| Project/Area Number |
07304018
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 総合 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Tokyo |
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Principal Investigator |
NAMBA Knaji Univ.of Tokyo, Dept.Math.Sci.Prof., 大学院・数理科学研究科, 教授 (40015524)
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| Co-Investigator(Kenkyū-buntansha) |
KAKEHI Katsuhiko Waseda Univ.Facl.Sci.Tech., Prof., 理工学部, 教授 (20062672)
USHIJIMA Kazuo Kyushu Univ.Facl.Tech., Prof., 工学部, 教授 (40037750)
ENOMOTO Hikoe Keio Univ.Facl.Sci.Tech., Prof., 理工学部, 教授 (00011669)
DOI Norihisa Keio Univ.Facl.Sci.Tech., Prof., 理工学部, 教授 (50051553)
KOBAYASHI Kojiro Tokyo Inst.of Tech, Facl.Inf.Tech., Prof., 情報理工学科, 教授 (00016148)
川合 慧 東京大学, 教養学部, 教授 (50011664)
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| Project Period (FY) |
1995 – 1996
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| Project Status |
Completed (Fiscal Year 1996)
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| Budget Amount *help |
¥7,200,000 (Direct Cost: ¥7,200,000)
Fiscal Year 1996: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 1995: ¥4,000,000 (Direct Cost: ¥4,000,000)
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| Keywords | computational complexity / finite field / elliptic curve / Fuchsean function / factorization / polynomial time / cyclic othogonal matrix / 情報構造 / 複雑性問題 / 多項式時間アルゴリズム / 計算量の下限 / 整数の素因数分解 / 有限体の上の楕円曲線 / 可換群の位数 / フックスの関数 / 複雑性の問題 / 有限体上の楕円曲線 / 高速フーリェ変換 |
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| Research Abstract |
Information structures and the notion of computational complexity are one of representative problems of modern informatics and mathematics. Fast discrete Fourier Transform(FFT), efficient matrix multiplication, determination of primality and factorization of integers, such problems relates to the ranges of Greek math. to most modern P=NP problems. Results obtained by this research project are :
1)Results concerning to elliptic curves over finite fields :
Consider Weierstrass family of elliptic curves y^2=x^3+ax+b and their invariant z=-27b^2/4a^3. Then the Fuchsean function a_<12>(z)=z^<[(p+1)/4]>F(1/12,5/12,1,1-z) computed as the least absolute value residue has both analytic and algebraic properties. Algebraic structures of Monodromy group of differntial equations are obtained.
2)Consider the sequence, for example a<@D212@>D2(b<@D1k@>D1), where b is a primitive root mod.p, k=0, ・・・, p-1, then the minimal polinomial of cyclic matrix A=(a<@D212@>D2(b<@D1i-j@>D1))is(x<@D12@>D1-p<@D12@>D1)(x<@D12@>D1-p)and the dimension of eigen space of (]SY.+-。[) p is 2 or 4.and the cyclotomic projection generates large class of cyclic orthogonal matrices. To measure location of points or time shifting, it might be used.
3)There is a probabilistic polynomial time algorithm to find a primes of the form p=3n^2+3n+1 factor of composite number.
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