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Í­¸ÂÂξå¤ÎÂʱ߶ÊÀþ¤Ë´Ø¤¹¤ë¤¤¤¯¤Ä¤«¤Î´ðËÜŪ¤Ê±é»»¤¬, ÁȤ߹þ¤ß´Ø¿ô¤È¤·¤Æ Ä󶡤µ¤ì¤Æ¤¤¤ë.

Âʱ߶ÊÀþ¤Î»ØÄê¤Ï, Ťµ 2 ¤Î¥Ù¥¯¥È¥ë [a b] ¤Ç¹Ô¤¦. a, b ¤ÏÍ­¸ÂÂΤ赤Ç, setmod_ff ¤ÇÄêµÁ¤µ¤ì¤Æ¤¤¤ëÍ­¸ÂÂΤ¬ÁÇÂΤξì¹ç, y^2=x^3+ax+b, ɸ¿ô 2 ¤ÎÂΤξì¹ç y^2+xy=x^3+ax^2+b ¤òɽ¤¹.

Âʱ߶ÊÀþ¾å¤ÎÅÀ¤Ï, ̵¸Â±óÅÀ¤â¹þ¤á¤Æ²ÃË¡·²¤ò¤Ê¤¹. ¤³¤Î±é»»¤Ë´Ø¤·¤Æ, ²Ã»» (ecm_add_ff()), ¸º»» (ecm_sub_ff()) ¤ª¤è¤ÓµÕ¸µ·×»»¤Î¤¿¤á¤Î ´Ø¿ô (ecm_chsgn_ff()) ¤¬Ä󶡤µ¤ì¤Æ¤¤¤ë. Ãí°Õ¤¹¤Ù¤­¤Ï, ±é»»¤ÎÂÐ¾Ý ¤È¤Ê¤ëÅÀ¤Îɽ¸½¤¬,

  • ̵¸Â±óÅÀ¤Ï 0.
  • ¤½¤ì°Ê³°¤ÎÅÀ¤Ï, Ťµ 3 ¤Î¥Ù¥¯¥È¥ë [x y z]. ¤¿¤À¤·, z ¤Ï 0 ¤Ç¤Ê¤¤.

¤È¤¤¤¦ÅÀ¤Ç¤¢¤ë. [x y z] ¤ÏÀƼ¡ºÂɸ¤Ë¤è¤ëɽ¸½¤Ç¤¢¤ê, ¥¢¥Õ¥£¥óºÂɸ ¤Ç¤Ï [x/z y/z] ¤Ê¤ëÅÀ¤òɽ¤¹. ¤è¤Ã¤Æ, ¥¢¥Õ¥£¥óºÂɸ [x y] ¤Ç ɽ¸½¤µ¤ì¤¿ÅÀ¤ò±é»»ÂоݤȤ¹¤ë¤Ë¤Ï, [x y 1] ¤Ê¤ë¥Ù¥¯¥È¥ë¤ò À¸À®¤¹¤ëɬÍפ¬¤¢¤ë. ±é»»·ë²Ì¤âÀƼ¡ºÂɸ¤ÇÆÀ¤é¤ì¤ë¤¬, z ºÂɸ¤¬ 1 ¤È¤Ï¸Â¤é¤Ê¤¤¤¿¤á, ¥¢¥Õ¥£¥óºÂɸ¤òµá¤á¤ë¤¿¤á¤Ë¤Ï x, y ºÂɸ¤ò z ºÂɸ¤Ç ³ä¤ëɬÍפ¬¤¢¤ë.


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