An Efficient Algorithm for Elliptic Curve Scalar Multiplication Using Extended Double-Base Chain and Point Halving

The elliptic curve cryptography is almost used in the literature to develop the public key cryptosystems. A new general-purpose symmetric key cryptosystem using elliptic prime curves is proposed in this paper. For a given prime number P, there are many elliptic prime curves on which each point in the quadrant (0,0) to (P-1,P-1) lays. The set of minimum number of eligible elliptic prime curves for a given prime number P constitutes a lookup table that is the secret key of the proposed cryptosystem. The sender can pick up one eligible elliptic curve for each 2n-bit plaintext block. The sender sends two integers cipher for each 2n-bit plaintext block with encryption rate = 0.8. Moreover, a small prime number can be used efficiently with high security and good robustness against brute force attack.

Elliptic curve cryptography (ECC) can ensure an equivalent security with much smaller key sizes. Elliptic curve scalar multiplication (ECSM) is a fundamental computation used in ECC. This paper focuses on ECSM resisting simple power attack and safe error attack of side-channel attack specifically. Elliptic curve complete addition (CA) formulae can achieve secure ECSM algorithms but are inefficient from memory and computational cost perspectives. Another secure ECSM, which uses (extended) affine, is more efficient for both memory and computational costs. However, it scans input scalars from right to left. In this paper, our developed scalar multiplication algorithms also use their extended affine, but scan from left to right (LR). We also prove the security of our LR ECSM algorithms and analyze them both theoretically and experimentally. Our new LR ECSM algorithms can reduce the amount of memory by \(37.5\%\) and reduce the computational time by more than \(40\%\) compared to Joye’s regular 2-ary LR algorithm with CA formulae.

The security of Internet of Things (IoT) devices is one of the most important problems to be addressed by the cryptographers and security engineers. The processing ability of IoT devices is limited, therefore light-weight and secure cryptographic tools are necessary for security of them. This paper shows the implementation of 256-bit Elliptic Curve Cryptography (ECC) on an 8-bit microcontroller. The proposed implementation applies towering technique for extension field of degree 32 with a certain 8-bit prime characteristic instead of the 256-bit prime characteristic. It enables to execute 256-bit ECC operations without complicated multiple-precision arithmetic on small computers like 8-bit microcontrollers. This approach efficiently realizes the scalability of the ECC encryption strength. In addition, the authors use a twisted Montgomery curve with a Montgomery ladder technique which enables fast calculations without inversions referring to Curve25519. It is considered resistant to the Side Channel Attack (SCA) since it applies the Montgomery ladder technique for scalar multiplication (SCM). This ECC implementation on Arduino UNO, an 8-bit microcontroller board, can be utilized for a key agreement protocol among IoT devices.

Wireless sensor networks (WSNs) pose a number of unique security challenges that demand innovation in several areas including the design of cryptographic primitives and protocols. Despite recent progress, the efficient implementation of Elliptic Curve Cryptography (ECC) for WSNs is still a very active research topic, and techniques to further reduce the time and energy cost of ECC are eagerly sought. This paper presents an optimized ECC implementation that we developed from scratch to comply with the severe resource constraints of 8‐bit sensor nodes such as the MICAz and IRIS motes. Our ECC software uses Optimal Prime Fields as underlying algebraic structure and supports two different families of elliptic curves, namely, Weierstraß‐form and twisted Edwards‐form curves. Due to the combination of efficient field arithmetic and fast group operations, we achieve an execution time of 5.3·106 clock cycles for a full 160‐bit scalar multiplication on an 8‐bit ATmega128 microcontroller, which is more than three times faster than the widely used TinyECC library. Our implementation also shows that the energy cost of scalar multiplication on a MICAz (or IRIS) mote amounts to just 17.34mJ when using a twisted Edwards curve over a 160‐bit Optimal Prime Field. This result further demonstrates the advantage of special family of elliptic curves for resource‐constrained environments. Copyright © 2015 John Wiley & Sons, Ltd.

Wireless Sensor Networks (WSNs) pose a number of unique security challenges that demand innovation in several areas including the design of cryptographic primitives and protocols. Despite recent progress, the efficient implementation of Elliptic Curve Cryptography (ECC) for WSNs is still a very active research topic and techniques to further reduce the time and energy cost of ECC are eagerly sought. This paper presents an optimized ECC implementation that we developed from scratch to comply with the severe resource constraints of 8-bit sensor nodes such as the MICAz and IRIS motes. Our ECC software uses Optimal Prime Fields (OPFs) as underlying algebraic structure and supports two different families of elliptic curves, namely Weierstraß-form and twisted Edwards-form curves. Due to the combination of efficient field arithmetic and fast group operations, we achieve an execution time of 5.8 · 106 clock cycles for a full 158-bit scalar multiplication on an 8-bit ATmega128 micro-controller, which is 2.78 times faster than the widely-used TinyECC library. Our implementation also shows that the energy cost of scalar multiplication on a MICAz (or IRIS) mote amounts to just 19~mJ when using a twisted Edwards curve over a 160-bit OPF. This result compares fairly well with the energy figures of two recently-presented hardware designs of ECC based on twisted Edwards curves.

In this paper, we research the operation rules and scalar multiplication of Twisted Edwards elliptic curve. Based on the properties of point addition and double point operations, we propose the fast continuous double point algorithm, continuous point addition algorithm and continuous double point-point addition algorithm on Twisted Edwards elliptic curve. These algorithms reduce the amount of operations significantly by reducing the inverse operations. For comparison purposes, the top-level operations of the new algorithm use the non-adjacent form(NAF) and double base number system(DBNS) scalar multiplication algorithms. In contrast, it effectively improves the computational efficiency of the Twisted Edwards elliptic curve scalar multiplication algorithm, in which it is more than 50% faster than the underlying operation using Horner's rule, continuous multiplier algorithm, and about 10% to 35% or more faster than the base operation using the 4-NNAF algorithm.

There are many threats from virus, hacker, electronic eavesdropper, and electronic deceiver in the network. The growth of the computer system and network makes the organizations and personal users to depend on the information that circulate in these systems more and more. This makes us be aware of that protecting the data and resources from leak, insuring the reliability of data and information, and protecting the system from suffering the hit from network are very important.Submit a cryptography that is affirmative by people to reach the demand on the safety of SET system. The algorithm submitted is public, and everyone can use and to verify its safety, using Elliptic Curve Cryptography to execute the key exchange and encryption decryption, and making sure its performance of SET is better while the safety of SET is same as before, and the overload of compute and transmission is yet lower.

This work presents an evaluation of different software implementations of algorithms to compute the most demanding operation in cryptographic schemes based on Elliptic Curve Cryptography (ECC), the scalar multiplication. Five different methods were studied, including traditional and more sophisticated methods for elliptic curve cryptography defined over prime and binary fields. For evaluation, the key generation algorithm in ECC consisting on a single scalar multiplication was implemented on a P500h LG smarthphone, which includes an ARM processor running at 600MHz. Both execution time and memory usage was evaluated. It was found that in general, scalar multiplication runs 8 times faster for ECC defined over prime fields, being the NAF the best performer method. For ECC over binary fields, the best performer method was wNAF. The results presented in this work could help a designer to select the most appropriate method when implementing ECC-based cryptographic schemes such as encryption or digital signatures on mobile devices like smartphones, meeting implementation requirements in terms of execution time and memory usage.

Cloud storage is a useful service of cloud computing which allows users to upload their data in the cloud and share it with others. Although, this new service is affordable and practical, but it is associated with several privacy and security challenges. Nowadays, Attribute Based Encryption (ABE) is widely used to provide secure data sharing in the distributed environment such as cloud computing. Unfortunately, most of the existing ABE schemes are not suitable for resource- constraint cloud systems, because they use expensive bilinear pairing operation and thus they cause a very high encryption and decryption computation overhead. In this paper, we propose an efficient no-pairing and revocable ABE data sharing scheme based on Elliptic Curve Cryptography (ECC) for cloud storage systems. Moreover, a comprehensive security and performance analysis shows that our scheme is both secure and efficient.

Elliptic curve cryptography is a rather new, efficient technology for security. However, its implementation is complex and software versions can be prohibitively slow. The main original contribution of this paper is the proposition of a highly parameterizable soft intellectual property core that implements all the operations needed to perform elliptic curve cryptography in hardware. This core supports several standardized elliptic curves. Here, finite field operations use only affine coordinates, which keeps interoperability with software implementations. Also, core operators such as the finite field multiplier and inversion are highly configurable, to enable fulfilling constraints of area or performance, according to what is relevant to each application. The core can thus serve several different purposes, through the setting of parameters that pick the adequate elliptic curve and others that control the finite field multiplier and inversion operator characteristics. Results obtained by synthesizing the core with different parameters for Xilinx FPGAs and 65nm CMOS ASICs show that the exploitable design space is ample: implementations can take from 17 to 5,800μs to execute a point multiplication, taking from 5.7 to 162.2 thousand LUTs in FPGAs or from 16,6 to 360,7 thousand gates in ASICs.

Blommer, Otto, and Seifert presented a fault attack on elliptic curve scalar multiplication called the Sign Change Attack, which causes a fault that changes the sign of the accumulation point. As the use of a sign bit for an extended integer is highly unlikely, this appears to be a highly selective manipulation of the key stream. Here we describe two plausible fault attacks on a smart card implementation of elliptic curve cryptography.

For secure communications in particular, elliptic curve cryptography relies heavily on Elliptic Curve Scalar Multiplication (ECSM). The window approach is perfect for FPGA-based solutions since it increases ECSM efficiency. Verifying the accuracy of calculations made by hardware or software is essential to spotting possible mistakes. In this study, we introduce new defect detection strategies for scalar multiplication using the window method, which has not been studied in detail. Our strategy entails creating intricate algorithms and implementations to successfully minimize both transient and permanent faults. We validate our method's correctness by simulating a failure model, proving its dependability and broad error coverage in our assessments.

We implemented elliptic curve cryptography in the frequency domain on the MSP430 constrained microcontroller. Our implementation of 169-bit elliptic curve cryptography (ECC) on MSP430, one of the most popular microcontrollers for wireless sensor network (WSN) nodes, performs an ECC scalar point multiplication operation, for random points, in only 1.55 ms which is similar to or faster than existing implementations. To our knowledge, this work proposes the first ever software implementation of ECC in the frequency domain on a constrained low-power microcontroller.

With the rapid development of cloud computing, e-commerce, authentication among multi-robot systems, the highspeed implementation of Elliptic Curve Cryptography (ECC) is in widespread use. The performance of ECC is decided by the design of scalar point multiplier, which is one of the most time-consuming component. This paper presents a full set of methods to achieve an ultra high-speed scalar point multiplier, its Scalar Point Multiplication (SPM) is optimized comprehensively in terms of speed-first design approach by concurrently implementing the Point-add (PA) and Point-double (PD) algorithms, improving large integer modular inversion algorithm and large integer modular multiplication algorithm. Finally, Montgomery domain operation and Non-Adjacent Form (NAF) encoding theory are applied to enhance the speed of scalar point multiplier. In the VLSI design, a high-speed $\pmb{256\times 256}$ -bit scalar point multiplier is achieved based on SMIC's 65nm process. It can complete single calculation of SPM within 12.5us on average, in other word, 80,000 times of SPM can be computed in one second. Compared to the scalar point multiplier realized by other publications based on VLSI, the reports for circuits synthesis show that our multiplier is optimal in terms of $AT^{2}$ and ultrafast in terms of speed.

On the low-end embedded processors, the implementations of Elliptic Curve Cryptography (ECC) are considered to be a challenging task due to the limited computation power and storage of the low-end embedded processors. Particularly, the multi-precision multiplication and squaring operations are the most expensive operations for ECC implementations. In order to enhance the performance, many works presented efficient multiplication and squaring routines on the target devices. Recent works show that 128-bit security level ECC is available within a second and this is practically fast enough for IoT services. However, previous approaches missed the other important storage issues (i.e., program size, ROM). Considering that the embedded processors only have a few KB ROM, we need to pay attention to the compact ROM size with reasonable performance. In this article, we present very compact and generic implementations of multiplication and squaring operations on the 16-bit MSP430X processors for the ECC. The implementations utilize the new 32-bit multiplier and advanced multiplication and squaring routines. Since the proposed routines are generic, the arbitrary length of operand is available with high-speed and small code size. With proposed multiplication and squaring routines, we implemented Curve25519 on the MSP430X processors. The scalar multiplication is performed within 6,666,895 clock cycles and 4,054 bytes. Compared with previous works based on the speed-optimized version, our memory-efficient version reduces the code size by 59.8%, sacrificing the execution timing by 20.5%.