IEICE TRANSACTIONS on Fundamentals
New Explicit Conditions of Elliptic Curve Traces for FR-Reduction
Summary :
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.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E84-A No.5 pp.1234-1243
- Publication Date
- 2001/05/01
- Publicized
- Online ISSN
- DOI
- Type of Manuscript
- Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
- Category
Authors
Keyword
Latest Issue
Copyrights notice of machine-translated contents
The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.
Cite this
Copy
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
Copy
@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},}
Copy
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 -
English
