Annotation of OpenXM/doc/sci-semi2001/factorb.tex, Revision 1.3
1.3 ! noro 1: % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.2 2001/07/24 08:02:47 noro Exp $
1.1 noro 2:
1.3 ! noro 3: \Large
1.1 noro 4: \parskip 0pt
5:
6: \begin{slide}{}
1.2 noro 7: \fbox{\sc 1. $B$O$8$a$K(B}
1.1 noro 8:
9: computer = compute $B$9$k$?$a$N$b$N(B
10:
1.2 noro 11: compute = {\ec $B7W;;(B}$B$9$k(B
1.1 noro 12:
1.2 noro 13: $B:G6a$G$O(B {\ec $B%G%8%?%k>pJsDL?.(B} $B$N<jCJ$H$J$C$F$7$^$C$?(B
1.1 noro 14:
15: $\Rightarrow$ $B!V7W;;!W$K;H$C$F$$$k?M$O$4$/>/?t(B
16:
1.2 noro 17: {\bf $BNc(B} : email, $B%&%'%V(B {\eec $B!V%$%s%?!<%M%C%H$9$k!W(B}
1.1 noro 18:
19: $B$7$+$7(B, $B7W;;5!$NG=NO$O0[MM$K8~>e(B
20:
1.2 noro 21:
22: {\ec $B7W;;$K;H$o$J$$$N$O$b$C$?$$$J$$(B}($B$H;W$&(B)
1.1 noro 23: \end{slide}
24:
25: \begin{slide}{}
1.2 noro 26: \fbox{\sc 2. $B%3%s%T%e!<%?$K$D$$$F$N%$%m%O(B}
1.1 noro 27:
28: \begin{itemize}
1.2 noro 29: \item {\eec CPU}
1.1 noro 30:
31: $B%W%m%0%i%`$K=>$C$FL?Na$r<B9T(B
32:
1.2 noro 33: \item {\eec $B%a%b%j(B}
1.1 noro 34:
35: $B%W%m%0%i%`(B, $B%G!<%?$rCV$/>l=j(B. $B>l=j(B ($BHVCO(B) $B$r;XDj$7$F9bB.$K=P$7F~$l$,$G$-$k(B.
36:
1.2 noro 37: \item {\eec $B%l%8%9%?(B}
1.1 noro 38:
39: CPU $B$,;}$C$F$$$kFCJL$J%a%b%j$G(B, $B1i;;$NBP>]$K$J$k(B. $B?t$OB?$/$J$/(B, $BBg$-$5(B
40: ($BD9$5(B) $B$b>.$5$$(B.
41: \end{itemize}
42:
43: \end{slide}
44:
45: \begin{slide}{}
1.2 noro 46: \underline{\uc $BL?Na$NNc(B}
1.1 noro 47: \begin{itemize}
48: \item 10 $BHVCO$+$i(B 1 $B%P%$%HFI$s$G(B A $B%l%8%9%?$KF~$l$h(B
49: \item A, B $B%l%8%9%?$NCM$rB-$7$F(B C $B%l%8%9%?$KF~$l$h(B
50: \item A $B%l%8%9%?$NCM$,(B 0 $B$J$i(B 100 $BHVCO@h$KHt$Y(B
51: \end{itemize}
52:
1.2 noro 53: \underline{\uc $B07$($k?t(B}
1.1 noro 54:
1.2 noro 55: $B%l%8%9%?$NBg$-$5(B = $B07$($k?t$NHO0O(B
1.1 noro 56:
57: 32$B%S%C%H%l%8%9%?(B $\Rightarrow$ 0 $B$+$i(B $2^{32}-1$ $B$^$G$N@0?t$7$+07$($J$$(B
58:
59: \end{slide}
60:
61: \begin{slide}{}
1.2 noro 62: \underline{\uc $B?t3X$K;H$&>l9g$r9M$($k$H(B...}
1.1 noro 63:
64: $11111111111 \times 11111111111$
65:
1.2 noro 66: $\Rightarrow$ {\ec 1332508849} ??? (=$B7k2L$r(B $2^{32}$ $B$G3d$C$?M>$j(B)
1.1 noro 67:
68: $B$+$H$$$C$F(B
69:
70: $11111111111 \times 11111111111$
71:
72: $\Rightarrow 1.234567 \times 10^{20}$
73:
74: $B$b:$$k(B
75:
1.2 noro 76: {\ec $B8m:9$,F~$k$H?t3X$H$7$F$OL50UL#$J7W;;(B}
1.1 noro 77: \end{slide}
78:
79: \begin{slide}{}
1.2 noro 80: \underline{\uc $B$H$j$"$($:Bg$-$J@0?t$O07$($J$$$H$$$1$J$$(B}
1.1 noro 81:
1.2 noro 82: $\Rightarrow$ {\eec $B%W%m%0%i%`(B}$B$r=q$1$P$h$$(B
1.1 noro 83:
1.2 noro 84: $B%a%b%j>e$K@0?t$rJB$Y$F(B, $B%3%s%T%e!<%?$K(B {\eec $B!VI.;;!W(B}$B$r$5$;$l$P$h$$(B
1.1 noro 85:
86: \begin{itemize}
1.2 noro 87: \item {\eec $B?M4V(B}
1.1 noro 88:
89: $B$R$H$1$?(B : 0 $B0J>e(B 9 $B0J2<(B
90:
1.2 noro 91: \item {\eec $B%3%s%T%e!<%?(B}
1.1 noro 92:
93: $B$R$H$1$?(B : 0 $B0J>e(B $2^{32}-1$ $B0J2<(B
94: \end{itemize}
95: \end{slide}
96:
97: \begin{slide}{}
1.2 noro 98: \underline{\uc $BNc(B : $B@0?t$NB-$7;;(B}
1.1 noro 99:
1.2 noro 100: \begin{tabular}{ccccc} \\
1.1 noro 101: & 5 & 4001257187 & 1914644777 & (= $3^{42}$) \\
102: + & & 2830677074 & 689956897 & (= $3^{40}$) \\ \hline
103: & 6 & 2536966965 & 2604601674 &
104: \end{tabular}
105:
1.2 noro 106: \vskip 1cm
1.1 noro 107:
1.2 noro 108: \underline{\uc $B0lJQ?tB?9`<0(B}
1.1 noro 109:
1.2 noro 110: $B3F<!?t$N78?t$rJB$Y$l$P$h$$(B
111:
112: $\Rightarrow$ $B$3$l$G(B{\ec $B@0?t78?t$NB?9`<0$r?t3XE*$K07$($k(B}
1.1 noro 113:
114: \end{slide}
115:
116: \begin{slide}{}
1.2 noro 117: \fbox{\sc 3. $BB?9`<0$N0x?tJ,2r(B --- $BCf3X9b9;E*J}K!(B}
118: {
119: \Large\parskip 0pt
1.1 noro 120:
121: \begin{enumerate}
1.2 noro 122: \item {\eec $B4cNOK!(B} ($B2r$H78?t$N4X78(B)
1.1 noro 123:
124: $x^2+ax+b \Rightarrow$ $BB-$7$F(B $a$, $B$+$1$F(B $b$ $B$K$J$k?t$NAH(B
125:
126: $x^3+ax^2+bx+c$ $B$O$I$&$9$k(B?
127:
1.2 noro 128: \item {\eec $B0x?tDjM}(B}
1.1 noro 129:
130: $BBeF~$7$F(B 0 $B$K$J$k?t$rC5$9(B ($B$I$&$d$C$FC5$9(B?)
131:
1.2 noro 132: \item {\eec $B2r$N8x<0(B}
1.1 noro 133:
1.2 noro 134: $x^2+ax+b=0$ $B$N:,(B ${-b \pm \sqrt{a^2-4b}} \over 2$
1.1 noro 135:
136: $\Rightarrow$ $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D4$Y$k(B
137: \end{enumerate}
1.2 noro 138: }
1.1 noro 139: \end{slide}
140:
141: \begin{slide}{}
1.2 noro 142: \underline{\uc $B4cNOK!$OLdBj$rFq$7$/$7$F$$$k(B}
1.1 noro 143:
144: $BNc(B : $x^2+11508x+28386587$
145:
1.2 noro 146: $28386587=3581\times 7927$ $B$,4cNO$GJ,$+$k?M$O>/$J$$(B($B$H;W$&(B)
147:
148: \vskip 1cm
1.1 noro 149:
1.2 noro 150: \underline{\uc $B2r$N8x<0K!$OM-K>(B}
1.1 noro 151:
152: $(a^2-4b)/4 = 4717584 = 2172^2$ $B$J$i2?$H$+$J$k(B?
153:
1.2 noro 154: $\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$$(B}
1.1 noro 155: \end{slide}
156:
157: \begin{slide}{}
1.2 noro 158: \underline{\uc 3 $B<!0J2<$NB?9`<0(B}
1.1 noro 159:
1.2 noro 160: {\eec $B@0?t>e$GJ,2r$G$-$k$J$i(B, $B0l<!0x;R$r;}$D(B}
1.1 noro 161:
1.2 noro 162: $\Rightarrow$ {\ec $B:,$rC5$9J}K!$,E,MQ$G$-$k(B}
1.1 noro 163:
1.2 noro 164: {\eec $B:,5r(B : $BCf4VCM$NDjM}(B}
1.1 noro 165: $B!V(B$f(a) < 0, f(b) > 0$ $B$J$i(B $a$, $b$ $B$N4V$K(B $f(c)=0$ $B$J$k(B $c$ $B$,$"$k(B.$B!W(B
166:
167: \begin{itemize}
1.2 noro 168: \item {\eec $BFsJ,K!(B}
1.1 noro 169:
170: $B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B
171:
1.2 noro 172: \item {\eec Newton $BK!(B}
1.1 noro 173:
174: $BFsJ,K!$h$j$:$C$H9bB.(B
175: \end{itemize}
176:
177: \end{slide}
178:
179: \begin{slide}{}
1.2 noro 180: \underline{\uc 4 $B<!0J>e$N>l9g(B}
1.1 noro 181:
182: $B$$$m$$$m$JJ,2r%Q%?!<%s$,$"$jF@$k(B
183:
184: 4 $B<!(B = 2 $B<!(B $\times$ 2 $B<!(B
185:
1.2 noro 186: $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
1.1 noro 187:
1.2 noro 188: \vskip 1cm
1.1 noro 189:
1.2 noro 190: \underline{\uc $B%3%s%T%e!<%?$K$ONO5;(B($B7+$jJV$7(B)$B$,;w9g$&(B}
1.1 noro 191:
1.2 noro 192: {\eec $BCf4VCM$NDjM}(B} = {\eec $B<B?t$K$*$1$k6a;w(B} $B$NMxMQ(B
193:
194: $BJL$N6a;w(B $\Rightarrow$ {\ec $B3d$C$?M>$j(B}$B$KCmL\(B
1.1 noro 195: \end{slide}
196:
197: \begin{slide}{}
1.2 noro 198: \fbox{\sc 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}
199: {\Large\parskip 0pt
1.1 noro 200:
1.2 noro 201: \underline{\uc $B86M}(B} : {\eec $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$
1.1 noro 202:
1.2 noro 203: {\eec $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k(B}
1.1 noro 204:
1.2 noro 205: ({\eec $m$ $B$O==J,Bg$-$$@0?t$G3d$j@Z$l$k(B})
1.1 noro 206:
207: $B$?$H$($P(B,
208:
209: \begin{enumerate}
210: \item $B:G=i(B, $f(x)-g_1(x)h_1(x)$ $B$N78?t$,@0?t(B $p$ $B$G3d$j@Z$l$k$h$&$J(B $g_1$,
211: $h_1$ $B$r8+$D$1$k(B.
212:
213: \item $f(x)-g_k(x)h_k(x)$ $B$N78?t$,(B $p^k$ $B$G3d$j@Z$l$k$h$&$K(B $g_k$, $h_k$ $B$r(B
214: $B:n$C$F$$$/(B ($k=1,2,\ldots$)
215:
1.2 noro 216: \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.
217: \end{enumerate}}
1.1 noro 218: \end{slide}
219:
220: \begin{slide}{}
1.2 noro 221: \underline{\uc $B8@$$$+$($l$P(B...}
222:
223: $B0J2<(B, {\ec $B4JC1$N$?$a(B}, $f(x)$ $B$*$h$S0x;R$N78?t$OA4$F@5$G$"$k$H$9$k(B.
1.1 noro 224:
1.2 noro 225: {\eec $f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$}
1.1 noro 226:
1.2 noro 227: $B$H!V$Y$-5i?tE83+!W(B ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
1.1 noro 228:
1.2 noro 229: {\ec $g(x) = b_0(x)+p\cdot b_1(x)+p^2\cdot b_2(x)+\cdots$}
1.1 noro 230:
1.2 noro 231: {\ec $h(x) = c_0(x)+p\cdot c_1(x)+p^2\cdot c_2(x)+\cdots$}
1.1 noro 232:
1.2 noro 233: ($b_i$, $c_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
1.1 noro 234:
235: $B$H$*$$$F(B $f(x)-g(x)h(x)=0$ $B$+$i(B $b_i$, $c_i$ $B$r7h$a$k(B.
236: \end{slide}
237:
238: \begin{slide}{}
1.2 noro 239: \underline{\uc $B5-9f(B $a \equiv b \bmod M$}
1.1 noro 240:
1.2 noro 241: $M$ $B$r@0?t$H$9$k(B. {\eec $a \equiv b \bmod M$} $B$H$O(B
1.1 noro 242:
243: \begin{itemize}
244: \item $a,b$ $B$,@0?t$N$H$-(B,
245:
1.2 noro 246: {\eec $a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k(B}$B$3$H(B
1.1 noro 247:
248: \item $a,b$ $B$,@0?t78?tB?9`<0$N$H$-(B
249:
1.2 noro 250: {\eec $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k(B}$B$3$H(B
1.1 noro 251: \end{itemize}
252:
1.2 noro 253: \vskip 1cm
254:
255: \underline{\uc $a$ $B$r(B $M$ $B$G3d$C$?M>$j$b(B $a \bmod M$ $B$H=q$/(B}
1.1 noro 256: \end{slide}
257:
258: \begin{slide}{}
1.2 noro 259: \underline{\uc $b_0(x)$, $c_0(x)$ $B$+$i%9%?!<%H(B}
260:
261: $f-gh$
262:
263: $\quad = a_0-${\ec $b_0c_0$} + ($p$$B$G3d$j@Z$l$kB?9`<0(B)
1.1 noro 264:
1.2 noro 265: $B$@$+$i(B, $f=gh$ $B$J$i(B
1.1 noro 266:
1.2 noro 267: $a_0 \equiv$ {\ec $b_0c_0$} $\bmod p$ $B$N$O$:(B
1.1 noro 268:
1.2 noro 269: \underline{\uc $BNc(B}
1.1 noro 270:
1.2 noro 271: {\eec
1.1 noro 272: \begin{tabbing}
273: $f(x)=$ \= $x^4+17056x^3+72658809x^2$ \\
274: \> $+3504023212x+30603759869$
1.2 noro 275: \end{tabbing}}
1.1 noro 276:
277: $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
278: \end{slide}
279:
280: \begin{slide}{}
1.2 noro 281: \underline{\uc $B0l<!0x;R$,$"$k$+(B?}
1.1 noro 282:
1.2 noro 283: {\ec $b_0(x) = x+q$},
284: {\ec $c_0(x) = x^3+rx^2+sx+t$} $B$H$*$/(B.
1.1 noro 285:
286: $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
287:
1.2 noro 288: {\ec
1.1 noro 289: $\left\{
290: \parbox[c]{6in}{
1.2 noro 291: $q+r \equiv 1 \bmod 3$ \\
292: $qr+s \equiv 0 \bmod 3$ \\
293: $qs+t \equiv 1 \bmod 3$ \\
294: $qt \equiv 2 \bmod 3$}
295: \right.$\\}
1.1 noro 296:
1.2 noro 297: $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$r$I$&F~$l$F$b%@%a(B.
1.1 noro 298:
1.2 noro 299: $B$h$C$F(B, {\eec $B0l<!0x;R$O$J$$(B}.
1.1 noro 300:
301: \end{slide}
302:
303: \begin{slide}{}
1.2 noro 304: \underline{\uc $BFs<!0x;R$O$"$k$+(B? --- $B$^$:(B $b_0$, $c_0$ $B$rC5$9(B}
1.1 noro 305:
1.2 noro 306: {\ec $b_0(x) = x^2+qx+r$},
307: {\ec $c_0(x) = x^2+sx+t$}
1.1 noro 308:
1.2 noro 309: $B$H$*$/$H(B, $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
1.1 noro 310:
1.2 noro 311: {\ec
1.1 noro 312: $\left\{
313: \parbox[c]{6in}{
1.2 noro 314: $q+s \equiv 1 \bmod 3$ \\
315: $qs+r+t \equiv 0 \bmod 3$ \\
316: $qt+rs \equiv 1 \bmod 3$ \\
317: $tr \equiv 2 \bmod 3$}
318: \right.$\\}
1.1 noro 319:
1.2 noro 320: $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$l$P(B
1.1 noro 321:
1.2 noro 322: {\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}
1.1 noro 323:
1.2 noro 324: $B0lJ}$,(B $b_0$, $BB>J}$,(B $c_0$ $\Rightarrow$ $B$3$l$i$OF1$8$b$N(B
1.1 noro 325: \end{slide}
326:
327: \begin{slide}{}
1.2 noro 328: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9@-<A(B}
1.1 noro 329:
1.2 noro 330: {\Large\parskip 0pt
331: {\eec $b_0 = x^2+1$},
332: {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B
1.1 noro 333:
1.2 noro 334: \centerline{\eec $f-b_0c_0 \equiv 0 \bmod 3$}
1.1 noro 335:
1.2 noro 336: $f-gh \equiv a_0-b_0c_0+p(a_1-$
337: $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3^2$
1.1 noro 338:
339: $B$h$j(B, $BN>JU$r(B 3 $B$G3d$C$F(B
340:
1.2 noro 341: ${{f-gh}\over 3} \equiv {{a_0-b_0c_0}\over 3}+(a_1-$
342: $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3$
1.1 noro 343:
344: $B:8JU$O(B $3$ $B$G2?2s$G$b3d$l$k(B $\Rightarrow$ $B1&JU$O(B $3$ $B$G3d$l$k(B
345:
1.2 noro 346: $BJd@59`(B {\ec $b_1$}, {\ec $c_1$} : $x^2$ $B$N78?t$O(B 0 $B$H$7$F$h$$(B}
1.1 noro 347:
348: \end{slide}
349:
350: \begin{slide}{}
1.2 noro 351: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}
1.1 noro 352:
1.2 noro 353: {\Large\parskip 0pt
354: {\ec $b_1 = qx+r$},
355: {\ec $c_1 = sx+t$} $B$H$*$/(B.
1.1 noro 356:
357: \begin{tabbing}
1.2 noro 358: $B1&JU(B = \= {\ec $-(q+s)x^3-(q+r+t+1)x^2$}\\
359: \> {\ec $-(2q+r+s-1)x-(2r+t)$}
1.1 noro 360: \end{tabbing}
361:
362: $$B1&JU(B \equiv 0 \bmod 3$ $B$h$j(B\\
363:
1.2 noro 364: {\ec
1.1 noro 365: $\left\{
366: \parbox[c]{6in}{
1.2 noro 367: $q+s \equiv 0 \bmod 3$ \\
368: $q+r+t+1 \equiv 0 \bmod 3$ \\
369: $2q+r+s-1 \equiv 0 \bmod 3$ \\
370: $2r+t \equiv 0 \bmod 3$}
371: \right.$\\}
1.1 noro 372:
373: $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B
374:
1.2 noro 375: {\eec $(q,r,s,t) = (0,1,0,1)$} $B$9$J$o$A(B {\eec $b_1 = 1$}, {\eec $c_1 = 1$}}
1.1 noro 376:
377: \end{slide}
378:
379: \begin{slide}{}
1.2 noro 380: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}
1.1 noro 381:
1.2 noro 382: {\Large\parskip 0pt
1.1 noro 383: $B$3$l$G(B,
384:
1.2 noro 385: \centerline{\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
1.1 noro 386:
387: $B0J2<F1MM$K(B,
388:
1.2 noro 389: \centerline{\ec $b_k = qx+r, c_k = sx+t$}
1.1 noro 390:
1.2 noro 391: $B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r2r$1$P(B
1.1 noro 392:
1.2 noro 393: \centerline{\eec $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}
1.1 noro 394:
395: $B$9$J$o$A(B
396:
1.2 noro 397: \centerline{\eec $f \equiv g_kh_k \bmod 3^k$}
1.1 noro 398:
1.2 noro 399: $B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B. }
1.1 noro 400:
401: \end{slide}
402:
403: \begin{slide}{}
1.2 noro 404: \underline{\uc $(g_k, h_k)$ $B$NI=(B}
1.1 noro 405:
406: {\large
407: \begin{tabular} { c | c c }
408: $k$ & $g_k$ & $h_k$ \\ \hline
409: 1&$x^2+1$&$x^2+x+2$\\ \hline
410: 2&$x^2+4$&$x^2+x+5$\\ \hline
411: 3&$x^2+18x+4$&$x^2+x+5$\\ \hline
412: 4&$x^2+45x+4$&$x^2+x+59$\\ \hline
413: 5&$x^2+45x+166$&$x^2+x+140$\\ \hline
414: 6&$x^2+531x+409$&$x^2+487x+626$\\ \hline
415: 7&$x^2+1260x+1867$&$x^2+487x+1355$\\ \hline
416: 8&$x^2+1260x+4054$&$x^2+2674x+1355$\\ \hline
417: 9&$x^2+7821x+10615$&$x^2+9235x+7916$\\ \hline
418: 10&$x^2+7821x+30298$&$x^2+9235x+47282$\\ \hline
419: 11&$x^2+7821x+89347$&$x^2+9235x+165380$\\ \hline
1.2 noro 420: 12&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
421: 13&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
1.1 noro 422: \end{tabular}}
423: \end{slide}
424:
425: \begin{slide}{}
1.2 noro 426: \underline{\uc $\bmod 3^k$ $B$G$N0x;R$+$i??$N0x;R$X(B}
1.1 noro 427:
1.2 noro 428: $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}
1.1 noro 429:
1.2 noro 430: $\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0!}
1.1 noro 431:
1.2 noro 432: {\eec
433: $f(x) = (x^2+7821x+89347) \times$
1.1 noro 434:
1.2 noro 435: $(x^2+9235x+342527)$}
1.1 noro 436:
1.2 noro 437: \underline{\uc $B<B:]$K$O(B...}
1.1 noro 438:
439: \begin{itemize}
440: \item $BIi$N78?t$N>l9g$r07$&$?$a$N9)IW$,I,MW(B
441:
442: \item $B<:GT$N2DG=@-$b$"$k$N$G(B, $k$ $B$r$I$3$^$G>e$2$l$P$$$$$+$N>e8B$,I,MW(B
443: \end{itemize}
444: \end{slide}
445:
446: \begin{slide}{}
1.2 noro 447: \underline{\uc $\bmod p$ $B$G$NJ,2r$,0lHVBg@Z(B}
1.1 noro 448:
1.2 noro 449: $B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B
1.1 noro 450:
451: \begin{itemize}
1.2 noro 452: \item {\eec $k > 1$}
1.1 noro 453:
1.2 noro 454: $BO"N)0l<!J}Dx<0(B ($B<B:]$K$O9gF1<0(B)
1.1 noro 455:
1.2 noro 456: \item {\eec $k = 1$}
1.1 noro 457:
1.2 noro 458: $B0l<!J}Dx<0$G$J$$(B
459:
460: $\Rightarrow$ $B$7$i$_$D$V$7$G2r$/$N$O$"$^$j$K8zN((B
1.1 noro 461: $B$,$o$k$$(B ($B$$$/$i%3%s%T%e!<%?$G$b(B)
462: \end{itemize}
463: \end{slide}
464:
465: \begin{slide}{}
1.2 noro 466: \fbox{\sc 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }
467:
468: $p$ $B$,(B{\ec $BAG?t(B}$B$N$H$-(B,
1.1 noro 469:
1.2 noro 470: {\eec $GF(p) = \{0,1,\cdots,p-1\}$} $B$K(B, $+$, $-$, $\times$ $B$r(B
471: {\eec $B!V7k2L$r(B $p$ $B$G(B $B3d$C$?M>$j!W(B}$B$GDj5A$9$k$H(B
1.1 noro 472:
473: \begin{enumerate}
474: \item $B2C8:>h;;$GJD$8$F$$$k(B.
1.2 noro 475: \item {\eec 0 $B0J30$N85$G3d;;$,$G$-$k(B. }
1.1 noro 476:
1.2 noro 477: $B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$J$k(B $b$ $B$,$"$k!W(B
1.1 noro 478: \end{enumerate}
479:
1.2 noro 480: $B$9$J$o$A(B, {\eec $GF(p)$ $B$OBN(B($B%?%$(B)}
1.1 noro 481:
1.2 noro 482: $B85$N8D?t$,M-8B8D(B ($p$ $B8D(B)$B$J$N$G(B {\ec $BM-8BBN(B} $B$H$h$V(B.
1.1 noro 483:
484: \end{slide}
485:
486: \begin{slide}{}
1.2 noro 487: \underline{\uc $k=1$ $\Rightarrow$ $BM-8BBN>e$G$N0x?tJ,2r(B}
1.1 noro 488:
489: $a_0 \equiv f \bmod p$ $B$r(B $GF(p)$ $B78?tB?9`<0$H$_$k(B.
490:
491: $\Rightarrow$ $a_0 \equiv b_0c_0 \bmod p$ $B$H$J$k(B $b_0$, $c_0$ $B$r(B
492: $B5a$a$k$3$H$O(B, $GF(p)$ $B>e$G$N0x?tJ,2r$KAjEv(B
493:
1.2 noro 494: $\Rightarrow$ {\eec $B<B$O$h$$%"%k%4%j%:%`$,$"$k(B}
495:
496: \vskip 1cm
1.1 noro 497:
1.2 noro 498: \underline{\uc $k > 1$ $\Rightarrow$ $BM-8BBN>e$G$NO"N)0l<!J}Dx<05a2r(B}
1.1 noro 499:
500: $B<B:]$K$O(B, $k=1$ $B$N7k2L$+$i5!3#E*$K7W;;$G$-$k(B.
501: \end{slide}
502:
503: \begin{slide}{}
1.2 noro 504: \underline{\uc $B0x?tJ,2r$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}
1.1 noro 505:
506: \begin{enumerate}
1.2 noro 507: \item {\eec $B$h$$AG?t(B $p$ $B$rA*$s$G(B $f \bmod p$ $B$r0x?tJ,2r(B}
1.1 noro 508:
1.2 noro 509: {\eec $B!V$h$$!W(B} $B$H$O(B
1.1 noro 510:
1.2 noro 511: \begin{itemize}
512: \item $f$ $B$N:G9b<!78?t$r3d$i$J$$(B
513:
514: \item $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.
515: \end{itemize}
1.1 noro 516:
1.2 noro 517: \item {\eec $B<!$r7+$jJV$7(B}
1.1 noro 518:
519: \begin{enumerate}
520: \item $GF(p)$ $B>e$N0x;R$rFsAH$KJ,$1$k(B
521:
522: \item $B3FAH$N@Q$r(B $g_1$, $h_1$ $B$H$9$k(B.
523:
524: \item $f \equiv g_kh_k \bmod p^k$ $B$J$k(B $g_k$, $h_k$ $B$r:n$k(B
525:
526: \item $B78?t$N@5Ii$rD4@a$7$F;n$73d$j(B
527: \end{enumerate}
528:
529: \end{enumerate}
530: \end{slide}
531:
532: \begin{slide}{}
1.2 noro 533: \underline{\uc $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}
1.1 noro 534:
535: \begin{itemize}
1.2 noro 536: \item {\eec $BBN$N>e$G$N0x?tJ,2r$N0l0U@-(B}
1.1 noro 537:
538: $BBN>e$NB?9`<04D$N@-<A(B
539:
1.2 noro 540: \item {\eec $BM-8BBN>e$G$N0x?tJ,2r%"%k%4%j%:%`(B}
1.1 noro 541:
542: Berlekamp $B%"%k%4%j%:%`(B
543:
1.2 noro 544: \item {\eec $\bmod p$ $B$+$i(B $\bmod p^k$ $B$X$N;}$A>e$2(B}
1.1 noro 545:
546: Euclid $B$N8_=|K!(B, Hensel $B$NJdBj(B
547: \end{itemize}
548:
1.2 noro 549: $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
550:
551: {\ec $B?t3X$r$&$^$/;H$C$?%"%k%4%j%:%`@_7W$,I,MW(B}
1.1 noro 552:
553: \end{slide}
554:
555: \begin{slide}{}
1.2 noro 556: \fbox{\sc 6. $BM-8BBN$N1~MQNc(B : $B0E9f(B}
1.1 noro 557:
1.2 noro 558: \underline{\uc $B%M%C%H%o!<%/>e$G$NDL?.$O4pK\E*$KE{H4$1(B}
1.1 noro 559:
1.2 noro 560: $B<+J,$N?H$O<+J,$G<i$k(B $\Rightarrow$ $BDL?.FbMF$r(B{\ec $B0E9f(B}$B2=(B
1.1 noro 561:
1.2 noro 562: \underline{\uc $B0E9f2=DL?.$N0lNc(B}
1.1 noro 563:
564: \begin{enumerate}
1.2 noro 565: \item $B0E9f2=(B/$BI|9f2=(B{\ec $B80(B}$B$r(B{\ec $B6&M-(B}$B$9$k(B.
1.1 noro 566:
567: \item $BAw?.B&(B : $B80$G0E9f2=(B $\Rightarrow$ $B<u?.B&(B : $B80$GI|9f2=(B
568: \end{enumerate}
569:
1.2 noro 570: \underline{\uc $BLdBj(B : $BDL?.O)$,E{H4$1$N$H$-$K(B,}
571:
572: \underline{\uc $B$I$&$d$C$F80$r6&M-(B?}
1.1 noro 573: \end{slide}
574:
575: \begin{slide}{}
1.2 noro 576: {\Large\parskip 0pt
577: \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}
1.1 noro 578:
579: \begin{itemize}
1.2 noro 580: \item {\eec $B8x3+>pJs(B}
1.1 noro 581:
582: $BBg$-$$AG?t(B $p$, $0 < g < p$ $B$J$k@0?t(B $g$
583:
1.2 noro 584: \item {\eec A $B$5$s$N;E;v(B}
1.1 noro 585:
586: \begin{enumerate}
1.2 noro 587: \item $0 < s_A < p$ $B$J$k@0?t(B {\eec $s_A$} ($BHkL)(B) $B$r:n$k(B.
588: \item $w_A =$ {\eec $g^{s_A} \bmod p$} $B$r(B B $B$5$s$KAw$k(B.
589: \item $s =$ {\eec $w_B^{s_A} \bmod p$} $B$r:n$k(B.
1.1 noro 590: \end{enumerate}
591:
1.2 noro 592: \item {\eec B $B$5$s$N;E;v(B}
1.1 noro 593:
594: \begin{enumerate}
1.2 noro 595: \item $0 < s_B < p$ $B$J$k@0?t(B {\eec $s_B$} ($BHkL)(B) $B$r:n$k(B.
596: \item $w_B =$ {\eec $g^{s_B} \bmod p$} $B$r(B A $B$5$s$KAw$k(B.
597: \item $s =$ {\eec $w_A^{s_B} \bmod p$} $B$r:n$k(B.
1.1 noro 598: \end{enumerate}
599:
1.2 noro 600: \end{itemize}}
1.1 noro 601: \end{slide}
602:
603: \begin{slide}{}
1.2 noro 604: \underline{\uc $BBg;v$JE@(B}
1.1 noro 605:
606: \begin{itemize}
1.2 noro 607: \item {\eec $w_B^{s_A} = w_A^{s_B} \bmod p$}
1.1 noro 608:
609: $B$3$l$G80$,6&M-$G$-$?(B
610:
1.2 noro 611: \item {\eec $w_A$, $w_B$ $B$O0E9f2=$5$l$J$$(B}
1.1 noro 612:
613: $g^{s_A} \bmod p$ $B$+$i(B $s_A$ $B$r5a$a$k$N$OFq$7$$(B
614:
1.2 noro 615: {\ec ($BM-8BBN$N>hK!72$K$*$1$kN%;6BP?tLdBj(B)}
1.1 noro 616:
1.2 noro 617: \item $\overline{a^b} = a^b \bmod p$ $B$O(B {\eec $p$ $BDxEY$N?t$N$+$1;;3d;;$K5"Ce(B}
1.1 noro 618:
619: $\overline{a^{100}} = \overline{(\overline{a^{50}})^2}$,
620: $\overline{a^{50}} = \overline{(\overline{a^{25}})^2}$,
621: $\overline{a^{25}} = \overline{\overline{(\overline{a^{12}})^2} \times \overline{a}}$,
622: $\overline{a^{12}} = \overline{(\overline{a^{6}})^2}$,
623: $\overline{a^{6}} = \overline{(\overline{a^{3}})^2}$,
624: $\overline{a^{3}} = \overline{\overline{(\overline{a})^2} \times \overline{a}}$
625:
626: \end{itemize}
627: \end{slide}
628:
629: \begin{slide}{}
1.2 noro 630: \underline{\uc $BB>$K$b$$$m$$$m$"$k(B}
1.1 noro 631:
632: \begin{itemize}
1.2 noro 633: \item {\eec RSA $B0E9f(B}
1.1 noro 634:
1.2 noro 635: {\eec $BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5(B}$B$rMxMQ(B
1.1 noro 636:
1.2 noro 637: \item {\eec $BBJ1_6J@~0E9f(B}
1.1 noro 638:
639: $BM-8BBN>e$G(B $y^2=x^3+ax+b$ $B$N2r(B $P=(x,y)$ $B$r9M$($k$H(B,
640: $k$ $BG\;;(B $kP$ $B$,Dj5A$G$-$k(B.
641:
1.2 noro 642: {\eec $kP$ $B$+$i(B $k$ $B$r5a$a$k7W;;$NFq$7$5(B}$B$rMxMQ(B
1.1 noro 643: \end{itemize}
644:
1.2 noro 645: $\Rightarrow$ {\ec $BD>@\(B, $B4V@\$K@0?t$N>jM>1i;;$,4XM?(B}
1.1 noro 646: \end{slide}
647:
648: \begin{slide}{}
1.2 noro 649: \fbox{\sc 7. $B$^$H$a(B}
1.1 noro 650:
651: \begin{enumerate}
1.2 noro 652: \item {\eec $BB?9`<00x?tJ,2rDxEY$G$b(B, $B8zN($h$$<B8=$OBgJQ(B}
653:
654: $B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$K(B{\ec $BM-8BBN(B}
655:
656: \item {\eec $B$G$bM-8BBN$J$s$FB>$K2?$NLr$KN)$D$N(B?}
657:
658: $B<B$O(B IT $B<R2q$rN"$G;Y$($F$$$?$j$9$k(B.
1.1 noro 659:
1.2 noro 660: \item {\eec $B?t3X$N2{$N?<$5(B}
1.1 noro 661:
1.2 noro 662: $B8e$K$J$C$F$H$s$G$b$J$$$H$3$m$K1~MQ$5$l$k2DG=@-(B
1.1 noro 663:
1.2 noro 664: $B7W;;$NFq$7$5$,Lr$KN)$D$3$H$b$"$k$H$$$&IT;W5D(B
1.1 noro 665:
666:
667: \end{enumerate}
668:
669: \end{slide}
670:
671: %\begin{slide}{}
672: %\fbox{\bf}
673: %\end{slide}
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