, or SCADA, systems. These systems monitor critical infrastructures, such as factories, power generation stations, nuclear power reactors etc. in order to detect malfunctions and ignite alerts and corrective actions. However, due to their criticality, SCADA systems are vulnerable to attacks that range from simple eavesdropping on operation parameters, exchanged messages, and valuable infrastructure information to malicious modification of vital infrastructure data towards infliction of damage. Thus, the SCADA research community has been active over strengthening SCADA systems with suitable data protection mechanisms based, to a large extend, on cryptographic methods for data encryption, device authentication, and message integrity protection. However, due to the limited computation power of many SCADA sensor and embedded devices, the usual public key cryptographic methods are not appropriate due to their high computational requirements. As an alternative, Elliptic Curve Cryptography has been proposed, which requires smaller key sizes and, thus, less demanding cryptographic operations. Until now, however, no such implementation has been proposed in the SCADA literature, to the best of our knowledge. In order to fill this gap, our methodology was focused on integrating Modbus, a frequently used SCADA communication protocol, with Elliptic Curve based cryptography and develop a server/client application to demonstrate the proof of concept. For the implementation we deployed two C language libraries, which were suitably modify in order to be successfully integrated: libmodbus (https://github.com/stephane/libmodbus) and ecc-lib https://www.ceid.upatras.gr/webpages/faculty/zaro/software/ecc-lib/). The first library provides a C implementation of the Modbus/TCP protocol while the second one offers the functionality to develop cryptographic protocols based on Elliptic Curve Cryptography. These two libraries were combined, after suitable modifications and enhancements, in order to give a modified version of the Modbus/TCP protocol focusing on the security of the data exchanged among the devices and the supervisory servers. The mechanisms we implemented include key generation, key exchange/sharing, message authentication, data integrity check, and encryption/decryption of data. The key generation and key exchange protocols were implemented with the use of Elliptic Curve Cryptography primitives. The keys established by each device are saved in their local memory and are retained during the whole communication session and are used in encrypting and decrypting exchanged messages as well as certifying entities and the integrity of the messages. Finally, the modified library was compiled for the Android environment in order to run the server application as an Android app. The client program runs on a regular computer. The communication between these two entities is an example of the successful establishment of an Elliptic Curve Cryptography based, secure Modbus wireless communication session between a portable device acting as a supervisor station and a monitoring computer. Our first performance measurements are, also, very promising and demonstrate the feasibility of embedding Elliptic Curve Cryptography into SCADA systems, filling in a gap in the relevant scientific literature.

Keywords: elliptic curve cryptography, ICT security, modbus protocol, SCADA, TCP/IP protocol

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Authors: Chiou-Yng Lee, Wen-Yo Lee, Chieh-Tsai Wu, Cheng-Chen Yang

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Shifted polynomial basis (SPB) is a variation of polynomial basis representation. SPB has potential for efficient bit-level and digit-level implementations of multiplication over binary extension fields with subquadratic space complexity. For efficient implementation of pairing computation with large finite fields, this paper presents a new SPB multiplication algorithm based on Karatsuba schemes, and used that to derive a novel scalable multiplier architecture. Analytical results show that the proposed multiplier provides a trade-off between space and time complexities. Our proposed multiplier is modular, regular, and suitable for very-large-scale integration (VLSI) implementations. It involves less area complexity compared to the multipliers based on traditional decomposition methods. It is therefore, more suitable for efficient hardware implementation of pairing based cryptography and elliptic curve cryptography (ECC) in constraint driven applications.

Keywords: digit-serial systolic multiplier, elliptic curve cryptography (ECC), Karatsuba algorithm (KA), shifted polynomial basis (SPB), pairing computation

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Authors: Shunsuke Miyoshi, Yasuyuki Nogami, Takuya Kusaka, Nariyoshi Yamai

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Elliptic curve discrete logarithm problem (ECDLP) is one of problems on which the security of pairing-based cryptography is based. This paper considers Pollard's rho method to evaluate the security of ECDLP on Barreto-Naehrig (BN) curve that is an efficient pairing-friendly curve. Some techniques are proposed to make the rho method efficient. Especially, the group structure on BN curve, distinguished point method, and Montgomery trick are well-known techniques. This paper applies these techniques and shows its optimization. According to the experimental results for which a large-scale parallel system with MySQL is applied, 94-bit ECDLP was solved about 28 hours by parallelizing 71 computers.

Keywords: Pollard's rho method, BN curve, Montgomery multiplication

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Authors: Berhane Tewelday Weldhiwot

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This research investigates the pivotal role of modular forms in modern cryptographic systems, particularly focusing on their applications in secure communications and data integrity. Modular forms, which are complex analytic functions with rich arithmetic properties, have gained prominence due to their connections to number theory and algebraic geometry. This study begins by outlining the fundamental concepts of modular forms and their historical development, followed by a detailed examination of their applications in cryptographic protocols such as elliptic curve cryptography and zero-knowledge proofs. By employing techniques from analytic number theory, the research delves into how modular forms can enhance the efficiency and security of cryptographic algorithms. The findings suggest that leveraging modular forms not only improves computational performance but also fortifies security measures against emerging threats in digital communication. This work aims to contribute to the ongoing discourse on integrating advanced mathematical theories into practical applications, ultimately fostering innovation in cryptographic methodologies.

Keywords: modular forms, cryptography, elliptic curves, applications, mathematical theory

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Authors: Safa Saoudi, Souheib Yousfi, Riadh Robbana

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E-passport is a relatively new electronic document which maintains the passport features and provides better security. It deploys new technologies such as biometrics and Radio Frequency identification (RFID). The international civil aviation organization (ICAO) and the European union define mechanisms and protocols to provide security but their solutions present many threats. In this paper, a new mechanism is presented to strengthen e-passport security and authentication process. We propose a new protocol based on Elliptic curve, identity based encryption and shared secret between entities. Authentication in our contribution is formally proved with BAN Logic verification language. This proposal aims to provide a secure data storage and authentication.

Keywords: e-passport, elliptic curve cryptography, identity based encryption, shared secret, BAN Logic

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Authors: Shiji Hu, Lei Li, Wanting Zhou, DaoHong Yang

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The data encryption, is the foundation of today’s communication. On this basis, how to improve the speed of data encryption and decryption is always a problem that scholars work for. In this paper, we proposed an elliptic curve crypto processor architecture based on SM2 prime field. In terms of hardware implementation, we optimized the algorithms in different stages of the structure. In finite field modulo operation, we proposed an optimized improvement of Karatsuba-Ofman multiplication algorithm, and shorten the critical path through pipeline structure in the algorithm implementation. Based on SM2 recommended prime field, a fast modular reduction algorithm is used to reduce 512-bit wide data obtained from the multiplication unit. The radix-4 extended Euclidean algorithm was used to realize the conversion between affine coordinate system and Jacobi projective coordinate system. In the parallel scheduling of point operations on elliptic curves, we proposed a three-level parallel structure of point addition and point double based on the Jacobian projective coordinate system. Combined with the scalar multiplication algorithm, we added mutual pre-operation to the point addition and double point operation to improve the efficiency of the scalar point multiplication. The proposed ECC hardware architecture was verified and implemented on Xilinx Virtex-7 and ZYNQ-7 platforms, and each 256-bit scalar multiplication operation took 0.275ms. The performance for handling scalar multiplication is 32 times that of CPU(dual-core ARM Cortex-A9).

Keywords: Elliptic curve cryptosystems, SM2, modular multiplication, point multiplication.

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Authors: Luis A. Lizama-Perez, Manuel J. Linares, Mauricio Lopez

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We introduce a method to perform remote user authentication on what we call non-invertible cryptography. It exploits the fact that the multiplication of an invertible integer and a non-invertible integer in a ring Zn produces a non-invertible integer making infeasible to compute factorization. The protocol requires the smallest key size when is compared with the main public key algorithms as Diffie-Hellman, Rivest-Shamir-Adleman or Elliptic Curve Cryptography. Since we found that the unique opportunity for the eavesdropper is to mount an exhaustive search on the keys, the protocol seems to be post-quantum.

Keywords: invertible, non-invertible, ring, key transfer

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Authors: Amit K. Singh, Subhankar Sen

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The bifurcation curve of separation in steady two-dimensional viscous flow past an elliptic cylinder is studied by varying the angle of incidence (α) with different aspect ratio (ratio of minor to major axis). The solutions are based on numerical investigation, using finite element analysis, of the Navier-Stokes equations for incompressible flow. Results are presented for Reynolds number up to 50 and angle of incidence varies from 0° to 90°. Range of aspect ratio (Ar) is from 0.1 to 1 (in steps of 0.1) and flow is considered as unbounded flow. Bifurcation curve represents the locus of Reynolds numbers (Res) at which flow detaches or separates from the surface of the body at a given α and Ar. In earlier studies, effect of Ar on laminar separation curve or bifurcation curve is limited for Ar = 0.1, 0.2, 0.5 and 0.8. Some results are also available at α = 90° and 45°. The present study attempts to provide a systematic data and clear understanding on the effect of Ar at bifurcation curve and its point of maxima. In addition, issues regarding location of separation angle and maximum ratio of coefficient of lift to drag are studied. We found that nature of curve, separation angle and maximum ratio of lift to drag changes considerably with respect to change in Ar.

Keywords: aspect ratio, bifurcation curve, elliptic cylinder, GMRES, stabilized finite-element

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Authors: Emily Stamm, Neil Smyth, Elizabeth O'Sullivan

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In Identity-Based Encryption (IBE), an identity, such as a username, email address, or domain name, acts as the public key. IBE consolidates the PKI by eliminating the repetitive process of requesting public keys for each message encryption. Two of the most popular schemes are Sakai-Kasahara (SAKKE), which is based on elliptic curve pairings, and the Ducas, Lyubashevsky, and Prest lattice scheme (DLP- Lattice), which is based on quantum-secure lattice cryptography. In or- der to embed the schemes in a standard enterprise setting, both schemes are implemented as shared system libraries and integrated into a REST service that functions at the enterprise level. The performance of both schemes as libraries and services is compared, and the practicalities of implementation and application are discussed. Our performance results indicate that although SAKKE has the smaller key and ciphertext sizes, DLP-Lattice is significantly faster overall and we recommend it for most enterprise use cases.

Keywords: identity-based encryption, post-quantum cryptography, lattice-based cryptography, IBE

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Authors: Merad Boudia Omar Rafik, Feham Mohammed

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Data aggregation in wireless sensor networks (WSNs) provides a great reduction of energy consumption. The limited resources of sensor nodes make the choice of an encryption algorithm very important for providing security for data aggregation. Asymmetric cryptography involves large ciphertexts and heavy computations but solves, on the other hand, the problem of key distribution of symmetric one. The latter provides smaller ciphertexts and speed computations. Also, the recent researches have shown that achieving the end-to-end confidentiality and the end-to-end integrity at the same is a challenging task. In this paper, we propose (SA-SPKC), a novel security protocol which addresses both security services for WSNs, and where only the base station can verify the individual data and identify the malicious node. Our scheme is based on stateful public key encryption (StPKE). The latter combines the best features of both kinds of encryption along with state in order to reduce the computation overhead. Our analysis

Keywords: secure data aggregation, wireless sensor networks, elliptic curve cryptography, homomorphic encryption

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Authors: Sidra Shabbir, Ayesha Manzoor, Mehreen Sirshar

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The use of the network communication has imposed serious threats to the security of assets over the network. Network security is getting more prone to active and passive attacks which may result in serious consequences to data integrity, confidentiality and availability. Various cryptographic techniques have been proposed in the past few years to combat with the concerned problem by ensuring quality but in order to have a fully secured network; a framework of new cryptosystem was needed. This paper discusses certain cryptographic techniques which have shown far better improvement in the network security with enhanced quality assurance. The scope of this research paper is to cover the security pitfalls in the current systems and their possible solutions based on the new cryptosystems. The development of new cryptosystem framework has paved a new way to the widespread network communications with enhanced quality in network security.

Keywords: cryptography, network security, encryption, decryption, integrity, confidentiality, security algorithms, elliptic curve cryptography

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Authors: R. Rama Kishore, Sunesh

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Recent development in the usage of internet for different purposes creates a great threat for the copyright protection of the digital images. Digital watermarking can be used to address the problem. This paper presents detailed review of the different watermarking techniques, latest trends in the field of secured, robust and imperceptible watermarking. It also discusses the different optimization techniques used in the field of watermarking in order to improve the robustness and imperceptibility of the method. Different measures are discussed to evaluate the performance of the watermarking algorithm. At the end, this paper proposes a watermarking algorithm using (2, 2) share visual cryptography and Bezier curve based algorithm to improve the security of the watermark. The proposed method uses fractional transformation to improve the robustness of the copyright protection of the method. The algorithm is optimized using fuzzy entropy for better results.

Keywords: digital watermarking, fractional transform, visual cryptography, Bezier curve, fuzzy entropy

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Authors: Hyun-Ho Lee, Kee-Won Kim

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The arithmetic operations over GF(2m) have been extensively used in error correcting codes and public-key cryptography schemes. Finite field arithmetic includes addition, multiplication, division and inversion operations. Addition is very simple and can be implemented with an extremely simple circuit. The other operations are much more complex. The multiplication is the most important for cryptosystems, such as the elliptic curve cryptosystem, since computing exponentiation, division, and computing multiplicative inverse can be performed by computing multiplication iteratively. In this paper, we present a parallel computation algorithm that operates Montgomery multiplication over finite field using redundant basis. Also, based on the multiplication algorithm, we present an efficient semi-systolic multiplier over finite field. The multiplier has less space and time complexities compared to related multipliers. As compared to the corresponding existing structures, the multiplier saves at least 5% area, 50% time, and 53% area-time (AT) complexity. Accordingly, it is well suited for VLSI implementation and can be easily applied as a basic component for computing complex operations over finite field, such as inversion and division operation.

Keywords: finite field, Montgomery multiplication, systolic array, cryptography

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Authors: Edamana Krishnan, Khalil Al-Ghafri

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In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed.

Keywords: travelling wave solutions, solitary wave solutions, compactons, Jacobi elliptic functions, mapping methods

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Authors: G. Akgun, I. Algul, H. Kurtaran

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In this paper geometrically nonlinear static behavior of laminated composite hollow super-elliptic beams is investigated using generalized differential quadrature method. Super-elliptic beam can have both oval and elliptic cross-sections by adjusting parameters in super-ellipse formulation (also known as Lamé curves). Equilibrium equations of super-elliptic beam are obtained using the virtual work principle. Geometric nonlinearity is taken into account using von-Kármán nonlinear strain-displacement relations. Spatial derivatives in strains are expressed with the generalized differential quadrature method. Transverse shear effect is considered through the first-order shear deformation theory. Static equilibrium equations are solved using Newton-Raphson method. Several composite super-elliptic beam problems are solved with the proposed method. Effects of layer orientations of composite material, boundary conditions, ovality and ellipticity on bending behavior are investigated.

Keywords: generalized differential quadrature, geometric nonlinearity, laminated composite, super-elliptic cross-section

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Authors: Saif Akram, E. Rathakrishnan

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The mixing promoting efficiency of two identical sharp and truncated vertex triangular tabs offering geometrical blockage of 2.5% each, placed at the exit of a Mach 1.5 elliptic nozzle was studied experimentally. The effectiveness of both the tabs in enhancing the mixing of jets with the ambient air are determined by measuring the Pitot pressure along the jet axis and the jet spread in both the minor and major axes of the elliptic nozzle, covering marginally overexpanded to moderately underexpanded levels at the nozzle exit. The results reveal that both the tabs enhance mixing characteristics of the uncontrolled elliptic jet when placed at minor axis. A core length reduction of 67% is achieved at NPR 3 which is the overexpanded state. Similarly, the core length is reduced by about 67%, 50% and 57% at NPRs of 4, 5 and 6 (underexpanded states) respectively. However, unlike the considerable increment in mixing promoting efficiency by the use of truncated vertex tabs for axisymmetric jets, the effect is not much pronounced for the case of supersonic elliptic jets. The CPD plots for both the cases almost overlap, especially when tabs are placed at minor axis, at all the pressure conditions. While, when the tabs are used at major axis, in the case of overexpanded condition, the sharp vertex triangular tabs act as a better mixing enhancer for the supersonic elliptic jets. For the jet controlled with truncated vertex triangular tabs, the core length reductions are of the same order as those for the sharp vertex triangular tabs. The jet mixing is hardly influenced by the tip effect in case of supersonic elliptic jet.

Keywords: elliptic jet, tabs, truncated, triangular

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Authors: William Wong, Zakaria Alomari, Hon Ching Lai, Zhida Li

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Modern-day information security depends on implementing Diffie-Hellman, which requires the generation of prime numbers. Because the number of primes is infinite, it is impractical to store prime numbers for use, and therefore, primality tests are indispensable in modern-day information security. A primality test is a test to determine whether a number is prime or composite. There are two types of primality tests, which are deterministic tests and probabilistic tests. Deterministic tests are adopting algorithms that provide a definite answer whether a given number is prime or composite. While in probabilistic tests, a probabilistic result would be provided, there is a degree of uncertainty. In this paper, we review three probabilistic tests: the Fermat Primality Test, the Miller-Rabin Test, and the Baillie-PSW Test, as well as one deterministic test, the Agrawal-Kayal-Saxena (AKS) Test. Furthermore, we do an analysis of these tests. All of the reviews discussed are not based on the Elliptic Curve. The analysis demonstrates that, in the majority of real-world scenarios, the Baillie- PSW test’s favorability stems from its typical operational complexity of O(log 3n) and its capacity to deliver accurate results for numbers below 2^64.

Keywords: primality tests, Fermat’s primality test, Miller-Rabin primality test, Baillie-PSW primality test, AKS primality test

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Authors: Lívia B. Meirelles, Erika C. A. N. Chrisman, Flávia B. de Andrade, Lilian C. M. de Oliveira

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True Boiling Point distillation (TBP) is one of the most common experimental techniques for the determination of petroleum properties. This curve provides information about the performance of petroleum in terms of its cuts. The experiment is performed in a few days. Techniques are used to determine the properties faster with a software that calculates the distillation curve when a little information about crude oil is known. In order to evaluate the accuracy of distillation curve prediction, eight points of the TBP curve and specific gravity curve (348 K and 523 K) were inserted into the HYSYS Oil Manager, and the extended curve was evaluated up to 748 K. The methods were able to predict the curve with the accuracy of 0.6%-9.2% error (Software X ASTM), 0.2%-5.1% error (Software X Spaltrohr).

Keywords: distillation curve, petroleum distillation, simulation, true boiling point curve

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