锘??xml version="1.0" encoding="utf-8" standalone="yes"?>狠狠色综合日日,一区二区三区欧美激情,欧美中日韩免费视频http://www.shnenglu.com/qinqing1984/zh-cnTue, 02 Dec 2025 23:31:43 GMTTue, 02 Dec 2025 23:31:43 GMT60鍏充簬LLL綆楁硶鐨勮ˉ鍏呰瘉鏄?/title><link>http://www.shnenglu.com/qinqing1984/archive/2025/09/28/230467.html</link><dc:creator>鏄ョ鍗佷簩鏈?/dc:creator><author>鏄ョ鍗佷簩鏈?/author><pubDate>Sun, 28 Sep 2025 09:43:00 GMT</pubDate><guid>http://www.shnenglu.com/qinqing1984/archive/2025/09/28/230467.html</guid><wfw:comment>http://www.shnenglu.com/qinqing1984/comments/230467.html</wfw:comment><comments>http://www.shnenglu.com/qinqing1984/archive/2025/09/28/230467.html#Feedback</comments><slash:comments>0</slash:comments><wfw:commentRss>http://www.shnenglu.com/qinqing1984/comments/commentRss/230467.html</wfw:commentRss><trackback:ping>http://www.shnenglu.com/qinqing1984/services/trackbacks/230467.html</trackback:ping><description><![CDATA[<div>鍏堟憳褰曟枃鐚甗1]涓殑LLL綆楁硶鎻忚堪嫻佺▼錛屽強LLL綰﹀寲鍩虹殑瀹氫箟 <br />  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/LLL-algorithm1.png" width="497" height="133" alt="" /><br />  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/LLL-algorithm2.png" width="492" height="571" alt="" /><br /> <br /> <div>LLL綰﹀寲鍩虹殑瀹氫箟濡備笅錛堟枃鐚甗1]瀹氫箟13.12錛?br />  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/LLL-Reduced-basis.png" width="438" height="90" alt="" /><br /> <br /> <div>鍐嶈瘉鏄庝笂鍥劇孩鑹叉柟妗嗕笁琛屼吉浠ｇ爜鐨勬紜э紙鍏跺畠閮ㄥ垎鏂囩尞[1]宸茶寰楁瘮杈冨叿浣擄級<br />  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/LLL-algorithm-additional-proof.jpg" alt="" /><br /> <br /> <div><strong style="font-size: 12pt;"><br /> 鍙傝冩枃鐚?/strong> </div> <div>  [1] 綆楁硶鏁拌        瑁村畾涓 紲濊穬椋?/div> <div>  [2] 楂樼瓑浠ｆ暟        涓樼淮澹?nbsp;</div> </div> </div> </div><img src ="http://www.shnenglu.com/qinqing1984/aggbug/230467.html" width = "1" height = "1" /><br><br><div align=right><a style="text-decoration:none;" href="http://www.shnenglu.com/qinqing1984/" target="_blank">鏄ョ鍗佷簩鏈?/a> 2025-09-28 17:43 <a href="http://www.shnenglu.com/qinqing1984/archive/2025/09/28/230467.html#Feedback" target="_blank" style="text-decoration:none;">鍙戣〃璇勮</a></div>]]></description></item><item><title>鍏充簬鍒嗗渾鍩熺殑涓鑸粨璁?/title><link>http://www.shnenglu.com/qinqing1984/archive/2025/07/28/230452.html</link><dc:creator>鏄ョ鍗佷簩鏈?/dc:creator><author>鏄ョ鍗佷簩鏈?/author><pubDate>Mon, 28 Jul 2025 04:01:00 GMT</pubDate><guid>http://www.shnenglu.com/qinqing1984/archive/2025/07/28/230452.html</guid><wfw:comment>http://www.shnenglu.com/qinqing1984/comments/230452.html</wfw:comment><comments>http://www.shnenglu.com/qinqing1984/archive/2025/07/28/230452.html#Feedback</comments><slash:comments>0</slash:comments><wfw:commentRss>http://www.shnenglu.com/qinqing1984/comments/commentRss/230452.html</wfw:commentRss><trackback:ping>http://www.shnenglu.com/qinqing1984/services/trackbacks/230452.html</trackback:ping><description><![CDATA[<img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/cyclotomic-field-general-conclusion1.jpg" width="622" height="955" alt="" /><br /> <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/cyclotomic-field-general-conclusion2.jpg" width="652" height="835" alt="" /><br /> <br /> <div><strong style="font-size: 11pt;"><br />鍙傝冩枃鐚?/strong></div> <div>  [1]浠ｆ暟涓庢暟璁?nbsp;      鏉庤秴     鍛ㄦ偊</div> <div>  [2]鎶借薄浠ｆ暟II       寰愭槑鏇?nbsp; 璧墊槬鏉?/div><img src ="http://www.shnenglu.com/qinqing1984/aggbug/230452.html" width = "1" height = "1" /><br><br><div align=right><a style="text-decoration:none;" href="http://www.shnenglu.com/qinqing1984/" target="_blank">鏄ョ鍗佷簩鏈?/a> 2025-07-28 12:01 <a href="http://www.shnenglu.com/qinqing1984/archive/2025/07/28/230452.html#Feedback" target="_blank" style="text-decoration:none;">鍙戣〃璇勮</a></div>]]></description></item><item><title>涓涓鎷夋暟鏁撮櫎闂鐨勪袱縐嶈瘉娉?/title><link>http://www.shnenglu.com/qinqing1984/archive/2025/06/20/230447.html</link><dc:creator>鏄ョ鍗佷簩鏈?/dc:creator><author>鏄ョ鍗佷簩鏈?/author><pubDate>Fri, 20 Jun 2025 10:41:00 GMT</pubDate><guid>http://www.shnenglu.com/qinqing1984/archive/2025/06/20/230447.html</guid><wfw:comment>http://www.shnenglu.com/qinqing1984/comments/230447.html</wfw:comment><comments>http://www.shnenglu.com/qinqing1984/archive/2025/06/20/230447.html#Feedback</comments><slash:comments>0</slash:comments><wfw:commentRss>http://www.shnenglu.com/qinqing1984/comments/commentRss/230447.html</wfw:commentRss><trackback:ping>http://www.shnenglu.com/qinqing1984/services/trackbacks/230447.html</trackback:ping><description><![CDATA[<img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/two-way-proofs-of-one-euler-number-proposition1.jpg" alt="" /><br /> <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/two-way-proofs-of-one-euler-number-proposition2.jpg" alt="" /><br /> <br /> <div><strong style="font-size: 11pt;"><br /> 鍙傝冩枃鐚?/strong></div> <div>  [1] 浠ｆ暟瀛﹀熀紜涓庢湁闄愬煙       鏋椾笢宀?/div> <div>  [2] 鎶借薄浠ｆ暟                     璧墊槬鏉?寰愭槑鏇?/div><img src ="http://www.shnenglu.com/qinqing1984/aggbug/230447.html" width = "1" height = "1" /><br><br><div align=right><a style="text-decoration:none;" href="http://www.shnenglu.com/qinqing1984/" target="_blank">鏄ョ鍗佷簩鏈?/a> 2025-06-20 18:41 <a href="http://www.shnenglu.com/qinqing1984/archive/2025/06/20/230447.html#Feedback" target="_blank" style="text-decoration:none;">鍙戣〃璇勮</a></div>]]></description></item><item><title>鏈夐檺鍩熶笂鐨勭壒寰佷笌鎸囨暟鍜屼箣鎵╁睍http://www.shnenglu.com/qinqing1984/archive/2025/06/05/230441.html鏄ョ鍗佷簩鏈?/dc:creator>鏄ョ鍗佷簩鏈?/author>Thu, 05 Jun 2025 01:30:00 GMThttp://www.shnenglu.com/qinqing1984/archive/2025/06/05/230441.htmlhttp://www.shnenglu.com/qinqing1984/comments/230441.htmlhttp://www.shnenglu.com/qinqing1984/archive/2025/06/05/230441.html#Feedback0http://www.shnenglu.com/qinqing1984/comments/commentRss/230441.htmlhttp://www.shnenglu.com/qinqing1984/services/trackbacks/230441.html絎﹀彿鍚箟 
 

鍏充簬鐗瑰緛鐨勭粨璁?/strong> 
  
  
  
鍏充簬鎸囨暟鍜岀殑緇撹 
  
鍙傝冩枃鐚?/strong>
   [1] 浠ｆ暟瀛﹀熀紜涓庢湁闄愬煙         鏋椾笢宀?/div>
   [2] 浠ｆ暟涓庢暟璁?nbsp;                   鏉庤秴 鍛ㄦ偊
   [3] 鍏充簬緹ょ殑涓浜涚粨璁哄強搴旂敤   鏈漢


]]>浜屽厓浜屾鍨嬬殑鐩鎬技鍙樻崲銆佹瀹氭т笌姝ｄ氦鍒嗚Вhttp://www.shnenglu.com/qinqing1984/archive/2025/04/25/230436.html鏄ョ鍗佷簩鏈?/dc:creator>鏄ョ鍗佷簩鏈?/author>Fri, 25 Apr 2025 11:05:00 GMThttp://www.shnenglu.com/qinqing1984/archive/2025/04/25/230436.htmlhttp://www.shnenglu.com/qinqing1984/comments/230436.htmlhttp://www.shnenglu.com/qinqing1984/archive/2025/04/25/230436.html#Feedback0http://www.shnenglu.com/qinqing1984/comments/commentRss/230436.htmlhttp://www.shnenglu.com/qinqing1984/services/trackbacks/230436.html   鏈枃涓昏闃愯堪鐢ㄤ袱縐嶆柟娉曞垽鏂粰瀹氫袱涓簩鍏冧簩嬈″瀷鏄惁鐩鎬技錛岀浉浼兼儏鍐典笅鐨勫叿浣撳彉鎹€?br /> 鐩鎬技鍙樻崲濡傛灉紜畾浜嗭紝涔熷埄浜庡垽鏂瀹氭э紝鍥犱負鐩鎬技浜屾鍨嬬殑姝ｅ畾鎬х浉鍚屻傛渶鍚庤鍒頒簡姝ｄ氦鍒嗚В錛?br /> 緇欏嚭鎬庝箞姹傜浉浼肩殑鏁存暟瀵硅鐭╅樀

鍩烘湰瀹氫箟
  涓嬭堪瀹氫箟鏉ヨ嚜鏂囩尞[1] 12.1鑺傦紝鏈夋墍鎵╁睍 
  

鍙樻崲姹傝В
  鍏堟潵鐪嬭繍鐢ㄨВ鏂圭▼鐨勬柟娉?nbsp;
  

 
 鍐嶆潵鐪嬬敤鐭╅樀鐨勮鐐規柟娉曪紝姹傝В鍙樻崲銆傝繖縐嶆柟娉曟洿閫傚悎姹傝В鍒板瑙掑瀷鐨勫彉鎹?br />  
 
 

姝ｄ氦鍒嗚В 
   

  

鍙傝冩枃鐚?/strong> 
   [1] 鍗庣綏搴氭枃闆嗘暟璁哄嵎2
   [2] 楂樼瓑浠ｆ暟                 涓樼淮澹?/div>


]]>鍏充簬緹ょ殑涓浜涚粨璁哄強搴旂敤http://www.shnenglu.com/qinqing1984/archive/2025/04/22/230435.html鏄ョ鍗佷簩鏈?/dc:creator>鏄ョ鍗佷簩鏈?/author>Tue, 22 Apr 2025 13:18:00 GMThttp://www.shnenglu.com/qinqing1984/archive/2025/04/22/230435.htmlhttp://www.shnenglu.com/qinqing1984/comments/230435.htmlhttp://www.shnenglu.com/qinqing1984/archive/2025/04/22/230435.html#Feedback0http://www.shnenglu.com/qinqing1984/comments/commentRss/230435.htmlhttp://www.shnenglu.com/qinqing1984/services/trackbacks/230435.html銆?strong>鍛介1銆?鎵鏈夌兢鍚屾佺殑鍘熷儚涓暟鐩稿悓錛屽嵆涓烘牳鐨勫ぇ灝?/span> 
    

  涓嬮潰鐪嬩笅榪欎釜緇撹鍦ㄦ枃鐚甗1]涓?.2鑺傜殑搴旂敤 
     

銆?strong>鍛介2銆戞墍鏈夊厓绱犻樁灝忎簬絳変簬2 鐨勭兢涓轟氦鎹㈢兢錛屼笖鍏墮樁涓?鐨勬暣鏁板箓 
  
 璇ョ粨璁哄湪https://zhuanlan.zhihu.com/p/644888274涓殑鎺ㄨ2.2璇佹槑涓敤鍒?br />
銆?strong>鍛介3銆戠兢涓換涓鍏冪殑鐩稿浜庢瑙勫瓙緹ょ殑鎸囨暟嬈″箓灞炰簬姝ｈ瀛愮兢錛?闃舵瑙勫瓙緹ゅ繀
灞炰簬緹ょ殑涓績
 

銆愬畾鐞?銆?/strong>妯″鍚堟暟鐨勬棦綰︿箻娉曠兢錛屽叾涓泤鍙瘮絎﹀彿涓?鐨勫厓绱犳瀯鎴愬畠鐨勫瓙緹わ紝鍏墮樁涓?/span>
鏃㈢害涔樻硶緹ょ兢闃剁殑涓鍗?br />     

銆愬畾鐞?銆?/strong>璁綠鏄兢錛孒銆並鏄湁闄愬瓙緹わ紝鍒橦K鐨勫ぇ灝忕瓑浜嶩鐨勯樁涓嶬鐨勯樁涔樼Н闄や互H涓嶬浜ょ兢鐨勯樁 
  


鍙傝冩枃鐚?/strong>
  [1] 妞渾鏇茬嚎鍙婂叾鍦ㄥ瘑鐮佸涓殑搴旂敤—瀵煎紩  Andreas Enge
  [2] 鎶借薄浠ｆ暟I                                      璧墊槬鏉?寰愭槑鏇?br />   [3] 鍗庣綏搴氭枃闆嗘暟璁哄嵎2
  [4] 緇勫悎鏁板                                       鍐崳鏉?瀹嬫槬浼?/div>


]]>涓嶅畾鏂圭▼鐨勪唬鏁版暟璁鴻В娉?/title><link>http://www.shnenglu.com/qinqing1984/archive/2024/12/23/230411.html</link><dc:creator>鏄ョ鍗佷簩鏈?/dc:creator><author>鏄ョ鍗佷簩鏈?/author><pubDate>Mon, 23 Dec 2024 03:33:00 GMT</pubDate><guid>http://www.shnenglu.com/qinqing1984/archive/2024/12/23/230411.html</guid><wfw:comment>http://www.shnenglu.com/qinqing1984/comments/230411.html</wfw:comment><comments>http://www.shnenglu.com/qinqing1984/archive/2024/12/23/230411.html#Feedback</comments><slash:comments>0</slash:comments><wfw:commentRss>http://www.shnenglu.com/qinqing1984/comments/commentRss/230411.html</wfw:commentRss><trackback:ping>http://www.shnenglu.com/qinqing1984/services/trackbacks/230411.html</trackback:ping><description><![CDATA[<div><strong style="font-size: 12pt;">絎﹀彿鍚箟涓庨傜敤鍓嶆彁</strong><br /> <p>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-term-premise.png" width="591" height="246" alt="" /><br /> <br /> <span style="font-size: 12pt;"><strong><br /> 浜屾鍩熺殑鍩烘湰緇撹</strong></span> </p> <p>   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-quadratic-field-conclusion.png" width="600" height="551" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-quadratic-field-conclusion2.png" width="599" height="272" alt="" /><br /> <br /> </p> <div><strong style="font-size: 12pt;">x<sup>2</sup>-dy<sup>2</sup>=</strong><strong style="font-size: 12pt;">±1</strong>   </div> <div>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-pell-solve.jpg" alt="" /><br /> <br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-pell-solve2.jpg" width="635" height="955" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-pell-solve3.jpg" width="623" height="1026" alt="" /><strong style="font-size: 12pt;"><br /> x<sup>2</sup> + d = y<sup>3</sup></strong>  </div> <div>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-x2-d-y3-solve.jpg" width="586" height="802" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-x2-d-y3-solve2.jpg" width="574" height="1000" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-x2-d-y3-solve3.jpg" alt="" /> <br /> <br /> <div><strong style="font-size: 12pt;"><br /> x<sup>2</sup> + y<sup>2</sup> = n</strong>   </div> <div>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-two-quadratic-sum-solve.jpg" alt="" /><br /> <br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/diophantine-equation-two-quadratic-sum-solve2.jpg" alt="" /><br /> <br /> <br /> <div><strong>鍙傝冩枃鐚?/strong> <strong><br /> </strong><strong>    </strong>[1] 浠ｆ暟涓庢暟璁?nbsp;          鏉庤秴  鍛ㄦ偊</div> </div> </div> </div><img src ="http://www.shnenglu.com/qinqing1984/aggbug/230411.html" width = "1" height = "1" /><br><br><div align=right><a style="text-decoration:none;" href="http://www.shnenglu.com/qinqing1984/" target="_blank">鏄ョ鍗佷簩鏈?/a> 2024-12-23 11:33 <a href="http://www.shnenglu.com/qinqing1984/archive/2024/12/23/230411.html#Feedback" target="_blank" style="text-decoration:none;">鍙戣〃璇勮</a></div>]]></description></item><item><title>鍏充簬妞渾鏇茬嚎鐨勯獙璇佽綆?/title><link>http://www.shnenglu.com/qinqing1984/archive/2024/11/10/230402.html</link><dc:creator>鏄ョ鍗佷簩鏈?/dc:creator><author>鏄ョ鍗佷簩鏈?/author><pubDate>Sun, 10 Nov 2024 13:45:00 GMT</pubDate><guid>http://www.shnenglu.com/qinqing1984/archive/2024/11/10/230402.html</guid><wfw:comment>http://www.shnenglu.com/qinqing1984/comments/230402.html</wfw:comment><comments>http://www.shnenglu.com/qinqing1984/archive/2024/11/10/230402.html#Feedback</comments><slash:comments>0</slash:comments><wfw:commentRss>http://www.shnenglu.com/qinqing1984/comments/commentRss/230402.html</wfw:commentRss><trackback:ping>http://www.shnenglu.com/qinqing1984/services/trackbacks/230402.html</trackback:ping><description><![CDATA[<span style="font-size: 13pt;"><strong>絎﹀彿鍚箟</strong></span>  <div>    E            琛ㄧず婊¤凍妞渾鏇茬嚎Weierstrass鏂圭▼涓婄殑鐐圭兢</div> <div>    K            浠ｆ暟闂煙錛岀敤鏉ラ檺鍒禬eierstrass鏂圭▼鐨勭郴鏁頒笌E涓殑鐐?/div> <div>    E(K)        瀹氫箟鍦↘涓婄殑鐐圭兢E</div> <div>    E/K         瀹氫箟鍦↘涓婄殑妞渾鏇茬嚎E</div> <div>    End(E)    E涓婄殑鑷悓鎬佺幆<br /> <br /> <br /> <span style="font-size: 13pt;"><strong>鍩熸墿寮犲垎鏋?/strong></span><span style="background-color: #c0c0c0; font-size: 1px;"> </span><span style="font-size: 13pt;"><strong><br /></strong></span>   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-field-extention-analysis.png" alt="" /><br /> <br /> <div><span style="font-size: 13pt;"><strong>End(E)妯′笌Z浠ｆ暟</strong></span> <span style="font-size: 14pt;"><br /> </span><span>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-ende-module-z-algebra.png" width="550" height="760" alt="" /><br /> <br /> </span> <div><span style="font-size: 13pt;"><strong>鏋佺偣棣栭」緋繪暟</strong></span> <span style="font-size: 14pt;"><br /> </span><span style="font-size: 14pt;">  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-acnode-leading-term-coefficient-theory.jpg" width="571" height="970" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-acnode-leading-term-coefficient-example.png" width="562" height="831" alt="" /><br /> </span> <div><span style="font-size: 13pt;"><br /> <strong> 闄ゅ瓙鏄犲皠鍙婂悓鏋?/strong></span> <strong style="font-size: 1px;"> </strong></div> <div>   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-divisor-map-isomorphism.jpg" width="560" height="918" alt="" /><br /> <br /> <div><span style="font-size: 13pt;"><strong>鍚岀鏄犲皠鍚屾佹х殑瑙ｉ噴</strong></span> <span style="font-size: 14pt;"><br /> </span><span>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-isogeny-homomorphism-theorem.PNG" width="453" height="388" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-principal-divisor-corollary.PNG" width="454" height="218" alt="" /><br />   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-isogeny-map-homomorphism-explain.png" width="531" height="238" alt="" /><br /> <br /> </span> <div><span style="font-size: 13pt;"><strong>Hasse瀹氱悊涔嬪紩鐞嗚瘉鏄庣殑琛ュ厖</strong></span>  </div> <div>  <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-hasse-lemma-proof-supplement.jpg" width="591" height="958" alt="" /><br /> <br /> <div><span style="font-size: 13pt;"><strong>鎸犳洸綰垮強鍏朵釜鏁?/strong></span>   </div>   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-torsional-curve-and-num.jpg" width="584" height="978" alt="" /><br /> <br /> <div><strong><span style="font-size: 13pt;">鏈夐檺鍩熶笂鐨勬き鍦嗘洸綰?/span> </strong>  <div><span style="font-size: 11pt;">  <strong>涓縐嶇‘瀹氬瀷緹ら樁璁＄畻娉?/strong></span> <span style="font-size: 12pt;"><br /> </span><span>    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-odd-prime-field-calc-ecgrouporder-algo.png" width="563" height="507" alt="" /><br /> </span><span style="font-size: 12pt;"> <br /> </span><span style="font-size: 11pt;">  <strong>濂囩礌鍩熶笂鐨勭畻娉曞簲鐢?/strong></span><strong> </strong><span style="font-size: 12pt;"><br /> </span><span>    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-odd-prime-field-calc-ecgrouporder-algo-apply.png" width="571" height="380" alt="" /><br />    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-odd-prime-field-calc-ecgrouporder-algo-apply2.jpg" width="571" height="1019" alt="" /><br /> <br /> </span> <div><span style="font-size: 11pt;"> <strong>GF鍩熶笂鐨勭兢闃惰綆?/strong></span>  </div> <div><span>   <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-gf-calc-grouporder-weil-example.jpg" width="543" height="995" alt="" /><br />    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-gf-calc-grouporder-weil-example2.jpg" width="571" height="1019" alt="" /><br /> <br /> </span> <div><span style="font-size: 13pt;"><strong>Schoof綆楁硶姝ｇ‘鎬ф牴鏈?/strong></span>   </div> <div>    涓縐嶈綆楁き鍦嗘洸綰跨兢鐨勯樁鐨勭‘瀹氬瀷澶氶」寮忔椂闂寸畻娉曪紝紜畾鍨嬫槸鍥犱負綆楁硶鍐呴儴娌℃湁闅忔満閫夋嫨/姒傜巼鎶涘竵鎿嶄綔錛屽欏瑰紡鏃墮棿鏄洜涓哄煙<em>k</em>鐨勪箻娉曚笌姹傞嗘繪鏁版槸O((logq)^6)<br /> 錛?em>q</em>涓?em>k</em>鐨勫ぇ灝忥紝涔樻硶涓庢眰閫嗙浉瀵瑰姞鍑忚繍綆楁樉钁楄楁椂錛夈傚叿浣撳師鐞嗗強嫻佺▼璇﹁鍙傝冩枃鐚甗1]涓?.2鑺傘傝繖閲岀粰鍑虹瑪鑰呯殑涓浜涙濊? <div>​     1. Hasse瀹氱悊錛團robenius鑷悓鎬佹柟紼嬪紡錛夊湪鎵偣緹や笂鐨勯檺鍒朵害鎴愮珛錛岃繖鍐沖畾浜?em>t</em>妯?em>l</em>鐨勪竴涓悓浣欐柟紼嬫垚绔嬶紝涓斿湪妯鐨勬渶灝忛潪璐熷墿浣欑郴涓嬭В鏄敮涓鐨?/div> <div>​     2. 瀛欏瓙瀹氱悊淇濊瘉浜嗘煇鍙栧艱寖鍥村唴鐨勪竴涓?em>t</em>妯?em>L</em>錛?em>L</em>涓哄悇绱犲洜瀛恖鐨勪箻縐級鐨勫敮涓瑙ｏ紝鍗崇敱<em>t</em>妯?em>L</em>鍚勪釜绱犲洜瀛?em>l</em>鐨勫悓浣欐柟紼嬫瀯鎴愮殑鍚屼綑鏂圭▼緇勭殑瑙ｆ槸鍞竴鐨?/div> <div>​     3. <em>L</em>蹇呴』澶т簬<em>t</em>鍙栧間笂闄愮殑2鍊嶃傝繖鏄負浜嗙畻娉曟眰寰楃殑瑙ｆ弧瓚充笂榪?錛堝惁鍒欏湪鏇村皬鐨凩鍐呭緱鍒扮殑瑙ｄ笉鍞竴錛屽洜<em>L</em>涓?em>t</em>涓婇檺鎴栦笅闄愰棿鐨勬煇鏁板彲浠ヤ笌<em>t</em>妯?em>L</em>鍚屼綑錛?/div> <div>​     4. 绱犲洜瀛?em>l</em>鐨勯夋嫨鎺掗櫎2涓庢き鍦嗘洸綰跨壒寰?em>p</em>銆傝繖鏄洜涓虹畻娉曟瀯閫犳墍渚濊禆鐨勪竴涓紩鐞嗕箣鍓嶆彁鏉′歡錛氫負濂囩礌鏁頒繚璇?em>l</em>嬈￠櫎瀛愬欏瑰紡灞炰簬<em>k</em>[X]錛屽嵆寮曠悊璁烘柇鏈夋剰涔夛紱<br />        涓嶇瓑浜?em>p</em>淇濊瘉媯嫻嬩竴涓欏瑰紡f鏄惁闆跺欏瑰紡鐨勫厖瑕佹潯浠舵垚绔嬶紝鍗沖彲浠ョ敤<em>l</em>嬈￠櫎瀛愬欏瑰紡鍘繪暣闄?em>f</em>鏉ュ垽鏂傚彟l涓虹礌鏁頒繚璇佷簡涓庡叾瀹冮櫎瀛愬欏瑰紡錛堝強鍏跺箓嬈★級浜掔礌<br />      鍙﹀鍙戠幇浜嗙畻娉曠殑涓澶勭憰鐤碉紝鍗崇4姝ラ璁＄畻闄ゅ瓙澶氶」寮忎笌Frobenius鑷悓鎬佺殑澶嶅悎灝戜簡涓や釜鍊鹼紝榪欏鑷寸5姝ュ彲鑳藉穿婧冿紝褰撲緷璧栫殑鍚庣畫涓や釜澶嶅悎澶氶」寮忔病琚綆楁椂銆?br />   榪欎釜綰犳鍙氳繃淇敼絎?姝ユ墿澶?涓鹼紝鎴栫5姝ラ氳繃闄ゅ瓙澶氶」寮忕殑閫掓帹鍏紡鎸夐渶璁＄畻</div> <br /> <div><strong><span style="font-size: 13pt;">鎵偣鐨勯樁璁＄畻姝ｇ‘鎬ф牴鏈?/span></strong> <strong> </strong></div>     <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-calc-torsion-point-order-algo-correctness-analysis.png" width="599" height="374" alt="" /><br /> <br /> <div><span style="font-size: 13pt;"><strong>鍦ㄥ瘑鐮佸涓殑搴旂敤</strong></span><span style="background-color: #c0c0c0; font-size: 1px;"> </span> </div><div>    <span style="font-size: 11pt;"> <strong>閫夊彇鍘熷垯</strong></span>  </div> <div>        1. 鎺掗櫎瓚呭寮傛き鍦嗘洸綰褲傝繖鏄負閬垮厤<span style="color: #ff6600;">MOV</span>絳夌害鍖栨敾鍑伙紝綰﹀寲鏀誨嚮鏃墮棿澶嶆潅搴︽槸浜氭寚鏁?/div> <div>        2. 鏈夐檺鍩熺殑閫夋嫨瑕佷嬌<em>E</em>(<em>F</em><sub><em>q</em></sub>)鐨勭兢闃惰凍澶熷ぇ銆傝繖鏄負浜嗙紦瑙?span style="color: #ff00ff;">Shanks</span>鍙?span style="color: #ff00ff;">Pollard ρ</span>鏀誨嚮</div> <div>        3. <em>E</em>(<em>F</em><sub><em>q</em></sub>)瀛樺湪闃朵負澶х礌鏁扮殑瀛愮兢銆傝繖鏄負浜嗘姷鎶?span style="color: #ff00ff;">Pohlig-Hellman</span>鏀誨嚮</div> <div>      瀵逛簬絎?鐐癸紝灝辨帓闄や簡char(<em>K</em>)=2鎴?涓?em>j</em>(<em>E</em>)=0瀵瑰簲鐨勫涓嬫爣鍑嗗艦寮忔洸綰?/div> <div>           Y<sup>2</sup>+α<sub>3</sub>Y=X<sup>3</sup>+α<sub>4</sub>X+α<sub>6</sub>錛?#945;<sub>3</sub>≠0錛?涓?nbsp; Y<sup>2</sup>=X<sup>3</sup>+α<sub>4</sub>X+α<sub>6</sub> <br />      <br />     <span style="font-size: 11pt;"> <strong>涓縐嶅吀鍨嬫柟妗?/strong></span><span style="background-color: #c0c0c0; font-size: 1px;"><strong> </strong></span><span style="font-size: 12pt;"><br /> </span>           妞渾鏇茬嚎鍙婃湁闄愬煙鐨勯夋嫨浣垮緱|<em>E</em>(<em>F<sub>q</sub></em>)|=cm錛屼笖char(<em>F<sub>q</sub></em>) ∤ <em>q</em>+1-<em>cm</em>銆傚叾涓?em>m</em>鏄竴涓ぇ绱犳暟錛堥氬父涓嶄綆浜?56浣嶄簩榪涘埗闀垮害錛屾彁渚涗腑闀挎湡瀹夊叏鎬э級錛?em>c</em>灝忎簬<em>m</em>銆?br />          m闃跺瓙緹ょ殑鐢熸垚鍏冨彲閫氳繃浠ヤ笅鏂規硶紜畾錛氶殢鏈洪夋嫨<em>E</em>涓婄殑涓涓湁鐞嗙偣<em>P</em>錛屽鏋?em>Q</em>=<em>cP</em>涓洪浂鍏冿紙鍗蟲棤絀瘋繙鐐癸級錛屽垯閲嶅閫夋嫨錛岀洿鍒板叾涓嶇瓑浜庨浂鍏冦?br />          涓鏃︽壘鍒頒簡鐢熸垚鍏冿紝閭ｄ箞瀛愮兢灝卞彲浠ユ瀯閫犲嚭鏉ヤ簡銆備笅闈㈠垎鏋愭紜?nbsp; </div> <div>           <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/ec-cryptography-apply-typical-scheme.png" width="472" height="364" alt="" /><br /> <br /> <br /> <div><span style="font-size: 12pt;"><strong>鍙傝冩枃鐚?/strong></span></div> <div>  [1] 妞渾鏇茬嚎鍙婂叾鍦ㄥ瘑鐮佸涓殑搴旂敤—瀵煎紩      Andreas Enge</div> <div>  [2] 綆楁硶鏁拌                                           瑁村畾涓銆佺璺冮 </div> <div>  [3] The Arithmetic of Elliptic Curves        Joseph H. Silverman</div> <div>  [4] 鏍囪瘑瀵嗙爜瀛?nbsp;                                       紼嬫湞杈?/div> <div>  [5] 浠ｆ暟瀛﹀熀紜涓庢湁闄愬煙                             鏋椾笢宀?br /> <div>  [6] 鎶借薄浠ｆ暟I                                          璧墊槬鏉?寰愭槑鏇?/div> <div>  [7] 浠ｆ暟涓庢暟璁?nbsp;                                       鏉庤秴   鍛ㄦ偊</div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div><img src ="http://www.shnenglu.com/qinqing1984/aggbug/230402.html" width = "1" height = "1" /><br><br><div align=right><a style="text-decoration:none;" href="http://www.shnenglu.com/qinqing1984/" target="_blank">鏄ョ鍗佷簩鏈?/a> 2024-11-10 21:45 <a href="http://www.shnenglu.com/qinqing1984/archive/2024/11/10/230402.html#Feedback" target="_blank" style="text-decoration:none;">鍙戣〃璇勮</a></div>]]></description></item><item><title>涓嶅彲綰﹀欏瑰紡鍒ゅ埆綆楁硶鐨勬敼姝?/title><link>http://www.shnenglu.com/qinqing1984/archive/2024/09/07/230392.html</link><dc:creator>鏄ョ鍗佷簩鏈?/dc:creator><author>鏄ョ鍗佷簩鏈?/author><pubDate>Sat, 07 Sep 2024 15:07:00 GMT</pubDate><guid>http://www.shnenglu.com/qinqing1984/archive/2024/09/07/230392.html</guid><wfw:comment>http://www.shnenglu.com/qinqing1984/comments/230392.html</wfw:comment><comments>http://www.shnenglu.com/qinqing1984/archive/2024/09/07/230392.html#Feedback</comments><slash:comments>0</slash:comments><wfw:commentRss>http://www.shnenglu.com/qinqing1984/comments/commentRss/230392.html</wfw:commentRss><trackback:ping>http://www.shnenglu.com/qinqing1984/services/trackbacks/230392.html</trackback:ping><description><![CDATA[<div><strong style="font-size: 12pt;">鍘熸湰綆楁硶<br /> </strong> <div>    鎽樻妱<em>鍙傝冩枃鐚?</em>涓檮褰曠殑綆楁硶嫻佺▼濡備笅</div> <strong style="font-size: 12pt;">    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/Irreducible-polynomial-decide-original-algo.PNG" width="547" height="298" alt="" /><br /> <br /> </strong> <div><strong style="font-size: 12pt;">渚嬪瓙嫻嬮獙</strong><br />    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/Irreducible-polynomial-decide-example-test-prime-field.PNG" width="558" height="442" alt="" /><br />     <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/Irreducible-polynomial-decide-example-test-extend-field.PNG" width="556" height="367" alt="" /><br /> <div><br /> <strong style="font-size: 12pt;">鏀規鍚庣殑綆楁硶</strong></div> <div>       鏀規涔嬪墠錛屽厛鐞嗘竻鍘熸湰綆楁硶鍒ゅ埆涓嶅彲綰﹀欏瑰紡鎵鐢ㄧ殑鍘熺悊銆傚叾鍘熺悊鏄嫢f(x)鍙害錛屽綋涓斾粎褰撳瓨鍦ㄦ鏁癷<=d=[deg(f(x))/2]鐨勪笉鍙害鍥犲瓙g(x)錛岃屾鏃秅cd(x<sup>q^i</sup>-x, f(x))≠1銆?br />    鏍規嵁<em>鍙傝冩枃鐚?</em>錛堣瑙佸涓嬪畾鐞嗭級錛寈<sup>q^i</sup>-x鏄墍鏈塱嬈′笉鍙害澶氶」寮忕殑涔樼Н錛屽洜姝ゅ畠蹇呭畾鍖呭惈g(x)鑰屼笌f(x)瀛樺湪鍏洜瀛愩備笉鍙害鍒ゅ埆綆楁硶鐨勬濇兂搴旇鏄亶鍘嗘鏁?鍒癲鐨勬墍鏈変笉鍙害澶氶」寮?br />  錛堟病蹇呰媯嫻嬪ぇ浜巇鐨勪笉鍙害澶氶」寮忥紝鍥犱負鑻(x)鍙害鍒欏叾鍒嗚В鍥犲瓙涓繀瀹氬瓨鍦ㄤ笉澶т簬d鐨勪笉鍙害澶氶」寮忥級錛屾嫻嬭緭鍏ュ欏瑰紡涓庡畠浠槸鍚﹀瓨鍦ㄥ叕鍥犲瓙銆傛墍浠ヨ繖涓師鐞嗘槸姝ｇ‘鐨勶紝鍙槸瀹炵幇涓嶅錛?br />    鐣ヤ綔鏀規濡備笅錛堢被c璇█鎻忚堪錛?br />    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/Irreducible-polynomial-decide-correct-algo.PNG" width="602" height="420" alt="" /><br /> <div><br /> <strong style="font-size: 12pt;">閲嶆柊嫻嬮獙</strong><br />    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/Irreducible-polynomial-decide-example-retest-prime-field.PNG" alt="" /><br /><br />    <img src="http://www.shnenglu.com/images/cppblog_com/qinqing1984/Irreducible-polynomial-decide-example-retest-extend-field.PNG" width="452" height="838" alt="" /><br /> <br /> <br /> <div><strong style="font-size: 11pt;">鍙傝冩枃鐚?/strong></div> <div>   [1] 綆楁硶鏁拌                 瑁村畾涓銆佺璺冮</div> <div>   [2] 浠ｆ暟瀛﹀熀紜涓庢湁闄愬煙   鏋椾笢宀?/div> </div> </div> </div> </div><img src ="http://www.shnenglu.com/qinqing1984/aggbug/230392.html" width = "1" height = "1" /><br><br><div align=right><a style="text-decoration:none;" href="http://www.shnenglu.com/qinqing1984/" target="_blank">鏄ョ鍗佷簩鏈?/a> 2024-09-07 23:07 <a href="http://www.shnenglu.com/qinqing1984/archive/2024/09/07/230392.html#Feedback" target="_blank" style="text-decoration:none;">鍙戣〃璇勮</a></div>]]></description></item><item><title>璁鴻瘉鏈夐檺鍩熶笂騫蟲柟鏍圭殑姹傝Вhttp://www.shnenglu.com/qinqing1984/archive/2024/08/30/230390.html鏄ョ鍗佷簩鏈?/dc:creator>鏄ョ鍗佷簩鏈?/author>Fri, 30 Aug 2024 14:22:00 GMThttp://www.shnenglu.com/qinqing1984/archive/2024/08/30/230390.htmlhttp://www.shnenglu.com/qinqing1984/comments/230390.htmlhttp://www.shnenglu.com/qinqing1984/archive/2024/08/30/230390.html#Feedback0http://www.shnenglu.com/qinqing1984/comments/commentRss/230390.htmlhttp://www.shnenglu.com/qinqing1984/services/trackbacks/230390.html閫氱敤綆楁硶
   鍏堟憳鎶勫弬鑰冩枃鐚甗1]涓殑綆楁硶嫻佺▼濡備笅
   

   姝ｇ‘鎬у垎鏋?br />       涓嬮潰璇佹槑浠ヤ笂綆楁硶鐢ㄥ埌鐨勪簨瀹炵粨璁猴紝鎻愮偧涓哄涓嬪嚑涓紩鐞?br />       
     

   綆楁硶鏋勯犳濇兂
         鐢ㄥ埌浜屾鍓╀綑鐭ヨ瘑錛屽嵆涓涓緟姹傚鉤鏂瑰厓蓱鍙互涓斿彧鑳借〃紺轟負涓や釜騫蟲柟鍥犲瓙鐨勪箻縐紝鍏朵腑涓鍥犲瓙涓轟換鎰忛殢鏈洪夊彇鐨勯潪騫蟲柟鍥犲瓙β鐨勫伓鏁板箓錛?br />       鍙︿竴鍥犲瓙涓哄彾瀛愮兢H鐨勪竴鍏冪礌r錛孒浣滀負闄泦鍒掑垎鏍圭兢錛堟湁闄愬煙涔樻硶緹わ級寰楀埌β鐢熸垚鐨勯泦鍚堝嵆鍟嗙兢G/H鐨勪竴涓唬琛ㄥ厓緋匯傝繖鏍蜂竴鏉ワ紝灝嗗紑鏂硅漿鍖栦負β涓巖鐨勪箻鏂硅繍綆楋紝
      榪唬鐨勮繃紼嬪氨鏄負姹傞偅涓叿浣撶殑浠ｈ〃鍏?#946;e涓殑鎸囨暟e錛堟敞鎰廵蹇呬負鍋舵暟錛夛紝浠嶨s-2鍒癎0=H錛岃凱浠ｇ粨鏉熷悗r琚敮涓紜畾錛宺鐨勫紑鏂圭瓑浜巖鐨?t+1)/2嬈℃柟錛堝洜涓簍鏄疕鐨勯樁涓斾負濂囨暟錛宺t+1=r錛夈?br />      瑙傚療綆楁硶嫻佺▼錛屽彲浠ュ彂鐜板鏋滃垎瑙-1鍚庡緱鍒皊=1錛岄偅涔堝氨娌″繀瑕侀夊彇闈炲鉤鏂瑰厓β浜嗭紙榪欐椂浠?#946;=1錛夛紝鐩存帴璺沖埌絎?姝ュ緱鍒扮粨鏋溿備粎褰搒≠1鎵嶉殢鏈洪夊彇β銆傝繖鏍鋒敼榪涘悗鍙姞蹇畻娉曡繍琛?br />
   渚嬪瓙嫻嬮獙
      
      
鐗規畩綆楁硶
   褰搎鏄礌鏁頒笖q≡3(mod 4)鏃訛紝瀛樺湪鏇村揩鐨勭畻娉曞強嫻嬮獙濡備笅 
   


鍙傝冩枃鐚?/strong>
   [1]  綆楁硶鏁拌   瑁村畾涓銆佺璺冮


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