Elliptic Curve Scalar Point Multiplication Research Articles

A High Speed and Area Efficient Processor for Elliptic Curve Scalar Point Multiplication for GF($2^m$)

A High Speed and Area Efficient Processor for Elliptic Curve Scalar Point Multiplication for GF($2^m$)

An Efficient and Secure Revocation-Enabled Attribute-Based Access Control for eHealth in Smart Society.

The ever-growing ecosystem of the Internet of Things (IoT) integrating with the ever-evolving wireless communication technology paves the way for adopting new applications in a smart society. The core concept of smart society emphasizes utilizing information and communication technology (ICT) infrastructure to improve every aspect of life. Among the variety of smart services, eHealth is at the forefront of these promises. eHealth is rapidly gaining popularity to overcome the insufficient healthcare services and provide patient-centric treatment for the rising aging population with chronic diseases. Keeping in view the sensitivity of medical data, this interfacing between healthcare and technology has raised many security concerns. Among the many contemporary solutions, attribute-based encryption (ABE) is the dominant technology because of its inherent support for one-to-many transfer and fine-grained access control mechanisms to confidential medical data. ABE uses costly bilinear pairing operations, which are too heavy for eHealth’s tiny wireless body area network (WBAN) devices despite its proper functionality. We present an efficient and secure ABE architecture with outsourcing intense encryption and decryption operations in this work. For practical realization, our scheme uses elliptic curve scalar point multiplication as the underlying technology of ABE instead of costly pairing operations. In addition, it provides support for attribute/users revocation and verifiability of outsourced medical data. Using the selective-set security model, the proposed scheme is secure under the elliptic curve decisional Diffie–Hellman (ECDDH) assumption. The performance assessment and top-ranked value via the help of fuzzy logic’s evaluation based on distance from average solution (EDAS) method show that the proposed scheme is efficient and suitable for access control in eHealth smart societies.

New Regular Sliding Window Algorithms for Elliptic Curve Scalar Point Multiplication

New Regular Sliding Window Algorithms for Elliptic Curve Scalar Point Multiplication

Elliptic Curve Cryptography for Wireless Sensor Networks Using the Number Theoretic Transform.

We implement elliptic curve cryptography on the MSP430 which is a commonly used microcontroller in wireless sensor network nodes. We use the number theoretic transform to perform finite field multiplication and squaring as required in elliptic curve scalar point multiplication. We take advantage of the fast Fourier transform for the first time in the literature to speed up the number theoretic transform for an efficient realization of elliptic curve cryptography. Our implementation achieves elliptic curve scalar point multiplication in only s and s for multiplication of fixed and random points, respectively, and has similar or better timing performance compared to previous works in the literature.

Security and Privacy enforced wireless mobile communication using PI-MAKA protocol design

The fast development of mobile networks and compact mobile devices bring attention to the users for wireless mobile communication. But providing security and protecting the privacy of users are the main challenges of wireless mobile communication. Recently, Authenticated Key Agreement protocols are used for secure and reliable communication in wireless mobile communication. However, the User-to-User Mutual Authentication and Key Agreement scheme is based on bilinear pairings, which involves relatively high computation cost when compared to elliptic curve scalar point multiplication. Hence, in this paper, we propose a Pairing-Free Identity-Based Mutual Authentication and Key Agreement protocol for wireless mobile communications. It consists of the following phases: Setup and Key generation, Mobility Management Entity authentication and Proxy Signature generation, Proxy Signature verification and User-to-User authentication. Experimental results show that Pairing-Free Identity-Based Mutual Authentication and Key Agreement attains significantly less computational and communication cost when compared to User-to-User Mutual Authentication and Key Agreement scheme.

Optimal Representation for Right-to-Left Parallel Scalar and Multi-Scalar Point Multiplication

This paper introduces an optimal representation for a right-to-left parallel elliptic curve scalar point multiplication. The right-to-left approach is easier to parallelize than the conventional left-to-right approach. However, unlike the left-to-right approach, there is still no work considering number representations for the right-to-left parallel calculation. By simplifying the implementation by Robert, we devise a mathematical model to capture the computation time of the calculation. Then, for any arbitrary amount of doubling time and addition time, we propose algorithms to generate representations which minimize the time in that model. As a result, we can show a negative result that a conventional representation like NAF is almost optimal. The parallel computation time obtained from any representation cannot be better than NAF by more than 1%. In addition to that, we devise a time model and propose an algorithm to generate optimal representations for multi-scalar point multiplication (under a condition). Similar to the result of scalar point multiplication, NAF is almost optimal also for multi-scalar point multiplication as the difference of parallel computation time obtained from optimal representation and NAF is less than 1% in all experimental settings.

An Efficient Authenticated Key Transfer Scheme in Client-Server Networks

In this paper, we presented a novel authenticated key transfer scheme in client-server networks, which can achieve two secure goals of remote user authentication and the session key establishment between the remote user and the server. Especially, the proposed scheme can subtly provide two fully different authentications: identity-base authentication and anonymous authentication, while the remote user only holds a private key. Furthermore, our scheme only needs to transmit 1-round messages from the remote user to the server, thus it is very efficient in communication complexity. In addition, the most time-consuming computation in our scheme is elliptic curve scalar point multiplication, so it is also feasible even for mobile devices.

FPGA Based High Speed SPA Resistant Elliptic Curve Scalar Multiplier Architecture

The higher computational complexity of an elliptic curve scalar point multiplication operation limits its implementation on general purpose processors. Dedicated hardware architectures are essential to reduce the computational time, which results in a substantial increase in the performance of associated cryptographic protocols. This paper presents a unified architecture to compute modular addition, subtraction, and multiplication operations over a finite field of large prime characteristicGF(p). Subsequently, dual instances of the unified architecture are utilized in the design of high speed elliptic curve scalar multiplier architecture. The proposed architecture is synthesized and implemented on several different Xilinx FPGA platforms for different field sizes. The proposed design computes a 192-bit elliptic curve scalar multiplication in 2.3 ms on Virtex-4 FPGA platform. It is 34%faster and requires 40%fewer clock cycles for elliptic curve scalar multiplication and consumes considerable fewer FPGA slices as compared to the other existing designs. The proposed design is also resistant to the timing and simple power analysis (SPA) attacks; therefore it is a good choice in the construction of fast and secure elliptic curve based cryptographic protocols.

Design of provably secure and efficient certificateless blind signature scheme using bilinear pairing

AbstractIn the literature, several pairing‐based blind signature schemes have been put forwarded using identity‐based cryptography. However, the private key escrow problem of these schemes makes them unsuitable in practical environments because the private key generator computes signer private key using signer public identity. Therefore, an untrusted/semitrusted private key generator may perform malicious activity on behalf of the signer. We took the advantage of certificateless public key cryptography and constructed a robust and efficient certificateless blind signature (CL‐BS) scheme using bilinear pairing. Furthermore, the probabilistic map‐to‐point hash function, which requires more computation time than the execution time of an elliptic curve scalar point multiplication, is circumvented in our scheme. We found that our scheme is unlinkable and provably secure against the adaptive chosen message and identity adversaries. From the perspective of computational costs, our scheme is more efficient and secure than other related schemes. Copyright © 2015 John Wiley & Sons, Ltd.

Design of an Elliptic Curve Cryptography processor for RFID tag chips.

Radio Frequency Identification (RFID) is an important technique for wireless sensor networks and the Internet of Things. Recently, considerable research has been performed in the combination of public key cryptography and RFID. In this paper, an efficient architecture of Elliptic Curve Cryptography (ECC) Processor for RFID tag chip is presented. We adopt a new inversion algorithm which requires fewer registers to store variables than the traditional schemes. A new method for coordinate swapping is proposed, which can reduce the complexity of the controller and shorten the time of iterative calculation effectively. A modified circular shift register architecture is presented in this paper, which is an effective way to reduce the area of register files. Clock gating and asynchronous counter are exploited to reduce the power consumption. The simulation and synthesis results show that the time needed for one elliptic curve scalar point multiplication over GF(2163) is 176.7 K clock cycles and the gate area is 13.8 K with UMC 0.13 μm Complementary Metal Oxide Semiconductor (CMOS) technology. Moreover, the low power and low cost consumption make the Elliptic Curve Cryptography Processor (ECP) a prospective candidate for application in the RFID tag chip.

Novel Architecture for Efficient FPGA Implementation of Elliptic Curve Cryptographic Processor Over ${\rm GF}(2^{163})$

A new and highly efficient architecture for elliptic curve scalar point multiplication is presented. To achieve the maximum architectural and timing improvements, we reorganize and reorder the critical path of the Lopez-Dahab scalar point multiplication architecture such that logic structures are implemented in parallel and operations in the critical path are diverted to noncritical paths. The results we obtained show that with G=55 our proposed design is able to compute scalar multiplication over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ) in 9.6 μs with the maximum achievable frequency of 250 MHz on Xilinx Virtex-4 (XC4VLX200), where G is the digit size of the underlying digit-serial finite-field multiplier. Another implementation variant for less resource consumption is also proposed; with G=33, the design performs the same operation in 11.6 μs at 263 MHz on the same platform. The results of synthesis show that, in the first implementation, 17 929 slices or 20% of the chip area is occupied, which makes it suitable for speed-critical cryptographic applications, while in the second implementation 14203 slices or 16% of the chip area is utilized, which makes it suitable for applications that may require speed-area tradeoff.

Elliptic Curve Cryptography in Securing Networks by Mobile Authentication

This paper proposes an enhanced authentication model, which is suitable for low-power mobile devices. It uses an Extended Password Key Exchange Protocols (2) and elliptic-curve-cryptosystem based trust delegation mechanism to generate a delegation pass code for mobile station authentication, and it can effectively defend all known attacks to mobile networks including the denial-of-service attack. Moreover, the mobile station only needs to receive one message and send one message to authenticate itself to a visitor's location register, and the model only requires a single elliptic-curve scalar point multiplication on a mobile device. Therefore, this model enjoys both computation efficiency and communication efficiency as compared to known mobile authentication models.

AN EFFICIENT AND SECURE STRONG DESIGNATED VERIFIER SIGNATURE SCHEME WITHOUT BILINEAR PAIRINGS

In literature, several strong designated verifier signature (SDVS) schemes have been devised using elliptic curve bilinear pairing and map-topoint (MTP) hash function. The bilinear pairing requires a super-singular elliptic curve group having large number of elements and the relative computation cost of it is approximately two to three times higher than that of elliptic curve point multiplication, which indicates that bilinear pairing is an expensive operation. Moreover, the MTP function, which maps a user identity into an elliptic curve point, is more expensive than an elliptic curve scalar point multiplication. Hence, the SDVS schemes from bilinear pairing and MTP hash function are not efficient in real environments. Thus, a cost-efficient SDVS scheme using elliptic curve cryptography with pairingfree operation is proposed in this paper that instead of MTP hash function uses a general cryptographic hash function. The security analysis shows that our scheme is secure in the random oracle model with the hardness assumption of CDH problem. In addition, the formal security validation of the proposed scheme is done using AVISPA tool (Automated Validation of Internet Security Protocols and Applications) that demonstrated that our scheme is unforgeable against passive and active attacks. Our scheme also satisfies the different properties of an SDVS scheme including strongness, source hiding, non-transferability and unforgeability. The comparison of our scheme with others are given, which shows that it outperforms in terms of security, computation cost and bandwidth requirement.

An Efficient and Provably-secure Digital signature Scheme based on Elliptic Curve Bilinear Pairings

We proposed an efficient and secure digital signature scheme using elliptic curve cryptography (ECC) and bilinear pairings in this paper. The proposed scheme employs the general cryptographic hash function (i.e., SHA-1) instead of map-to-point function, because the map-to-point is a cost-intensive operation and it is usually implemented as a probabilistic algorithm. Further, our scheme is computationally efficient as one bilinear paring and three elliptic curve scalar point multiplication operations are executed for signature generation and verification, and thus the scheme requires much lesser computation cost than other related schemes. In addition, in the random oracle model, our scheme is proven to be existential unforgeable against the adaptive chosen message and identity attacks (EUF-CMA) based on a variation of the collusion attack algorithm with ktraitors (k-CAA3) problem.

Elliptic Curve Scalar Point Multiplication Algorithm Using Radix-4 Booth’s Algorithm

The main back-bone operation in elliptic curve cryptosystems is scalar point multiplication. The most frequently used method implementing the scalar point multiplication, which is performed in the topmost level of GF multiplication and GF division, has been the double-and-add algorithm, which is being recently challenged by NAF (Non-Adjacent Format) algorithm. In this paper, we propose a more efficient and novel approach of a scalar multiplication method than the double-and-add by applying redundant recoding which originates from the radix-4 modified Booth’s algorithm. We call the novel algorithm quad-and-add. After deriving the algorithm, we created a new GF operation, named point quadruple, and verified with calculations of a real-world application to utilize it. Derived numerical expressions were verified using both C programs and HDL (Hardware Description Language). Proposed method of elliptic curve scalar point multiplication can be utilized in many elliptic curve security applications for handling efficient and fast calculations.