Note on scalar multiplication using division polynomials

Chen, Bing-Long; Hu, Chuangqiang; Zhao, Chang-An

doi:10.1049/iet-ifs.2015.0119

Cited by 10 publications

(17 citation statements)

References 16 publications

(46 reference statements)

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“…The main contribution of this paper is in providing this adaptation and a derivation of this improvement. We also note that Kanayama's algorithm was improved by Chen et al [7]. The adaptation in this paper improves Chen's version of Kanayama's algorithm as well as Kanayama's original ECSM algorithm.…”

Section: Introductionmentioning

confidence: 75%

“…If we denote the multiplication and squaring in the finite field under consideration by M and S, respectively, it can be seen that the number of operations for both the Double and DoubleAdd steps is 26M + 6S. Chen et al [7] improved Kanayama's algorithm with the new operation count being 18M + 10S. In the next section, we adapt the improvement from [6].…”

Section: Review Of Kanayama's Elliptic Curve Scalar Multiplication Almentioning

confidence: 99%

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Division polynomial‐based elliptic curve scalar multiplication revisited

SubramanyaRao

Zhao

2019

IET Information Security

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Section: Introductionmentioning

confidence: 75%

Section: Review Of Kanayama's Elliptic Curve Scalar Multiplication Almentioning

confidence: 99%

Division polynomial‐based elliptic curve scalar multiplication revisited

SubramanyaRao

Zhao

2019

IET Information Security

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“…In turn, this paper provides a comprehensive investigation of the mapping phase for several known and widely used elliptic curves. For instance, secp192k1, NIST-224, and secp256k1 are examined [44][45][46]. Finally, this paper proposes effective padding bit values and a method that guarantees successful mapping, as well as efficiency using the least number of bits.…”

Section: Preliminary: Elliptic Curve Cryptography (Ecc)mentioning

confidence: 99%

iTrust—A Trustworthy and Efficient Mapping Scheme in Elliptic Curve Cryptography

Almajed

Almogren

Alabdulkareem

2020

Sensors

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Recently, many platforms have outsourced tasks to numerous smartphone devices known as Mobile Crowd-sourcing System (MCS). The data is collected and transferred to the platform for further analysis and processing. These data needs to maintain confidentiality while moving from smartphones to the platform. Moreover, the limitations of computation resources in smartphones need to be addressed to balance the confidentiality of the data and the capabilities of the devices. For this reason, elliptic curve cryptography (ECC) is accepted, widespread, and suitable for use in limited resources environments such as smartphone devices. ECC reduces energy consumption and maximizes devices’ efficiency by using small crypto keys with the same strength of the required cryptography of other cryptosystems. Thus, ECC is the preferred approach for many environments, including the MCS, Internet of Things (IoT) and wireless sensor networks (WSNs). Many implementations of ECC increase the process of encryption and/or increase the space overhead by, for instance, incorrectly mapping points to EC with extra padding bits. Moreover, the wrong mapping method used in ECC results in increasing the computation efforts. This study provides comprehensive details about the mapping techniques used in the ECC mapping phase, and presents performance results about widely used elliptic curves. In addition, it suggests an optimal enhanced mapping method and size of padding bit to secure communications that guarantee the successful mapping of points to EC and reduce the size of padding bits.

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“…There is a different form of equation 2proposed by Ward [3]. Kanayama [7] and Chen [14] use this equivalent in their scalar multiplication. However, the initial value  (2) N of the sequences was not identical.…”

Section: A Elliptic Divisibility Sequencesmentioning

confidence: 99%

“…The following Table I compares the value of  (2) N for elliptic net scalar multiplication between Kanayama et al [7], Chen [14] and our method.…”

Section: B Comparison Of the Initial Valuesmentioning

confidence: 99%

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

Muslim*,

Said

2019

IJEAT

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Chord and tangent is a classical method to calculate the elliptic curve scalar multiplication. Alternatively, the scalar multiplication can be calculated by dividing polynomials over certain finite fields and the first elliptic net scalar multiplication was implemented on a short Weierstrass curve. The net was originated from non-linear recurrence sequences, namely as elliptic divisibility sequence. It is well known that the linear recurrence sequences have been applied in the cryptosystem as a cipher in the encryption and decryption process. From the perspective of cryptographic application, the elliptic divisibility sequence is used generally for integer factorization, solving elliptic curve discrete logarithm problem and computation of pairing or scalar multiplication. But there is a lack of contribution of these non-linear recurrence sequences in scalar multiplication. Therefore, this paper aims to discuss a generalization of the equivalent sequence of elliptic divisibility for computing scalar multiplication. The experimental results of scalar multiplication via the net and its coding in computer programming are presented. The future direction of scalar multiplication via the elliptic net is also discussed.

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Note on scalar multiplication using division polynomials

Cited by 10 publications

References 16 publications

Division polynomial‐based elliptic curve scalar multiplication revisited

Division polynomial‐based elliptic curve scalar multiplication revisited

iTrust—A Trustworthy and Efficient Mapping Scheme in Elliptic Curve Cryptography

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

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