no4.htm

常用经典算法及讲解

lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp;</span><span style="mso-spacerun:
yes">&nbsp;</span><b style='mso-bidi-font-weight:normal'>xor</b></span><b
style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>运算的性质</span></b><b style='mso-bidi-font-weight:normal'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><o:p></o:p></span></b></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><span style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>在下文中，</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>x,y,z</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>∈<span lang=EN-US>N<sup>*</sup>, x<sub>1</sub>,x<sub>2</sub>,</span></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,x<sub>n</sub>∈N<sup>*</sup>.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span></span><b style='mso-bidi-font-weight:normal'><span style='font-size:
12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>性质</span></b><b style='mso-bidi-font-weight:
normal'><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>3: </span></b><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>x xor 0=x.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span></span><b style='mso-bidi-font-weight:normal'><span style='font-size:
12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>性质</span></b><b style='mso-bidi-font-weight:
normal'><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>4: </span></b><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'>x xor y=0 的充分必要条件是x=y.</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp;&nbsp; </span></span><span style='font-size:
12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>利用定义<span
lang=EN-US>2容易证明性质3、4.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun:
yes">&nbsp;</span><span style="mso-spacerun: yes">&nbsp;</span></span><b
style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>性质</span></b><b style='mso-bidi-font-weight:normal'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>5</span></b><b
style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>：</span></b><b style='mso-bidi-font-weight:normal'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>（交换律） </span></b><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>x
xor y = y xoy x<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span>证明：设x=(b<sub>t</sub></span><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;mso-ascii-font-family:宋体'>…</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>b<sub>1</sub>b<sub>0</sub>)<sub>2</sub>,b<sub>i</sub>∈</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>{0,1}, i=0,1,…,t,
b<sub>t</sub>=1;<span style="mso-spacerun: yes">&nbsp; </span>y=(c<sub>r</sub>…c<sub>1</sub>c<sub>0</sub>)<sub>2</sub>,</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'> c<sub>i</sub>∈</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>{0,1},<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>i=0,1,…,r, <span
style="mso-spacerun: yes">&nbsp;</span>c<sub>r</sub>=1.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><span style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>不妨设</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>t</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>≥</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt'>r, </span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>可在</span><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>c<sub>r</sub> </span><span style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"'>之前补</span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>r-t</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>个零，即</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>c<sub>t</sub>
=…=c<sub>r+1</sub>=0,<span style="mso-spacerun: yes">&nbsp; </span></span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>由性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>1</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>，<span lang=EN-US> b<sub>i</sub>⊙c<sub>i</sub>=c<sub>i</sub>⊙b<sub>i</sub>,
i=1,</span></span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;mso-ascii-font-family:宋体'>…</span><span lang=EN-US style='font-size:
12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>,t,
所以</span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'>5</span><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>成立<span lang=EN-US>.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>6:</span></b><b style='mso-bidi-font-weight:normal'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>（结合律） </span></b><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>(x
xor y)xor z=x xor(y xor z)<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;
mso-hansi-font-family:"Times New Roman"'><span style="mso-spacerun: yes">&nbsp;
</span>证明与</span><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"'>性质</span><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>5</span><span style='font-size:12.0pt;mso-bidi-font-size:
10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>类似（利用性质<span
lang=EN-US>2）</span></span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt'><o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>7</span></b><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>:<b
style='mso-bidi-font-weight:normal'>（增广律）</b>若x=y,<span style="mso-spacerun:
yes">&nbsp; </span>则x xor z=y xor z.<o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>性质<span lang=EN-US>7可利用定义1、2直接导出.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>8</span></b><b style='mso-bidi-font-weight:normal'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>：</span></b><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>若<span lang=EN-US>x&lt;2<sup>t</sup>, 则x xor 2<sup>t</sup>=x+2<sup>t</sup></span></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>9</span></b><b style='mso-bidi-font-weight:normal'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"'>：</span></b><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>设<span lang=EN-US>x=(b<sub>t</sub></span></span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;mso-ascii-font-family:
宋体'>…</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt;
font-family:宋体;mso-hansi-font-family:"Times New Roman"'>b<sub>1</sub>b<sub>0</sub>)<sub>2</sub>,
b<sub>i</sub>∈</span><span lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:
10.0pt'>{0,1},<span style="mso-spacerun: yes">&nbsp; </span>i=0,1,…,t, b<sub>t</sub>=1;<span
style="mso-spacerun: yes">&nbsp; </span></span><span style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>则<span
lang=EN-US>x xor 2<sup>t</sup>=x-2<sup>t</sup></span></span><span lang=EN-US
style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:
"Times New Roman"'>性质<span lang=EN-US>8、9 可利用定义1、2直接导出.<o:p></o:p></span></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span
style="mso-spacerun: yes">&nbsp; </span></span><b style='mso-bidi-font-weight:
normal'><span style='font-size:12.0pt;mso-bidi-font-size:10.0pt;font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>性质</span></b><b
style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt'>10</span></b><span lang=EN-US style='font-size:12.0pt;
mso-bidi-font-size:10.0pt;font-family:宋体;mso-hansi-font-family:"Times New Roman"'>:
若x&lt;2<sup>t</sup>, 且y&lt;2<sup>t</sup>,<span style="mso-spacerun: yes">&nbsp;
</span>则(x xor y)+2<sup>t</sup>=x xor (y+2<sup>t</sup>).</span><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><o:p></o:p></span></p>

<p class=MsoNormal style='line-height:20.0pt;mso-line-height-rule:exactly'><span
lang=EN-US style='font-size:12.0pt;mso-bidi-font-size:10.0pt'><span