2001 : : IMS: ; NDC:418; keywords:, ; ( ) 2. ( ) 3. ( ) ( ) 4. ( ) 5.
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1 2001 : : IMS: ; NDC:418; keywords:, ; ( ) 2. ( ) 3. ( ) ( ) 4. ( ) 5. Karatsuba ( ) 6. ( ) 7. Karatsuba ( ) 8. Toom-Cook ( ) 9. ( ) 10. (1) ( ) 11. (1) ( ) ( ) 12. (1) ( ) ( ) 13. (1) ( ) 14. (2) ( ) 15. (2) ( ) ( ) 16. (3) ( ) 17. (4) ( ) 18. (5) ( ) ( ) bit - ( SS ) C
2 x bit SS SS SS 5 98 SS SS K = 2 k (n, m, ζ) (K, 2 2K + 1, 2 4 ) (n, m, ζ) SS n FFT m ( - ) - (n,m, ) i. n FFT ii. m 2 ( ) XeX C++ 11 Internet NTL[N1] C++ GPL(GNU General Public License) NTL 1 m +1 m -1 UBASIC NTL S0 m=10,000,000 2 (m=10,000,000) 2
3 3
4 UBASIC NTL S UBASIC NTL S ( ) ( ) NTL S0 NTL C++ S0 Pentium Pentium NTL NTL S0 3 (m=10,000,000) UBASIC NTL S UBASIC NTL S0 20 S0 NTL NTL 4 ( ) NTL SNBench S
5 0 SNBench NTL NTL SNBench 4 SNBench Ver.0.01(3, ) No. S0 NTL p= S0 n O(n 2 ) Karatsuba Ver.0.01 GPL 5 Karatsuba ( ) Karatsuba [S3] SNBench 5 SNBench Ver.0.01(3, ) 5
6 6
7 No. S0 NTL NTL Karatsuba S0 Knuth [S2] Toom-Cook NTL 6 ( ) 6 SNBench Ver.0.01(3, ) No. S0 NTL
8 % 1 2 Karatsuba 1 2 Karatsuba 7 Karatsuba ( ) Karatsuba 7 SNBench Ver.0.01(3, ) No. S0 NTL Toom-Cook 8
9 8 Toom-Cook ( ) Toom Toom-Cook S0/math/uint32 Cook 9 ( ) Pentium Karatsuba NTL Toom-Cook 10 (1) ( ) p i. F p CPU ii. F p p = ( [S5] (1) ) (1) ( ) ( ) F p 9
10 p 2 q 1 AND p F p n n 1 n p = 2 q 1 F p F p [i], i 2 = 1( ) n I. S. Reed T. K. Truong 1975 IEEE Trans-IT ( - ) 1990 J. Buhler, M. Shokrollahi, V. Stemann (1) F p [i] 11 (1) ( ) ( ) FFT C++ Karatsuba Karatsuba Pentium 8 ] CL KO KO/CL M1 M1/CL 2 4 = = = = [msec] 2 8 = = = = = (CL ) Karatsuba (KO ) (1)(M1 CL 2 12 M1 M1 10
11 CPU:Pentium-133MHz :48MB OS:Windows98 SE M MB 1GB 12 (1) ( ) ( ) ] CL KO KO/CL M1 M1/CL 2 4 = = = = [msec] 2 8 = = = = = (1) ( ) Pentium 14 (2) ( ) (2) (1) (1) M 31 = Z M31 Z M31 [i], i 2 = 1 11
12 ( [S6] (2) ) 15 (2) ( ) ( ) Z M31 Z M31 [i], i 2 = 1 Z[i] (M 31 ) 1 N 12 + i (1) 10 ] CL KO KO/CL M2 M2/CL 2 4 = = = = [msec] 2 8 = = = = = (3) ( ) (3) (2) M 89 = Z M89 [i], i 2 = 1 ( [S7] (3) ) 11 ] CL KO KO/CL M3 M3/CL [msec] 2 4 =
13 2 5 = = = = = = = = (2) 4 2 (3) 6 17 (4) ( ) (4) ] CL KO KO/CL M4 M4/CL 2 4 = = = = [msec] 2 8 = = = = = = (5) ( ) (5) ] CL KO KO/CL M5 M5/CL 2 4 = [msec] 2 5 = =
14 2 7 = = = = = = = [N1] Victor Shoup, NTL: A Library for doing Number Theory [S1],,,,, 1983 [S2] D. E. Knuth( ), /,, 1986 [S3], Karatsuba [S4], Toom-Cook [S5], (1) [S6], (2) [S7], (3) 14
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