• 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.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.8 no.6
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• pp.1212-1217
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• 2004

The main back-bone operation in elliptic curve cryptosystems is scalar point multiplication. The most frequently used method implementing the scalar point multiplication, which is performed in the upper level of GF multiplication and GF division, has been the double-and-add algorithm, which is recently challenged by NAF(Non-Adjacent Format) algorithm. In this paper, we propose a more efficient and novel scalar multiplication method than existing double-and-add by applying redundant receding which originates from radix-4 Booth's algorithm. After deriving the novel quad-and-add algorithm, we created a new operation, named point quadruple, and verified with real application calculation to utilize it. Derived numerical expressions were verified using both C programs and HDL (Hardware Description Language) in real applications. Proposed method of elliptic curve scalar point multiplication can be utilized in many elliptic curve security applications for handling efficient and fast calculations.

• 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.

• 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.

• 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.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2004.05a
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• pp.49-52
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• 2004

Scalar multiplication is the back-bone operation in the elliptic curve cryptosystem. Quad-and-add algorithm replaced the traditional double-and-add algorithm to compute the scalar multiplication. In this paper, we introduce the method of utilizing the point quadruple scalar operation in the elliptic curve cryptosystem. Induced expressions were applied to real cryptosystem and proven at C language level. Point quadruple operation can be utilized to fast and efficient computation in the elliptic curve cryptosystem.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.10
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• pp.588-592
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• 2004

Scalar multiplication is to compute $textsc{k}$P when an integer $textsc{k}$ and an elliptic curve point f are given. As a general method to accelerate scalar multiplication, Agnew, Mullin and Vanstone proposed to use $textsc{k}$'s with fixed Hamming weights. We suggest a new method that uses $textsc{k}$'s with fixed signed Hamming weights and show that this method is more secure.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2004.05b
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• pp.784-787
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• 2004

The most time-consuming back-bone operation in an elliptic curve cryptosystem is scalar multiplication. In this paper, we propose a method of inducing a GF operation named point quadruple operation to be used in the quad-and-add algorithm, whith was achieved by refining the traditional double-and-add algorithm. Induced expression of the algorithm was verified and proven by C program in a real model of calculation. The point quadruple operation can be used in fast and efficient implementation of scalar multiplication operation.

• JSTS:Journal of Semiconductor Technology and Science
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• v.16 no.1
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• pp.118-125
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• 2016

This paper presents a new high-efficient algorithm and architecture for an elliptic curve cryptographic processor. To reduce the computational complexity, novel modified Lopez-Dahab scalar point multiplication and left-to-right algorithms are proposed for point multiplication operation. Moreover, bit-serial Galois-field multiplication is used in order to decrease hardware complexity. The field multiplication operations are performed in parallel to improve system latency. As a result, our approach can reduce hardware costs, while the total time required for point multiplication is kept to a reasonable amount. The results on a Xilinx Virtex-5, Virtex-7 FPGAs and VLSI implementation show that the proposed architecture has less hardware complexity, number of clock cycles and higher efficiency than the previous works.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.26 no.8
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• pp.1172-1179
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• 2022

This paper describes a design of point scalar multiplier for public-key cryptography based on binary Edwards curves (BEdC). For efficient implementation of point addition (PA) and point doubling (PD) on BEdC, projective coordinate was adopted for finite field arithmetic, and computational performance was improved because only one inversion was involved in point scalar multiplication (PSM). By applying optimizations to hardware design, the storage and arithmetic steps for finite field arithmetic in PA and PD were reduced by approximately 40%. We designed two types of point scalar multipliers for BEdC, Type-I uses one 257-b×257-b binary multiplier and Type-II uses eight 32-b×32-b binary multipliers. Type-II design uses 65% less LUTs compared to Type-I, but it was evaluated that it took about 3.5 times the PSM computation time when operating with 240 MHz. Therefore, the BEdC crypto core of Type-I is suitable for applications requiring high-performance, and Type-II structure is suitable for applications with limited resources.