• ETRI Journal
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• v.35 no.5
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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.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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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.

• ETRI Journal
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• v.30 no.1
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• pp.68-80
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• 2008

Since many efficient algorithms for implementing pairings have been proposed such as ${\eta}_T$ pairing and the Ate pairing, pairings could be used in constraint devices such as smart cards. However, the secure implementation of pairings has not been thoroughly investigated. In this paper, we investigate the security of ${\eta}_T$ pairing over binary fields in the context of side-channel attacks. We propose efficient and secure ${\eta}_T$ pairing algorithms using randomized projective coordinate systems for computing the pairing.

• Journal of the Institute of Electronics and Information Engineers
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• v.50 no.2
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• pp.70-79
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• 2013

Recently, there has been introduced various types of pairing computations to implement ID based cryptosystem for mobile ad hoc network. The Miller algorithm is the most popular algorithm for the typical pairing computation such as Weil, Tate and Ate. In this paper, we analyze the feasibility of concrete data fault injection attack, which was proposed by Whelan and Scott, in terms of regardless of round positions during the execution of the Miller algorithm. As the simulation results, the proposed attack that can be employed to regardless of round positions and coordinate systems is effective and powerful.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.21 no.3
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• pp.11-25
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• 2011

The Miller algorithm is employed in the typical pairing computation such as Weil, Tate and Ate for implementing ID based cryptosystem. By analyzing the Mrabet's attack that is one of fault attacks against the Miller algorithm, this paper presents au efficient fault attack in Affine coordinate system, it is the most basic coordinates for construction of elliptic curve. The proposed attack is the effective model of a count check fault attack, it is verified to work well by practical fault injection experiments and can omit the probabilistic analysis that is required in the previous counter fault model.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.21 no.2
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• pp.83-90
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• 2011

Recently, pairings over elliptic curve have been applied for various ID-based encryption/signature/authentication/key agreement schemes. For efficiency, the $Eta_T$ pairings over GF($P^n$) (P = 2, 3) were invented, however, they are vulnerable to side channel attacks such as DPA because of their symmetric computation structure compared to other pairings such as Tate, Ate pairings. Several countermeasures have been proposed to prevent side channel attacks. Especially, Masaaki Shirase's method is very efficient with regard to computational efficiency, however, it has security flaws. This paper examines closely the security flaws of RVA-based countermeasure on $Eta_T$ Pairing algorithm from the implementation point of view.

• Journal of the Institute of Electronics and Information Engineers
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• v.50 no.7
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• pp.68-75
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• 2013

Recently, there has been introduced various types of pairing computations to implement ID based cryptosystem for mobile ad hoc network. According to spreading the applications of pairing computations, various fault attacks have been proposed. Among them, a counter fault attack has been considered the strongest threat. Thus this paper proposes a new countermeasure to prevent the counter fault attack on Miller's algorithm. The proposed method is able to reduce the possibility of fault propagation by a random index of intermediate values. Additionally, it is difficult to challenge fault attacks on the proposed method since a simple side channel leakage of 'if' branch is eliminated.