Scalar Multiplication Method Research Articles

Regular SPA Resistant Fixed-base Comb Method for Elliptic Curve Scalar Multiplication

Regular SPA Resistant Fixed-base Comb Method for Elliptic Curve Scalar Multiplication

The soft graphic integer sub-decomposition method for elliptic scalar multiplication

A new version of the ISD method for computing a scalar multiplication on elliptic curve defined over prime field has been proposed. This version is called the soft graphic ISD (SG-ISD) method. The SG-ISD method employed the connected sub-graphs of undirected simple graph, which formed a soft graph (F, A) generated by a set valued function F, to represent the sub-scalars of ISD method. Thus, undirected simple graph can be proved as the SG-ISD version. On the SG-ISD method, the binary representations, which are all edges of the ISD sub-scalars that forms a nonempty set A, has been done directly from the connected sub-graphs. So, every ISD version of a scalar k is proven mathematically as the SG-ISD version of k in an elliptic scalar multiplication kP. New experimental results on the proposed SG-ISD method are presented. The computational complexities of the original ISD and proposed SG-ISD methods are determined mathematically on the basis of the counting operations. These operations are elliptic curve and finite field operations. The comparison results on these computational complexities of the ISD and SG-ISD methods have been discussed. The experiments on the computational complexities reveal that the SG-ISD method is faster than the ISD method. Thus, the SG-ISD method is considered as an efficient algorithm in comparison with the original ISD for cryptographic applications.

Efficient Implementation of a Crypto Library Using Web Assembly

We implement a cryptographic library using Web Assembly. Web Assembly is expected to show better performance than Javascript. The proposed library provides comprehensive algorithm sets including revised CHAM, Hash Message Authentication Code (HMAC), and ECDH using the NIST P-256 curve to provide confidentiality, data authentication, and key agreement functions. To optimize the performance of revised CHAM in the proposed library, we apply an existing method that is a four-round combining method and additionally propose the precomputation method to CHAM-64/128. The proposed revised CHAM showed an approximate 2.06 times (CHAM-64/128), approximate 2.13 times (CHAM-128/128), and approximate 2.63 times (CHAM-128/256) performance improvement in Web Assembly compared to JavaScript. In addition, CHAM-64/128 applying the precomputation method showed an improved performance by approximately 1.2 times more than the existing CHAM-64/128. For the ECDH using P-256 curve, the naive implementation of ECDH is vulnerable to side-channel attacks (SCA), e.g., simple power analysis (SPA), and timing analysis (TA). Thus, we apply an SPA and TA resistant scalar multiplication method, which is a core operation in ECDH. We present atomic block-based scalar multiplication by revising the previous work. Existing atomic blocks show a performance overhead of 55%, 23%, and 37%, but atomic blocks proposed to use only P=(X,Y,Z) show 18%, 6%, and 11% performance overhead. The proposed Web Assembly-based crypto library provides enhanced performance and resistance against SCA thus, it can be used in various web-based applications.

Pseudo-random scalar multiplication based on group isomorphism

Elliptic curve cryptography is an essential cryptography which is widely used in data encryption, key agreement, digital signature, and other applications. Scalar multiplication is the fundamental operation in all ECC schemes, such as ECDSA, ECIES, and ECMQV. Even a slight improvement of scalar multiplication is precious. Various methods have been proposed for improving efficiency of the scalar multiplication, including the windows method, the NAF method, and the w-NAF method. However, the endeavour in this direction almost exhausted in the past several years since it is hard to find a method substantially better than the w-NAF method. This paper focuses on the scalar multiplication algorithm for the case when the scalar is a pseudo-random number. A faster pseudo-random scalar multiplication method is proposed based on a group isomorphism between the pseudo-random number group and elliptic curve point group. Experimental results show that the proposed method can considerably reduce the computation time compared with those traditional methods. The pseudo-random scalar multiplication accounts for a significant proportion of total scalar multiplication operations in almost all ECC schemes. Therefore, the proposed method is promising and applicable for various ECC applications in the fields such as Internet of things, edge computing, and swarm robotics.

New Quintupling Point Arithmetic 5P Formulas for Lopez-Dahab Coordinate over Binary Elliptic Curve Cryptography

In Elliptic Curve Cryptography (ECC), computational levels of scalar multiplication contains three levels: scalar arithmetic, point arithmetic and field arithmetic. To achieve an efficient ECC performance, precomputed points help to realize a faster computation, which takes away the need to repeat the addition process every time. This paper introduces new quintupling point (5P) formulas which can be precomputed once and can be reused at the scalar multiplication level. We considered mixed addition in Affine and Lŏpez-Dahab since the mixed addition computation cost is better than the traditional addition in Lŏpez-Dahab coordinates over binary curve. Two formulas are introduced for the point quintupling which (Double Double Add) and (Triple Add Double), the cost of the two formulas are 17 multiplication+12 squaringand 23 multiplication+13 squaring respectively. The two formulas are proven as valid points. The new quintupling point can be implemented with different scalar multiplication methods.

Euclidean addition chains scalar multiplication on curves with efficient endomorphism

Random Euclidean addition chain generation has proven to be an efficient low memory and SPA secure alternative to standard ECC scalar multiplication methods in the context of fixed base point [21]. In this work, we show how to generalize this method to random point scalar multiplication on elliptic curves with an efficiently computable endomorphism. In order to do so we generalize results from [21] on the relation of random Euclidean chains generation and elliptic curve point distribution obtained from those chains. We propose a software implementation of our method on various platforms to illustrate the impact of our approach. For that matter, we provide a comprehensive study of the practical computational cost of the modular multiplication when using Java and C standard libraries developed for the arithmetic over large integers.

Optimised elliptic curve digital signature on NIST compliant curves for authentication of MANET nodes

Secure routing protocols for mobile ad hoc networks (MANETs) use digital signatures for authentication which increases computational and communication overheads. Elliptical curve digital signature (ECDSA) uses much shorter keys resulting in smaller signatures, lower computational load, less memory and power requirements which are crucial to MANET nodes. The ECDSA has a characteristic that the signature generation is very fast but signature verification takes much longer time. Optimisation of point operations and scalar multiplication of points have been proposed for accelerating the signature generation and verification process. The proposed method has been software implemented by writing the code in Java using Big-integer class on a Linux platform for National Institute of Standards and Technology (NIST) compliant curves and has accelerated the signature verification process of ECDSA by approximately 27% over the sequential mixed Jacobian-Affine NAF scalar multiplication method of verification.

Optimised elliptic curve digital signature on NIST compliant curves for authentication of MANET nodes

Secure routing protocols for mobile ad hoc networks (MANETs) use digital signatures for authentication which increases computational and communication overheads. Elliptical curve digital signature (ECDSA) uses much shorter keys resulting in smaller signatures, lower computational load, less memory and power requirements which are crucial to MANET nodes. The ECDSA has a characteristic that the signature generation is very fast but signature verification takes much longer time. Optimisation of point operations and scalar multiplication of points have been proposed for accelerating the signature generation and verification process. The proposed method has been software implemented by writing the code in Java using Big-integer class on a Linux platform for National Institute of Standards and Technology (NIST) compliant curves and has accelerated the signature verification process of ECDSA by approximately 27% over the sequential mixed Jacobian-Affine NAF scalar multiplication method of verification.

High-Performance Generic-Point Parallel Scalar Multiplication

A high-performance parallel scalar multiplication method is presented in this paper. The proposed method takes a set of generic points and performs their scalar multiplications in parallel using parallel elliptic curve cryptoprocessors. The method requires neither precomputations nor postcomputations to speed up scalar multiplication and outperforms existing methods. Furthermore, it is scalable and its performance improves as the number of consecutive requests increases. An implementation of the proposed method on an FPGA is presented. The area-time\(^{2}\)\((\hbox {AT}^{2})\) performance metric of the implementation outperforms other recently proposed fast implementations. Accordingly, our method is very attractive for high-speed applications.

Faster elliptic curve arithmetic for triple-base chain by reordering sequences of field operations

In this work, we propose an algorithm to produce the Triple-base chain that optimize the time usage for computing an elliptic curve cryptosystem. Triple-base Chain is a scalar multiplication algorithm, which represents an integer k using three bases {2,3,5}. This paper provides a faster scalar multiplication method of elliptic curve based on {2,3,5} Triple-base Chain. The method proposed by this research speeds up the existing Triple-base Chain algorithm by optimizing the 5P operation of elliptic curve and reordering the operation order of base {2,3,5}. This method can improve the speed of operation from 4 to 6 % compared to the existing {2,3,5} Triple-base Chain.

Highly Efficient Generic-Point Parallel Scalar Multiplication using Concurrent Precomputations

A B S T R A C T An efficient precomputation-based generic-point parallel scalar multiplication method is presented in this study. The proposed method takes a set of generic points and performs their precomputations concurrently using an equivalent set of elliptic curve cryptoprocessors. Then, parallel scalar multiplications are performed sequentially for each of the generic points. The new method outperforms the existing postcomputation-based methods. Furthermore, it is scalable for any number of parallel processors and performs better as the number of consecutive requests increases. The proposed method has been implemented on an FPGA and its Area-Time 2 (AT 2 ) performance metric outperformed recent fast implementations.

GF(3<sup>n</sup>)椭圆曲线密码体制中标量乘快速算法研究

本文研究了特征为3的椭圆曲线密码体制中标量乘的快速算法，并针对底层运算和上层运算分别进行改进。在底层运算中，采用递推归纳、乘法代替求逆、立方代替乘法的思想提出在仿射坐标下直接计算3kP的算法，将求逆运算降至1次；在上层运算中，采用滑动窗口标量乘的方法，既有效减少了非零窗口的长度又减少了算法中3倍点总的运算量。 In this paper we investigate the fast algorithm of scalar multiplication based on Elliptic Curve Cryptography (ECC) over fields of characteristic three, and make improvements both in underlying operations and upper operations respectively. In the underlying operations, we deduce a formula of calculating 3kP directly under the affine coordinates based upon the idea of recursion, trading inversion for multiplication and trading multiplication for cube, which reduces the inversion to once; in the upper operations, we adopt the sliding window scalar multiplication method, which reduces the length of the non-zero windows and the total computations of 3P effectively.

The Construction of ElGamal over Koblitz Curve

Recently elliptic curve cryptosystems are widely accepted for security applications key generation, signature and verification. Cryptographic mechanisms based on elliptic curves depend on arithmetic involving the points of the curve. it is possible to use smaller primes, or smaller finite fields, with elliptic curves and achieve a level of security comparable to that for much larger integers. Koblitz curves, also known as anomalous binary curves, are elliptic curves defined over F2. The primary advantage of these curves is that point multiplication algorithms can be devised that do not use any point doublings. The ElGamal cryptosystem, which is based on the Discrete Logarithm problem can be implemented in any group. In this paper, we propose the ElGamal over Koblitz Curve Scheme by applying the arithmetic on Koblitz curve to the ElGamal cryptosystem. The advantage of this scheme is that point multiplication algorithms can be speeded up the scalar multiplication in the affine coodinate of the curves using Frobenius map. It has characteristic two, therefore it’s arithmetic can be designed in any computer hardware. Moreover, it has more efficient to employ the TNAF method for scalar multiplication on Koblitz curves to decrease the number of nonzero digits. It’s security relies on the inability of a forger, who does not know a private key, to compute elliptic curve discrete logarithm.

Implementation of an Elliptic Curve Scalar Multiplication Method Using Division Polynomials

Implementation of an Elliptic Curve Scalar Multiplication Method Using Division Polynomials

An efficient and scalable postcomputation-based generic-point parallel scalar multiplication method

An efficient and scalable postcomputation-based generic-point parallel scalar multiplication method

Fixed-Base Comb with Window-Non-Adjacent Form (NAF) Method for Scalar Multiplication

Elliptic curve cryptography (ECC) is one of the most promising public-key techniques in terms of short key size and various crypto protocols. For this reason, many studies on the implementation of ECC on resource-constrained devices within a practical execution time have been conducted. To this end, we must focus on scalar multiplication, which is the most expensive operation in ECC. A number of studies have proposed pre-computation and advanced scalar multiplication using a non-adjacent form (NAF) representation, and more sophisticated approaches have employed a width-w NAF representation and a modified pre-computation table. In this paper, we propose a new pre-computation method in which zero occurrences are much more frequent than in previous methods. This method can be applied to ordinary group scalar multiplication, but it requires large pre-computation table, so we combined the previous method with ours for practical purposes. This novel structure establishes a new feature that adjusts speed performance and table size finely, so we can customize the pre-computation table for our own purposes. Finally, we can establish a customized look-up table for embedded microprocessors.

Analysis of the width-[formula omitted] non-adjacent form in conjunction with hyperelliptic curve cryptography and with lattices

In this work the number of occurrences of a fixed non-zero digit in the width-w non-adjacent forms of all elements of a lattice in some region (e.g. a ball) is analysed. As bases, expanding endomorphisms with eigenvalues of the same absolute value are allowed. Applications of the main result are on numeral systems with an algebraic integer as base. Those come from efficient scalar multiplication methods (Frobenius-and-add methods) in hyperelliptic curves cryptography, and the result is needed for analysing the running time of such algorithms.The counting result itself is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange’s method.

Efficiently computable endomorphism for genus 3 hyperelliptic curve cryptosystems

Scalar multiplication methods using efficiently computable endomorphism are known for efficient methods to speed up (hyper)elliptic curve cryptosystems. In this paper we extend the results of Galbraith et al. (2009, 2011) [13,14] and Li et al. (2011) [16] to any genus 3 hyperelliptic curves over a finite field of even characteristic. For the quadratic twist of a genus 3 hyperelliptic curve, we give the explicit formulae for the efficiently computable endomorphism on the Jacobian and demonstrate that the endomorphism leads to 2-dimension GLV method. Our method is 49.9% faster than the previous best methods for the 128-bits point multiplication of genus 3 hyperelliptic curves.

An Innovative Scalar Multiplication Method Based on Improved m-ary

On purpose to elevate the efficiency of elliptic curve scalar multiplication in the device with weak computation power and to improve computational security, in this paper we pioneer a novel algorithm named Improved-m-ary, which is based on the depth first addition chain scheme and the improved m-ary mechanism compatible with a flexible width window. At first, we research and analyze the advantages of addition-chain-method, m-ary and other algorithms respectively in terms of speeding computation by comparison. It is discovered that the proportion of atomic operation and window width are 2 key factors which keep the speed of scalar multiplication and its computation cost in a leash. Then, an innovative scalar-point-multiplication algorithm is designed by the project crew on the basis of findings of project research. At last, the results of theoretical analysis and experimentation statistics demonstrate that by this algorithm the average of hamming weight of the scalar as a multiplicator could be undercut and the computation cost of point-scalar-multiplication could be lowered to an amazing extent. In addition, because of its built-in scheme whereby the window width is randomized constantly it presents a favorable strong immunity against most attack methods hinged on power analysis .As a whole, it is potential that Improved-m-ary be a practical and promising fast scalar multiplication method alternative.

Fast and scalable parallel processing of scalar multiplication in elliptic curve cryptosystems

ABSTRACTTo secure parallel systems in communication networks, in this paper, we propose a fast and scalable parallel scalar multiplication method over generic elliptic curves for elliptic curve cryptosystems, by means of our proposed scalar folding and unfolding techniques. In contrast to previous parallel scalar multiplication methods, our method can be implemented into scalable parallel computers. The optimal time complexity is k point doublings (D) plus log k point additions (A), denoted as kD + (log k)A, where k is the bit length of the scalar. If our method is applied to Koblitz curves, the optimal time complexity can be reduced to (log k)A. Furthermore, previous simple side‐channel‐protected scalar multiplication methods can be integrated into our method for resisting against simple side‐channel attacks. Copyright © 2011 John Wiley & Sons, Ltd.