The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. ex. Some numerals are expressed as "XNUMX".
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The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. Copyrights notice
Os criptossistemas de curva elíptica são baseados no problema do logaritmo discreto da curva elíptica (ECDLP). Se os criptossistemas de curva elíptica evitarem a redução de FR e a curva elíptica anômala ao longo Fq, então, com o conhecimento atual, podemos construir criptossistemas de curvas elípticas em um campo de definição menor. O ECDLP possui uma propriedade interessante de que a segurança depende profundamente de traços de curvas elípticas ao invés de campos de definição, o que não ocorre no caso do problema do logaritmo discreto (DLP). Portanto, é importante caracterizar explicitamente os traços da curva elíptica do ponto de vista da segurança. Quanto à redução de FR, curvas elípticas supersingulares ou curva elíptica E/Fq com rastreamento 2 foram relatados como vulneráveis. No entanto, infelizmente, estes foram apenas resultados que caracterizam explicitamente os traços da curva elíptica para reduções de FR e MOV. Mais importante ainda, o traço seguro contra a redução de FR não foi relatado. Curvas elípticas com traço seguro significam que o grau de extensão reduzido é sempre superior a um determinado nível. Neste artigo, pretendemos caracterizar traços de curvas elípticas por redução de FR e investigar condições explícitas de traços vulneráveis ou seguros contra redução de FR. Mostramos novas condições explícitas de traços de curvas elípticas para redução de FR. Apresentamos também algoritmos para construir tais curvas elípticas, que têm relação com problemas famosos da teoria dos números.
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Atsuko MIYAJI, Masaki NAKABAYASHI, Shunzou TAKANO, "New Explicit Conditions of Elliptic Curve Traces for FR-Reduction" in IEICE TRANSACTIONS on Fundamentals,
vol. E84-A, no. 5, pp. 1234-1243, May 2001, doi: .
Abstract: Elliptic curve cryptosystems are based on the elliptic curve discrete logarithm problem (ECDLP). If elliptic curve cryptosystems avoid FR-reduction and anomalous elliptic curve over Fq, then with current knowledge we can construct elliptic curve cryptosystems over a smaller definition field. ECDLP has an interesting property that the security deeply depends on elliptic curve traces rather than definition fields, which does not occur in the case of the discrete logarithm problem (DLP). Therefore it is important to characterize elliptic curve traces explicitly from the security point of view. As for FR-reduction, supersingular elliptic curves or elliptic curve E/Fq with trace 2 have been reported to be vulnerable. However unfortunately these have been only results that characterize elliptic curve traces explicitly for FR- and MOV-reductions. More importantly, the secure trace against FR-reduction has not been reported at all. Elliptic curves with the secure trace means that the reduced extension degree is always higher than a certain level. In this paper, we aim at characterizing elliptic curve traces by FR-reduction and investigate explicit conditions of traces vulnerable or secure against FR-reduction. We show new explicit conditions of elliptic curve traces for FR-reduction. We also present algorithms to construct such elliptic curves, which have relation to famous number theory problems.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e84-a_5_1234/_p
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@ARTICLE{e84-a_5_1234,
author={Atsuko MIYAJI, Masaki NAKABAYASHI, Shunzou TAKANO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={New Explicit Conditions of Elliptic Curve Traces for FR-Reduction},
year={2001},
volume={E84-A},
number={5},
pages={1234-1243},
abstract={Elliptic curve cryptosystems are based on the elliptic curve discrete logarithm problem (ECDLP). If elliptic curve cryptosystems avoid FR-reduction and anomalous elliptic curve over Fq, then with current knowledge we can construct elliptic curve cryptosystems over a smaller definition field. ECDLP has an interesting property that the security deeply depends on elliptic curve traces rather than definition fields, which does not occur in the case of the discrete logarithm problem (DLP). Therefore it is important to characterize elliptic curve traces explicitly from the security point of view. As for FR-reduction, supersingular elliptic curves or elliptic curve E/Fq with trace 2 have been reported to be vulnerable. However unfortunately these have been only results that characterize elliptic curve traces explicitly for FR- and MOV-reductions. More importantly, the secure trace against FR-reduction has not been reported at all. Elliptic curves with the secure trace means that the reduced extension degree is always higher than a certain level. In this paper, we aim at characterizing elliptic curve traces by FR-reduction and investigate explicit conditions of traces vulnerable or secure against FR-reduction. We show new explicit conditions of elliptic curve traces for FR-reduction. We also present algorithms to construct such elliptic curves, which have relation to famous number theory problems.},
keywords={},
doi={},
ISSN={},
month={May},}
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TY - JOUR
TI - New Explicit Conditions of Elliptic Curve Traces for FR-Reduction
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1234
EP - 1243
AU - Atsuko MIYAJI
AU - Masaki NAKABAYASHI
AU - Shunzou TAKANO
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E84-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 2001
AB - Elliptic curve cryptosystems are based on the elliptic curve discrete logarithm problem (ECDLP). If elliptic curve cryptosystems avoid FR-reduction and anomalous elliptic curve over Fq, then with current knowledge we can construct elliptic curve cryptosystems over a smaller definition field. ECDLP has an interesting property that the security deeply depends on elliptic curve traces rather than definition fields, which does not occur in the case of the discrete logarithm problem (DLP). Therefore it is important to characterize elliptic curve traces explicitly from the security point of view. As for FR-reduction, supersingular elliptic curves or elliptic curve E/Fq with trace 2 have been reported to be vulnerable. However unfortunately these have been only results that characterize elliptic curve traces explicitly for FR- and MOV-reductions. More importantly, the secure trace against FR-reduction has not been reported at all. Elliptic curves with the secure trace means that the reduced extension degree is always higher than a certain level. In this paper, we aim at characterizing elliptic curve traces by FR-reduction and investigate explicit conditions of traces vulnerable or secure against FR-reduction. We show new explicit conditions of elliptic curve traces for FR-reduction. We also present algorithms to construct such elliptic curves, which have relation to famous number theory problems.
ER -