A Brand-New, Area - Efficient Architecture for the FFT Algorithm Designed for Implementation of FPGAs

Elliptic curve cryptography, which is more commonly referred to by its acronym ECC, is widely regarded as one of the most effective new forms of cryptography developed in recent times. This is primarily due to the fact that elliptic curve cryptography utilises excellent performance across a wide range of hardware configurations in addition to having shorter key lengths. A High Throughput Multiplier design was described for Elliptic Cryptographic applications that are dependent on concurrent computations. A Proposed (Carry-Select) Division Architecture is explained and proposed throughout the whole of this work. Because of the carry-select architecture that was discussed in this article, the functionality of the divider has been significantly enhanced. The adder carry chain is reduced in length by this design by a factor of two, however this comes at the expense of additional adders and control. When it comes to designs for high throughput FFT, the total number of butterfly units that are implemented is what determines the amount of space that is needed by an FFT processor. In addition to blocks that may either add or subtract numbers, each butterfly unit also features blocks that can multiply numbers. The size of the region that is covered by these dual mathematical blocks is decided by the bit resolution of the models. When the bit resolution is increased, the area will also increase. The standard FFT approach requires that each stage contain  times as many butterfly units as the stage before it. This requirement must be met before moving on to the next stage

Implementation of Generic and Efficient Architecture of Elliptic Curve Cryptography over Various GF(p) for Higher Data Security

Elliptic Curve Cryptography (ECC) has recognized much more attention over the last few years and has time-honored itself among the renowned public key cryptography schemes. The main feature of ECC is that shorter keys can be used as the best option for implementation of public key cryptography in resource-constrained (memory, power, and speed) devices like the Internet of Things (IoT), wireless sensor based applications, etc. The performance of hardware implementation for ECC is affected by basic design elements such as a coordinate system, modular arithmetic algorithms, implementation target, and underlying finite fields. This paper shows the generic structure of the ECC system implementation which allows the different types of designing parameters like elliptic curve, Galois prime finite field GF(p), and input type. The ECC system is analyzed with performance parameters such as required memory, elapsed time, and process complexity on the MATLAB platform. The simulations are carried out on the 8th generation Intel core i7 processor with the specifications of 8 GB RAM, 3.1 GHz, and 64-bit architecture. This analysis helps to design an efficient and high performance architecture of the ECC system on Application Specific Integrated Circuit (ASIC) and Field Programmable Gate Array (FPGA).Elliptic Curve Cryptography (ECC) has recognized much more attention over the last few years and has time-honored itself among the renowned public key cryptography schemes. The main feature of ECC is that shorter keys can be used as the best option for implementation of public key cryptography in resource-constrained (memory, power, and speed) devices like the Internet of Things (IoT), wireless sensor based applications, etc. The performance of hardware implementation for ECC is affected by basic design elements such as a coordinate system, modular arithmetic algorithms, implementation target, and underlying finite fields. This paper shows the generic structure of the ECC system implementation which allows the different types of designing parameters like elliptic curve, Galois prime finite field GF(p), and input type. The ECC system is analyzed with performance parameters such as required memory, elapsed time, and process complexity on the MATLAB platform. The simulations are carried out on the 8th generation Intel core i7 processor with the specifications of 8 GB RAM, 3.1 GHz, and 64-bit architecture. This analysis helps to design an efficient and high performance architecture of the ECC system on Application Specific Integrated Circuit (ASIC) and Field Programmable Gate Array (FPGA)

Low-Latency Elliptic Curve Scalar Multiplication

This paper presents a low-latency algorithm designed for parallel computer architectures to compute the scalar multiplication of elliptic curve points based on approaches from cryptographic side-channel analysis. A graphics processing unit implementation using a standardized elliptic curve over a 224-bit prime field, complying with the new 112-bit security level, computes the scalar multiplication in 1.9ms on the NVIDIA GTX 500 architecture family. The presented methods and implementation considerations can be applied to any parallel 32-bit architectur

Theory and Practice of Cryptography and Network Security Protocols and Technologies

In an age of explosive worldwide growth of electronic data storage and communications, effective protection of information has become a critical requirement. When used in coordination with other tools for ensuring information security, cryptography in all of its applications, including data confidentiality, data integrity, and user authentication, is a most powerful tool for protecting information. This book presents a collection of research work in the field of cryptography. It discusses some of the critical challenges that are being faced by the current computing world and also describes some mechanisms to defend against these challenges. It is a valuable source of knowledge for researchers, engineers, graduate and doctoral students working in the field of cryptography. It will also be useful for faculty members of graduate schools and universities

Analysis of Parallel Montgomery Multiplication in CUDA

For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared

Efficient hardware prototype of ECDSA modules for blockchain applications

This paper concentrates on the hardware implementation of efficient and re- configurable elliptic curve digital signature algorithm (ECDSA) that is suitable for verifying transactions in Blockchain related applications. Despite ECDSA architecture being computationally expensive, the usage of a dedicated stand-alone circuit enables speedy execution of arithmetic operations. The prototype put forth supports N-bit elliptic curve cryptography (ECC) group operations, signature generation and verification over a prime field for any elliptic curve. The research proposes new hardware framework for modular multiplication and modular multiplicative inverse which is adopted for group operations involved in ECDSA. Every hardware design offered are simulated using modelsim register transfer logic (RTL) simulator. Field programmable gate array (FPGA) implementation of var- ious modules within ECDSA circuit is compared with equivalent existing techniques that is both hardware and software based to highlight the superiority of the suggested work. The results showcased prove that the designs implemented are both area and speed efficient with faster execution and less resource utilization while maintaining the same level of security. The suggested ECDSA structure could replace the software equivalent of digital signatures in hardware blockchain to thwart software attacks and to provide better data protection

Efficient Side-Channel Aware Elliptic Curve Cryptosystems over Prime Fields

Elliptic Curve Cryptosystems (ECCs) are utilized as an alternative to traditional public-key cryptosystems, and are more suitable for resource limited environments due to smaller parameter size. In this dissertation we carry out a thorough investigation of side-channel attack aware ECC implementations over finite fields of prime characteristic including the recently introduced Edwards formulation of elliptic curves, which have built-in resiliency against simple side-channel attacks. We implement Joye\u27s highly regular add-always scalar multiplication algorithm both with the Weierstrass and Edwards formulation of elliptic curves. We also propose a technique to apply non-adjacent form (NAF) scalar multiplication algorithm with side-channel security using the Edwards formulation. Our results show that the Edwards formulation allows increased area-time performance with projective coordinates. However, the Weierstrass formulation with affine coordinates results in the simplest architecture, and therefore has the best area-time performance as long as an efficient modular divider is available

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Efficient Algorithms for Elliptic Curve Cryptosystems

Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into low-level algorithms, which deal with arithmetic in the underlying finite field and high-level algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite fields GF((2n)m). The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)m). The third algorithm is an entirely new approach which accelerates the multiplication of points which is the core operation in elliptic curve public-key systems. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. Finally we apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide ablolute performance measures for the field operations and for an entire point multiplication. We also show the improvements gained by the new point multiplication algorithm in conjunction with the k-ary and improved k-ary methods for exponentiation

Elliptic curve cryptosystem over optimal extension fields for computationally constrained devices

Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller.Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006.Electrical, Electronic and Computer Engineeringunrestricte