| Project/Area Number |
09650424
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
情報通信工学
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| Research Institution | CHUO UNIVERSITY |
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Principal Investigator |
CHAO Jinhu Factulty of Science and Engineering Chuo University Professor, 理工学部, 教授 (60227345)
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| Co-Investigator(Kenkyū-buntansha) |
MOMOSE Fumiyuki Factulty of Science and Engineering Chuo University Professor, 理工学部, 教授 (80182187)
TSUJII Shigeo Factulty of Science and Engineering Chuo University Professor, 理工学部, 教授 (50020350)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,900,000 (Direct Cost: ¥1,900,000)
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| Keywords | Information security / Cryptosystems / Public key cryptosystems / Elliptic cryptosystems / Discrete logarithm problem / Finite fields / 公聞鍵暗号系 / 楕円曲線 |
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| Research Abstract |
Elliptic curves over finite fields have been used in recent public key clyptosysterns and authentication. The discrete logarithm problems over the elliptic curves can resist all known subexponential attacks which then can implement cryptographic schemes in higher speed and less key sizes while retain the same security comparing with traditional cryptographic functions. In this research, we propose efficient algorithms to construct secure elliptic arid hyperelliptic cryptosysterns.
The point-counting algorithms to construct explicitly secure elliptic curves for cryptosystems can find secure curves over finite fields from randomly selected elliptic curves, but are quite time consuming especially when one wishes to choose different curves for different users or periodically change curves over finite fields in the same cryptosystem, Elliptic curves over number fields with CM can be used to design non-isogenous elliptic cryptosystems over finite fields efficiently. The existing algorithm to build such CM curves, costing exponential time of computations OMICRON(2^<5h/2>h^<21/4>) where h is the class number of the endomorphism ring of the CM curve. Thus it carl only be used to construct CM elliptic curves with small class numbers.
We propose polynomial time algorithms in h to build CM elliptic curves over number fields : by lifting the ring class equations from small finite fields thus constructing CM curves. Its complexity is shown as in a polynomial time in h, i.e., . OMICRON(h^7). Furthermore, these algorithms are also extented to hyperelliptic cryptosystems, for which no efficient algorithm is known until now for construction of secure hyperelliptic curves. We propose efficient algorithms to construct secure discrete logarithm problems over hyperelliptic curves based on Weil elements. The lifting approach to build CM curves is also generalized to Jacobian varieties of algebraic curves of higher genera.
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