• KSII Transactions on Internet and Information Systems (TIIS)
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• 제11권7호
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• pp.3648-3663
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• 2017

With the development of cloud computing and widespread availability of mobile devices, outsourcing computation has gotten more and more attention in cloud computing services. The computation of bilinear pairing is the most expensive operation in pair-based cryptographic schemes. Currently, most of the algorithms for outsourcing bilinear pairing have small checkability or the outsourcers need to operate expensive computations. In this paper, we propose an efficient algorithm for outsourcing bilinear pairing with two servers, where the outsourcers can detect the errors with a probability of 1 if the cloud servers are dishonest, and the outsourcers are not involved in any complex computations. Finally, the performance evaluation demonstrates that the proposed algorithm is most efficient in all of fully verifiable outsourcing algorithms for bilinear pairing.

• ETRI Journal
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• 제35권5호
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• pp.880-888
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• 2013

The preexisting pairings ate, $ate_i$, R-ate, and optimal-ate use q-expansion, where q is the size of the defining field for the elliptic curves. Elliptic curves with small embedding degrees only allow a few of these pairings. In such cases, efficiently computable endomorphisms can be used, as in [11] and [12]. They used the endomorphisms that have characteristic polynomials with very small coefficients, which led to some restrictions in finding various pairing-friendly curves. To construct more pairing-friendly curves, we consider ${\mu}$-expansion using the Gallant-Lambert-Vanstone (GLV) decomposition method, where ${\mu}$ is an arbitrary integer. We illustrate some pairing-friendly curves that provide more efficient pairing from the ${\mu}$-expansion than from the ate pairing. The proposed method can achieve timing results at least 20% faster than the ate pairing.

• 한국정보통신학회논문지
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• 제11권2호
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• pp.301-308
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• 2007

본 논문에서는 Bilinear Pairing을 사용한 효율적인 신원기반 다중 수신자 암호 기법을 제 안 한다. 제안 기법은 암호화 과정에서 Pairing 연산을 필요로 하지 않으며 복호화 과정에서 단 한번의 Pairing 연산만을 요구한다. 뿐만 아니라, 제안기법을 이용하여 Subset-Cover framework 기반의 효율적인 스테이트리스 (stateless) 공개키 브로드캐스트 암호 기법을 제시한다.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제7권6호
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• pp.1480-1491
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• 2013

Certificate-based cryptography is a new cryptographic primitive which eliminates the necessity of certificates in the traditional public key cryptography and simultaneously overcomes the inherent key escrow problem suffered in identity-based cryptography. However, to the best of our knowledge, all existed constructions of certificate-based encryption so far have to be based on the bilinear pairings. The pairing calculation is perceived to be expensive compared with normal operations such as modular exponentiations in finite fields. The costly pairing computation prevents it from wide application, especially for the computation limited wireless sensor networks. In order to improve efficiency, we propose a new certificate-based encryption scheme that does not depend on the pairing computation. Based on the decision Diffie-Hellman problem assumption, the scheme's security is proved to be against the chosen ciphertext attack in the random oracle. Performance comparisons show that our scheme outperforms the existing schemes.

• ETRI Journal
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• 제33권5호
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• pp.780-790
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• 2011

This paper presents an efficient differential power analysis (DPA) countermeasure for the $Eta_T$ pairing algorithm over GF($2^n$). The proposed algorithm is based on a random value addition (RVA) mechanism. An RVA-based DPA countermeasure for the $Eta_T$ pairing computation over GF($3^n$) was proposed in 2008. This paper examines the security of this RVA-based DPA countermeasure and defines the design principles for making the countermeasure more secure. Finally, the paper proposes an efficient RVA-based DPA countermeasure for the secure computation of the $Eta_T$ pairing over GF($2^n$). The proposed countermeasure not only overcomes the security flaws in the previous RVAbased method but also exhibits the enhanced performance. Actually, on the 8-bit ATmega128L and 16-bit MSP430 processors, the proposed method can achieve almost 39% and 43% of performance improvements, respectively, compared with the best-known countermeasure.

• ETRI Journal
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• 제31권2호
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• pp.129-139
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• 2009

This paper provides an efficient algorithm for computing the ${\eta}_T$ pairing on supersingular elliptic curves over fields of characteristic two. In the proposed algorithm, we deploy a modified multiplication in $F_{2^{4n}}$ using the Vandermonde matrix. For F, G ${\in}$ $F_{2^{4n}}$ the proposed multiplication method computes ${\beta}{\cdot}F{\cdot}G$ instead of $F{\cdot}G$ with some ${\beta}$ ${\in}$ $F^*_{2n}$ because ${\beta}$ is eliminated by the final exponentiation of the ${\eta}_T$ pairing computation. The proposed multiplication method asymptotically requires only 7 multiplications in $F_{2^n}$ as n ${\rightarrow}$ ${\infty}$, while the cost of the previously fastest Karatsuba method is 9 multiplications in $F_{2^n}$. Consequently, the cost of the ${\eta}_T$ pairing computation is reduced by 14.3%.

• 대한수학회지
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• 제45권4호
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• pp.1057-1073
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• 2008

We present an explicit Eta pairing approach for computing the Tate pairing on general divisors of hyperelliptic curves $H_d$ of genus 2, where $H_d\;:\;y^2+y=x^5+x^3+d$ is defined over ${\mathbb{F}}_{2^n}$ with d=0 or 1. We use the resultant for computing the Eta pairing on general divisors. Our method is very general in the sense that it can be used for general divisors, not only for degenerate divisors. In the pairing-based cryptography, the efficient pairing implementation on general divisors is significantly important because the decryption process definitely requires computing a pairing of general divisors.

• 한국통신학회논문지
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• 제36권1C호
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• pp.14-21
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• 2011

겹선형 페어링(bilinear pairing)을 기반으로 하는 암호 프로토콜들은 이산 대수 문제를 기반으로 하는 전통적인 타원 곡선 암호시스템을 대신하여 여러 방면에의 응용성을 제공한다. 겹선형 페어링의 빠른 계산을 위하여 최근 활발한 연구가 진행 중이지만, 여전히 ECC 상수배 연산에 비해서 페어링 연산에 사용되는 계산 비용은 상당히 크다고 여겨진다. 그러나 이진 유한체상의 페어링 계산 연구는 최근 많은 발전이 이루어졌다. 본 논문에서는 이진 유한체상에서의 BLS 서명스킴과 ECDSA 서명 스킴의 복잡도를 비교한다. 공정한 비교를 위하여 1024-bit RSA와 같은 레벨의 보안성을 가지는 160-bit ECDSA와 250-bit BLS를 선택하였다. 분석결과 BLS 스킴은 ECDSA에 비해 하드웨어 복잡도 및 계산 지연시간의 측면에서 많은 차이가 나지 않음을 설명해준다.

• Journal of Communications and Networks
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• 제10권4호
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• pp.437-443
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• 2008

The randomized projective coordinate (RPC) method applied to a pairing computation algorithm is a good solution that provides an efficient countermeasure against side channel attacks. In this study, we investigate measures for increasing the efficiency of the RPC-based countermeasures and construct a method that provides an efficient RPC-based countermeasure against side channel attacks. We then apply our method to the well-known $\eta_T$ pairing algorithm over binary fields and obtain an RPC-based countermeasure for the $\eta_T$ pairing; our method is more efficient than the RPC method applied to the original $\eta_T$ pairing algorithm.

• 정보보호학회논문지
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• 제21권3호
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• pp.11-25
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• 2011

ID 기반 암호시스템의 구현을 위한 Weil, Tate, Ate와 같은 페어링 연산 기법에서는 밀러 알고리듬이 사용된다. 본 연구에서는 밀려 알고리듬에 대한 오류 공격의 하나인 Mrabet의 방법을 분석하여 타원곡선을 표현하는 가장 기본적인 좌표계인 아핀좌표계에서의 효과적인 오류주입공격 방법을 제안하였다. 제안하는 오류주입공격은 멀리 알고리듬의 루프 횟수를 판별하는 분기 구문에 오류를 주입하는 모델이며, 실제 레이저 주입 실험을 수행하여 검증하였다. 이 모델은 기존의 루프 횟수 오류 기법에서 요구하였던 확률적인 분석을 생략할 수 있어 효과적이다.