Division polynomial‐based elliptic curve scalar multiplication revisited

SubramanyaRao, SrinivasaRao; Hu, Zhi; Zhao, Chang-An

doi:10.1049/iet-ifs.2018.5361

Cited by 3 publications

(27 citation statements)

References 9 publications

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“…Notice that 4 multiplications can be saved at each iteration in the Double-and-Add algorithm if gcd(p − 1, 3) = 1 [6]. In this work, we will improve the Doubleand-Add algorithm for scalar multiplication in two situations in [32].…”

Section: Algorithmmentioning

confidence: 99%

“…In the second part, we will show the comparison of the efficiency between computing scalar multiplication in [32] and our work. Scalar multiplication algorithm with division polynomials is similar to the ladder algorithm, and this algorithm is easily coded compared to the traditional double-and-add algorithm.…”

Section: Implementation and Analysismentioning

confidence: 99%

“…Our work based on the scalar multiplication algorithm proposed in [32] is to replace 6 right-shift operations by 2 left-shift operations in each iteration. It seems that this improvement will not work obviously.…”

Section: Scalar Multiplicationmentioning

confidence: 99%

“…An optimized version of scalar multiplication algorithm using division polynomials was proposed in [6], which saved f our multiplications at each iteration by using the equivalence of elliptic nets. Based on these previous works, Rao et al [32] proposed a modified algorithm based on elliptic nets to compute scalar multiplication.…”

Section: Introductionmentioning

confidence: 99%

“…The implementation indicates that the IENA works well if it is further modified by this trick. Besides, this trick contributes to the scalar multiplication algorithm in [32].…”

Section: Introductionmentioning

confidence: 99%

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The Elliptic Net Algorithm Revisited

Cai¹,

Hu²,

Yao³

et al. 2021

Preprint

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Pairings have been widely used since their introduction to cryptography. They can be applied to identity-based encryption, tripartite Diffie-Hellman key agreement, blockchain and other cryptographic schemes. The Acceleration of pairing computations is crucial for these cryptographic schemes or protocols. In this paper, we will focus on the Elliptic Net algorithm which can compute pairings in polynomial time, but it requires more storage than Miller's algorithm. We use several methods to speed up the Elliptic Net algorithm. Firstly, we eliminate the inverse operation in the improved Elliptic Net algorithm. In some circumstance, this finding can achieve further improvements. Secondly, we apply lazy reduction technique to the Elliptic Net algorithm, which helps us achieve a faster implementation. Finally, we propose a new derivation of the formulas for the computation of the Optimal Ate pairing on the twisted curve. Results show that the Elliptic Net algorithm can be significantly accelerated especially on the twisted curve. The algorithm can be 80% faster than the previous ones on the twisted 381-bit BLS12 curve and 71.5% faster on the twisted 676-bit KSS18 curve respectively.

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Section: Algorithmmentioning

confidence: 99%

Section: Implementation and Analysismentioning

confidence: 99%

Section: Scalar Multiplicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The implementation indicates that the IENA works well if it is further modified by this trick. Besides, this trick contributes to the scalar multiplication algorithm in [32].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Elliptic Net Algorithm Revisited

Cai¹,

Hu²,

Yao³

et al. 2021

Preprint

Self Cite

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Implementation of Elliptic Net Scalar Multiplication Computation for NIST P-192 Curve using Python

Razali

Muslim

Kahar

2021

J. Phys.: Conf. Ser.

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Elliptic curve cryptography is one of the most efficient public-key cryptosystems compared to the Rivest-Shamir-Addleman scheme. One of the methods to compute elliptic curve scalar multiplication is division polynomials which utilize the non-linear recurrence relation also known as the elliptic net. Previous studies were related to elliptic curve scalar multiplication via elliptic net which implemented the use of C++, GP/PARI or Java programming. This study aims to implement the Python programming in the SageMath compiler for computing elliptic net scalar multiplication based on the National Institute of Standards and Technology curve of the type P-192. The implementation could be seen at three main processes which are the scalar decomposition, the blocks of elliptic net generation namely the initial and final blocks, as well as scalar multiplication computation. The base point and prime field of the short Weierstrass curve with a large scalar are the parameters used in the process. The results showed that the implemented programming was easier while working with a large prime number field, vulnerable to errors and can be easily verified. The implemented programming can also be used to compute scalar multiplication on other standardizations of elliptic curve cryptography such as Brainpool 384-bit prime field or NUMS 256-bit prime field curves.

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Enhanced Scalar Multiplication Algorithm over Prime Field Using Elliptic Net

Norliana Muslim,

Faridah Yunos,

Zuren Razali

et al. 2024

ARASET

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Scalar multiplication in elliptic curve cryptography is the most expensive and time-consuming operation. The elliptic curve cryptography attracted interest due to the development of modern technology since it could offer the equivalent high level of security with a reduced length of key. Therefore, improving elliptic curve scalar multiplication performance has always been the primary goal of cryptography. In this paper, a novel scalar multiplication algorithm based on the modified double and double add via elliptic net with Karatsuba method was proposed in order to enhance the efficiency of scalar multiplication. In the experimental results, the elliptic net equivalence sequence was applied to the Twisted Edwards curve together with safe curves of numsp384t1 and numsp512t1. At the point operational level, the proposed method reduced the cost of multiplication by 46.15% and 42.30% for double and double add, respectively, when compared to elliptic net using eight blocks method. The proposed double lowered the multiplication cost by 12.5% and the squaring cost by 20% when compared to elliptic net using ten temporary variables method. Following this, proposed double add cost reductions of 6.25% and 20% were obtained to multiplication and squaring. At the field operational level, in comparison to the binary method, the eight-block elliptic net method, and the elliptic net method with ten temporary variables for the 384 bits scenario, the developed scalar multiplication algorithm obtained cost reductions of 57.6%, 31.3%, and 13.2%, respectively. On 512 bits with similar comparison, the designed algorithm exhibited better performance by averages 59.2%, 31.0% and 13.2%. The results signified that the designed algorithm over prime field performed better at the point and field operational levels with larger scalar bit size.

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Division polynomial‐based elliptic curve scalar multiplication revisited

Cited by 3 publications

References 9 publications

The Elliptic Net Algorithm Revisited

The Elliptic Net Algorithm Revisited

Implementation of Elliptic Net Scalar Multiplication Computation for NIST P-192 Curve using Python

Enhanced Scalar Multiplication Algorithm over Prime Field Using Elliptic Net

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