Implementation of an Elliptic Curve Scalar Multiplication Method Using Division Polynomials

Kanayama, Naoki; Liu, Yang; Okamoto, Eiji; Saito, Kazutaka; Teruya, Tadanori; Uchiyama, Shigenori

doi:10.1587/transfun.e97.a.300

Cited by 10 publications

(37 citation statements)

References 5 publications

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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%

“…For example, Shipsey used them to solve the elliptic curve discrete logarithm problem or as an integer factorization [4]. Meanwhile, Stange implemented her net algorithm to calculate pairings [6] and Stange's algorithm was adapted to calculate elliptic curve scalar multiplication by Kanayama et al [7]. The earlier discussion of the elliptic net by Malaysian researchers can be seen in Muslim & Said [8]- [9].…”

Section: Introductionmentioning

confidence: 99%

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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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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%

Section: Introductionmentioning

confidence: 99%

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Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

Muslim*,

Said

2019

IJEAT

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“…The famous Weierstrass equation [8] known as an elliptic curve E of y 2 + axy +by = x 3 +cx 2 +dx +e and E is the general solutions of y 2 = x 3 + Ax +B. The curve E also has important auxiliary polynomials like φ m = x(ℽ m ) 2…”

Section: Weierstrass and Koblitz Curvementioning

confidence: 99%

“…Since then, the higher-ranked elliptic nets on Weierstrass curves have been applied in the computation of Tate and r − Ate pairings [6]. Years later, the same theory of elliptic nets which employed polynomials has been applied in the calculation of scalar multiplications [2,5,12]; this is the primary goal and interest in this current paper. In the attempt to prove the possibility of developing elliptic nets from a new cryptographic curve, Koblitz curve had been utilized.…”

Section: Introductionmentioning

confidence: 99%

Constructing Scalar Multiplication via Elliptic Net of Rank Two

Muslim¹,

Said²

2018

IJET

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Elliptic nets are a powerful method for computing cryptographic pairings. The theory of rank one nets relies on the sequences of elliptic divisibility, sets of division polynomials, arithmetic upon Weierstrass curves, as well as double and double-add properties. However, the usage of rank two elliptic nets for computing scalar multiplications in Koblitz curves have yet to be reported. Hence, this study entailed investigations into the generation of point additions and duplication of elliptic net scalar multiplications from two given points on the Koblitz curve. Evidently, the new net had restricted initial values and different arithmetic properties. As such, these findings were a starting point for the generation of higher-ranked elliptic net scalar multiplications with curve transformations. Furthermore, using three distinct points on the Koblitz curves, similar methods can be applied on these curves.

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An Improved EllipticNet Algorithm for Tate Pairing on Weierstrass’ Curves, Faster Point Arithmetic and Pairing on Selmer Curves and a Note on Double Scalar Multiplication

Rao

2016

Applications and Techniques in Information Security

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Implementation of an Elliptic Curve Scalar Multiplication Method Using Division Polynomials

Cited by 10 publications

References 5 publications

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

Constructing Scalar Multiplication via Elliptic Net of Rank Two

An Improved EllipticNet Algorithm for Tate Pairing on Weierstrass’ Curves, Faster Point Arithmetic and Pairing on Selmer Curves and a Note on Double Scalar Multiplication

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