• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.2
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• pp.3-10
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• 2002

For efficient implementation of scalar multiplication in Kobliz elliptic curves, Frobenius endomorphism is useful. Instead of binary expansion of scalar, using Frobenius expansion of scalar we can speed up scalar multiplication and so fast scalar multiplication is closely related to the expansion length of integral multipliers. In this paper we propose a new idea to reduce the length of Frobenius expansion of integral multipliers of scalar multiplication, which makes speed up scalar multiplication. By using the element whose norm is equal to a prime instead of that whose norm is equal to the order of a given elliptic curve we optimize the length of the Frobenius expansion. It can reduce more the length of the Frobenius expansion than that of Solinas, Smart.

• ETRI Journal
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• v.27 no.5
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• pp.617-627
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• 2005

This paper proposes an efficient scalar multiplication algorithm for hyperelliptic curves, which is based on the idea that efficient endomorphisms can be used to speed up scalar multiplication. We first present a new Frobenius expansion method for special hyperelliptic curves that have Gallant-Lambert-Vanstone (GLV) endomorphisms. To compute kD for an integer k and a divisor D, we expand the integer k by the Frobenius endomorphism and the GLV endomorphism. We also present improved scalar multiplication algorithms that use the new expansion method. By our new expansion method, the number of divisor doublings in a scalar multiplication is reduced to a quarter, while the number of divisor additions is almost the same. Our experiments show that the overall throughputs of scalar multiplications are increased by 15.6 to 28.3 % over the previous algorithms when the algorithms are implemented over finite fields of odd characteristics.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.15 no.3
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• pp.65-76
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• 2005

Speeding up scalar multiplication of an elliptic curve point has been a prime approach to efficient implementation of elliptic curve schemes such as EC-DSA and EC-ElGamal. Koblitz introduced a $base-{\phi}$ expansion method using the Frobenius map. Kobayashi et al. extended the $base-{\phi}$ scalar multiplication method to suit Optimal Extension Fields(OEF) by introducing the table reference method. In this paper we propose an efficient scalar multiplication algorithm on elliptic curve over OEF. The proposed $base-{\phi}$ scalar multiplication method uses an optimized batch technique after rearranging the computation sequence of $base-{\phi}$ expansion usually called Horner's rule. The simulation results show that the new method accelerates the scalar multiplication about $20\%{\sim}40\%$ over the Kobayashi et al. method and is about three times as fast as some conventional scalar multiplication methods.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.13 no.3
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• pp.65-70
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• 2003

In this paper, we propose a scalar multiplication method and its hardware architecture which is resistant to SPA while its computation speed is faster than Colon's. There were SPA-resistant scalar multiplication method which has performance problem. Due to this reason, the research about an efficient SPA-resistant scalar multiplication is one of important topics. The proposed architecture resists to SPA and is faster than Colon's method under the assumption that Colon's and the proposed method use same fmite field arithmetic units(multiplier and inverter). With n-bit scalar multiple, the computation cycle of the proposed is 2n·(Inversion cycle)+3(Aultiplication cycle).

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.4 no.4
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• pp.318-326
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• 2011

Public key cryptosystem applications are very difficult in Ubiquitos environments due to computational complexity, memory and power constrains. HECC offers the same of levels of security with much shorter bit-lengths than RSA or ECC. Scalar multiplication is the core operation in HECC. T.Lange proposed inverse free scalar multiplication on genus 2 HECC. However, further coordinate must be access to SCA and need more storage space. This paper developed secure scalar multiplication algorithm with simultaneous inversion algorithm in HECC. To improve the over all performance and security, the proposed algorithm adopt the comparable technique of the simultaneous inversion algorithm. The proposed algorithm is resistant to DPA and SPA.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2022.05a
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• pp.246-248
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• 2022

Binary Edwards curves (BEdC), a new form of elliptic curves proposed by Bernstein, satisfy the complete addition law without exceptions. This paper describes an efficient hardware implementation of point scalar multiplication on BEdC using projective coordinates. Modified Montgomery ladder algorithm was adopted for point scalar multiplication, and binary field arithmetic operations were implemented using 257-bit binary adder, 257-bit binary squarer, and 32-bit binary multiplier. The hardware operation of the BEdC crypto-core was verified using Zynq UltraScale+ MPSoC device. It takes 521,535 clock cycles to compute point scalar multiplication.

• East Asian mathematical journal
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• v.39 no.5
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• pp.529-536
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• 2023

This paper introduces the Frobenius endomophisms on the binary Hessian curves. It provides an efficient and computable homomorphism for computing point multiplication on binary Hessian curves. As an application, it is possible to construct the GLV method combined with the Frobenius endomorphism to accelerate scalar multiplication over the curve.

• ETRI Journal
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• v.26 no.3
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• pp.241-251
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• 2004

We propose two improved scalar multiplication methods on elliptic curves over $F_{{q}^{n}}$ $q= 2^{m}$ using Frobenius expansion. The scalar multiplication of elliptic curves defined over subfield $F_q$ can be sped up by Frobenius expansion. Previous methods are restricted to the case of a small m. However, when m is small, it is hard to find curves having good cryptographic properties. Our methods are suitable for curves defined over medium-sized fields, that is, $10{\leq}m{\leq}20$. These methods are variants of the conventional multiple-base binary (MBB) method combined with the window method. One of our methods is for a polynomial basis representation with software implementation, and the other is for a normal basis representation with hardware implementation. Our software experiment shows that it is about 10% faster than the MBB method, which also uses Frobenius expansion, and about 20% faster than the Montgomery method, which is the fastest general method in polynomial basis implementation.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.369-384
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• 2018

Multiplication is able to be described by using repeated addition, a Cartesian product, a scalar operation, rectangular array and area in many various context. Multiplication in various problem situations is learned by various of the teaching method and the order of teaching more than any other mathematical concepts and operations in elementary school. Nevertheless, the context of multiplication leaves further room for improvement. The purpose of this study is to examine the similarities and differences between the conceptual aspects of multiplication through the literature and to analyze the appropriateness of the teaching method and the order of teaching through textbook analysis. As a result of the study, it was found that multiplication of a scalar operation was introduced too early and did not properly reflect of meaning of multiplication as a scalar operation. There is also a need to use the concept of the rectangular array or area as a meaning of multiplication two quantities.

• Journal of Internet of Things and Convergence
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• v.6 no.2
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• pp.25-29
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• 2020

The elliptic curve crypto-algorithm is widely used in authentication for IoT environment, since it has small key size and low communication overhead compare to the RSA public key algorithm. If the scalar multiplication, a core operation of the elliptic curve crypto-algorithm, is not implemented securely, attackers can find the secret key to use simple power analysis or differential power analysis. In this paper, an elliptic curve scalar multiplication algorithm using a randomized scalar and an elliptic curve point blinding is suggested. It is resistant to power analysis but does not significantly reduce efficiency. Given a random r and an elliptic curve random point R, the elliptic scalar multiplication kP = u(P+R)-vR is calculated by using the regular variant Shamir's double ladder algorithm, where l+20-bit u≡rn+k(modn) and v≡rn-k(modn) using 2^lP=∓cP for the case of the order n=2^l±c.