Annotation of OpenXM/doc/sci-semi2001/factorb.tex, Revision 1.6
1.6 ! noro 1: % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.5 2001/07/26 07:55:05 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}
1.1 noro 118: \begin{enumerate}
1.2 noro 119: \item {\eec $B4cNOK!(B} ($B2r$H78?t$N4X78(B)
1.1 noro 120:
121: $x^2+ax+b \Rightarrow$ $BB-$7$F(B $a$, $B$+$1$F(B $b$ $B$K$J$k?t$NAH(B
122:
123: $x^3+ax^2+bx+c$ $B$O$I$&$9$k(B?
124:
1.2 noro 125: \item {\eec $B0x?tDjM}(B}
1.1 noro 126:
127: $BBeF~$7$F(B 0 $B$K$J$k?t$rC5$9(B ($B$I$&$d$C$FC5$9(B?)
128:
1.2 noro 129: \item {\eec $B2r$N8x<0(B}
1.1 noro 130:
1.2 noro 131: $x^2+ax+b=0$ $B$N:,(B ${-b \pm \sqrt{a^2-4b}} \over 2$
1.1 noro 132:
133: $\Rightarrow$ $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D4$Y$k(B
134: \end{enumerate}
135: \end{slide}
136:
137: \begin{slide}{}
1.2 noro 138: \underline{\uc $B4cNOK!$OLdBj$rFq$7$/$7$F$$$k(B}
1.1 noro 139:
140: $BNc(B : $x^2+11508x+28386587$
141:
1.2 noro 142: $28386587=3581\times 7927$ $B$,4cNO$GJ,$+$k?M$O>/$J$$(B($B$H;W$&(B)
143:
144: \vskip 1cm
1.1 noro 145:
1.2 noro 146: \underline{\uc $B2r$N8x<0K!$OM-K>(B}
1.1 noro 147:
148: $(a^2-4b)/4 = 4717584 = 2172^2$ $B$J$i2?$H$+$J$k(B?
149:
1.2 noro 150: $\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$$(B}
1.1 noro 151: \end{slide}
152:
153: \begin{slide}{}
1.2 noro 154: \underline{\uc 3 $B<!0J2<$NB?9`<0(B}
1.1 noro 155:
1.2 noro 156: {\eec $B@0?t>e$GJ,2r$G$-$k$J$i(B, $B0l<!0x;R$r;}$D(B}
1.1 noro 157:
1.2 noro 158: $\Rightarrow$ {\ec $B:,$rC5$9J}K!$,E,MQ$G$-$k(B}
1.1 noro 159:
1.5 noro 160: \begin{itemize}
161: \item {\eec $BFsJ,K!(B}
162:
1.2 noro 163: {\eec $B:,5r(B : $BCf4VCM$NDjM}(B}
1.1 noro 164: $B!V(B$f(a) < 0, f(b) > 0$ $B$J$i(B $a$, $b$ $B$N4V$K(B $f(c)=0$ $B$J$k(B $c$ $B$,$"$k(B.$B!W(B
165:
166: $B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B
167:
1.2 noro 168: \item {\eec Newton $BK!(B}
1.1 noro 169:
170: $BFsJ,K!$h$j$:$C$H9bB.(B
171: \end{itemize}
172:
173: \end{slide}
174:
175: \begin{slide}{}
1.2 noro 176: \underline{\uc 4 $B<!0J>e$N>l9g(B}
1.1 noro 177:
178: $B$$$m$$$m$JJ,2r%Q%?!<%s$,$"$jF@$k(B
179:
180: 4 $B<!(B = 2 $B<!(B $\times$ 2 $B<!(B
181:
1.2 noro 182: $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
1.1 noro 183:
1.2 noro 184: \vskip 1cm
1.1 noro 185:
1.4 noro 186: \underline{\uc $B%3%s%T%e!<%?$K9g$C$?J}K!$O(B?}
187:
188: \begin{itemize}
189: \item {\eec $B!V6a;w!W(B}$B$r$&$^$/;H$&(B
1.1 noro 190:
1.2 noro 191: {\eec $BCf4VCM$NDjM}(B} = {\eec $B<B?t$K$*$1$k6a;w(B} $B$NMxMQ(B
192:
193: $BJL$N6a;w(B $\Rightarrow$ {\ec $B3d$C$?M>$j(B}$B$KCmL\(B
1.4 noro 194:
195: \item $B%3%s%T%e!<%?$O(B{\eec $B7+$jJV$7(B}$B$,F@0U(B
196:
197: $B6a;w$r7+$jJV$7$F@:EY$r>e$2$k(B
198: \end{itemize}
1.1 noro 199: \end{slide}
200:
201: \begin{slide}{}
1.2 noro 202: \fbox{\sc 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}
1.1 noro 203:
1.2 noro 204: \underline{\uc $B86M}(B} : {\eec $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$
1.1 noro 205:
1.2 noro 206: {\eec $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k(B}
1.1 noro 207:
208: $B$?$H$($P(B,
209:
210: \begin{enumerate}
211: \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 $g_1$,
212: $h_1$ $B$r8+$D$1$k(B.
213:
214: \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
1.4 noro 215: $B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$)
1.1 noro 216:
1.2 noro 217: \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.
1.6 ! noro 218: \end{enumerate}
1.1 noro 219: \end{slide}
220:
221: \begin{slide}{}
1.2 noro 222: \underline{\uc $B8@$$$+$($l$P(B...}
223:
224: $B0J2<(B, {\ec $B4JC1$N$?$a(B}, $f(x)$ $B$*$h$S0x;R$N78?t$OA4$F@5$G$"$k$H$9$k(B.
1.1 noro 225:
1.2 noro 226: {\eec $f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$}
1.1 noro 227:
1.2 noro 228: $B$H!V$Y$-5i?tE83+!W(B ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
1.1 noro 229:
1.2 noro 230: {\ec $g(x) = b_0(x)+p\cdot b_1(x)+p^2\cdot b_2(x)+\cdots$}
1.1 noro 231:
1.2 noro 232: {\ec $h(x) = c_0(x)+p\cdot c_1(x)+p^2\cdot c_2(x)+\cdots$}
1.1 noro 233:
1.2 noro 234: ($b_i$, $c_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
1.1 noro 235:
236: $B$H$*$$$F(B $f(x)-g(x)h(x)=0$ $B$+$i(B $b_i$, $c_i$ $B$r7h$a$k(B.
237: \end{slide}
238:
239: \begin{slide}{}
1.2 noro 240: \underline{\uc $B5-9f(B $a \equiv b \bmod M$}
1.1 noro 241:
1.4 noro 242: $M$ $B$r@0?t$H$9$k(B.
1.1 noro 243:
244: \begin{itemize}
245: \item $a,b$ $B$,@0?t$N$H$-(B,
246:
1.4 noro 247: {\eec $a \equiv b \bmod M$} $\Leftrightarrow$
248: {\eec $a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k(B}
1.1 noro 249:
250: \item $a,b$ $B$,@0?t78?tB?9`<0$N$H$-(B
251:
1.4 noro 252: {\eec $a \equiv b \bmod M$} $\Leftrightarrow$
253: {\eec $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k(B}
254:
255: \item {\eec $a$ $B$r(B $M$ $B$G3d$C$?M>$j(B} $B$b(B {\eec $a \bmod M$} $B$H=q$/(B
256:
257: \item $\equiv$ $B$G7k$P$l$?<0(B : {\eec $BEy<0$HF1MM$K07$($k(B}
1.1 noro 258: \end{itemize}
259:
260: \end{slide}
261:
262: \begin{slide}{}
1.2 noro 263: \underline{\uc $b_0(x)$, $c_0(x)$ $B$+$i%9%?!<%H(B}
264:
265: $f-gh$
266:
267: $\quad = a_0-${\ec $b_0c_0$} + ($p$$B$G3d$j@Z$l$kB?9`<0(B)
1.1 noro 268:
1.2 noro 269: $B$@$+$i(B, $f=gh$ $B$J$i(B
1.1 noro 270:
1.2 noro 271: $a_0 \equiv$ {\ec $b_0c_0$} $\bmod p$ $B$N$O$:(B
1.1 noro 272:
1.2 noro 273: \underline{\uc $BNc(B}
1.1 noro 274:
1.2 noro 275: {\eec
1.1 noro 276: \begin{tabbing}
277: $f(x)=$ \= $x^4+17056x^3+72658809x^2$ \\
278: \> $+3504023212x+30603759869$
1.2 noro 279: \end{tabbing}}
1.1 noro 280:
281: $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
282: \end{slide}
283:
284: \begin{slide}{}
1.6 ! noro 285: \underline{\uc $f(x)$ $B$N(B $3$-$B?JE83+(B}
! 286:
! 287: $f(x)=(x^4+x^3+x+2)+3^1\cdot x+$
! 288:
! 289: $3^2(2x^3+x+2)+
! 290: 3^3(x^3+x^2+2x+2)+$
! 291:
! 292: $3^4(x^2+x+1)+
! 293: 3^5 \cdot x^3+
! 294: 3^6(2x^3+x+2)+$
! 295:
! 296: $3^7(x^3+x^2+x)+
! 297: 3^8(2x^3+x^2+2x)+$
! 298:
! 299: $3^9(x^2+2x+1)+
! 300: 3^{11}(2x^2+x+1)+$
! 301:
! 302: $3^{12}(x^2+2x+1)+
! 303: 3^{13}(x+1)+
! 304: 3^{14} \cdot 2+$
! 305:
! 306: $3^{15}(2x^2+x+2)+
! 307: 3^{16}(x^2+2)+
! 308: 3^{17} \cdot 2+$
! 309:
! 310: $3^{19} \cdot 2+
! 311: 3^{20}(x+2)+
! 312: 3^{21} \cdot 2$
! 313: \end{slide}
! 314:
! 315: \begin{slide}{}
1.2 noro 316: \underline{\uc $B0l<!0x;R$,$"$k$+(B?}
1.1 noro 317:
1.2 noro 318: {\ec $b_0(x) = x+q$},
319: {\ec $c_0(x) = x^3+rx^2+sx+t$} $B$H$*$/(B.
1.1 noro 320:
321: $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
322:
1.2 noro 323: {\ec
1.1 noro 324: $\left\{
325: \parbox[c]{6in}{
1.2 noro 326: $q+r \equiv 1 \bmod 3$ \\
327: $qr+s \equiv 0 \bmod 3$ \\
328: $qs+t \equiv 1 \bmod 3$ \\
329: $qt \equiv 2 \bmod 3$}
330: \right.$\\}
1.1 noro 331:
1.2 noro 332: $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$r$I$&F~$l$F$b%@%a(B.
1.1 noro 333:
1.2 noro 334: $B$h$C$F(B, {\eec $B0l<!0x;R$O$J$$(B}.
1.1 noro 335:
336: \end{slide}
337:
338: \begin{slide}{}
1.2 noro 339: \underline{\uc $BFs<!0x;R$O$"$k$+(B? --- $B$^$:(B $b_0$, $c_0$ $B$rC5$9(B}
1.1 noro 340:
1.2 noro 341: {\ec $b_0(x) = x^2+qx+r$},
342: {\ec $c_0(x) = x^2+sx+t$}
1.1 noro 343:
1.2 noro 344: $B$H$*$/$H(B, $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
1.1 noro 345:
1.2 noro 346: {\ec
1.1 noro 347: $\left\{
348: \parbox[c]{6in}{
1.2 noro 349: $q+s \equiv 1 \bmod 3$ \\
350: $qs+r+t \equiv 0 \bmod 3$ \\
351: $qt+rs \equiv 1 \bmod 3$ \\
352: $tr \equiv 2 \bmod 3$}
353: \right.$\\}
1.1 noro 354:
1.2 noro 355: $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$l$P(B
1.1 noro 356:
1.2 noro 357: {\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}
1.1 noro 358:
1.4 noro 359: ($b_0$,$c_0$) $B$N%Z%"$H$7$F$O$3$l$i$OF1$8$b$N(B
1.1 noro 360: \end{slide}
361:
362: \begin{slide}{}
1.4 noro 363: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9>r7o(B}
1.1 noro 364:
1.2 noro 365: {\eec $b_0 = x^2+1$},
366: {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B
1.1 noro 367:
1.4 noro 368: \centerline{\eec $f \equiv b_0c_0 \bmod 3$}
369:
370: $gh \equiv (b_0+3${\ec $b_1$}$)(c_0+3${\ec$c_1$}$) \bmod 3^2$ $B$h$j(B
1.1 noro 371:
1.4 noro 372: $f-gh \equiv a_0-b_0c_0+3(a_1-$
1.2 noro 373: $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3^2$
1.1 noro 374:
1.4 noro 375: $BN>JU$r(B 3 $B$G3d$C$F(B
1.1 noro 376:
1.2 noro 377: ${{f-gh}\over 3} \equiv {{a_0-b_0c_0}\over 3}+(a_1-$
378: $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3$
1.1 noro 379:
380: $B:8JU$O(B $3$ $B$G2?2s$G$b3d$l$k(B $\Rightarrow$ $B1&JU$O(B $3$ $B$G3d$l$k(B
381:
1.6 ! noro 382: $BJd@59`(B {\ec $b_1$}, {\ec $c_1$} : $x^2$ $B$N78?t$O(B 0 $B$H$7$F$h$$(B
1.1 noro 383:
384: \end{slide}
385:
386: \begin{slide}{}
1.2 noro 387: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}
1.1 noro 388:
1.2 noro 389: {\ec $b_1 = qx+r$},
390: {\ec $c_1 = sx+t$} $B$H$*$/(B.
1.1 noro 391:
392: \begin{tabbing}
1.2 noro 393: $B1&JU(B = \= {\ec $-(q+s)x^3-(q+r+t+1)x^2$}\\
394: \> {\ec $-(2q+r+s-1)x-(2r+t)$}
1.1 noro 395: \end{tabbing}
396:
397: $$B1&JU(B \equiv 0 \bmod 3$ $B$h$j(B\\
398:
1.2 noro 399: {\ec
1.1 noro 400: $\left\{
401: \parbox[c]{6in}{
1.2 noro 402: $q+s \equiv 0 \bmod 3$ \\
403: $q+r+t+1 \equiv 0 \bmod 3$ \\
404: $2q+r+s-1 \equiv 0 \bmod 3$ \\
405: $2r+t \equiv 0 \bmod 3$}
406: \right.$\\}
1.1 noro 407:
408: $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B
409:
1.6 ! noro 410: {\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 411:
412: \end{slide}
413:
414: \begin{slide}{}
1.6 ! noro 415: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_2$, $c_2$ $B$O(B $\bmod 3^3$ $B$G(B}
! 416:
! 417: $B$3$l$G(B, {\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
! 418:
! 419: $B<!$O(B $a_2$, $b_2$, $c_2$ $B$^$G$H$C$F(B $\bmod 3^3$ $B$G8+$k(B
! 420:
! 421: \centerline{\eec $f \equiv a_0+3a_1+3^2a_2 \bmod 3^3$}
! 422:
! 423: \centerline{\ec $f \equiv (b_0+3b_1+3^2b_2)(c_0+3c_1+3^2c_2) \bmod 3^3$}
! 424:
! 425: $B$+$i(B {$((a_0+3a_1)-(b_0+3b_1)(c_0+3c_1))+$}
! 426:
! 427: \centerline{$3^2(a_2-(c_0b_2+b_0c_2)) \equiv 0 \bmod 3^3$}
! 428:
! 429: $BN>JU$r(B $3^2$ $B$G3d$C$F(B, {\ec $b_2=qx+r$}, {\ec $c_2=sx+t$}
1.1 noro 430:
1.6 ! noro 431: $\Rightarrow$ $k=1$ $B$HF1MM$N(B{\eec $BO"N)0l<!9gF1<0(B}$B$rF@$k(B
1.1 noro 432:
1.6 ! noro 433: \end{slide}
! 434:
! 435: \begin{slide}{}
! 436: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}
1.1 noro 437:
438: $B0J2<F1MM$K(B,
439:
1.4 noro 440: \centerline{\ec $b_i = qx+r, c_i = sx+t$}
1.1 noro 441:
1.4 noro 442: ($i=2,3,\ldots$) $B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r=g<!(B
443: $B2r$$$F$$$1$P(B
1.1 noro 444:
1.2 noro 445: \centerline{\eec $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}
1.1 noro 446:
447: $B$9$J$o$A(B
448:
1.2 noro 449: \centerline{\eec $f \equiv g_kh_k \bmod 3^k$}
1.1 noro 450:
1.6 ! noro 451: $B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B.
1.1 noro 452: \end{slide}
453:
454: \begin{slide}{}
1.2 noro 455: \underline{\uc $(g_k, h_k)$ $B$NI=(B}
1.1 noro 456:
457: {\large
458: \begin{tabular} { c | c c }
459: $k$ & $g_k$ & $h_k$ \\ \hline
460: 1&$x^2+1$&$x^2+x+2$\\ \hline
461: 2&$x^2+4$&$x^2+x+5$\\ \hline
462: 3&$x^2+18x+4$&$x^2+x+5$\\ \hline
463: 4&$x^2+45x+4$&$x^2+x+59$\\ \hline
464: 5&$x^2+45x+166$&$x^2+x+140$\\ \hline
465: 6&$x^2+531x+409$&$x^2+487x+626$\\ \hline
466: 7&$x^2+1260x+1867$&$x^2+487x+1355$\\ \hline
467: 8&$x^2+1260x+4054$&$x^2+2674x+1355$\\ \hline
468: 9&$x^2+7821x+10615$&$x^2+9235x+7916$\\ \hline
469: 10&$x^2+7821x+30298$&$x^2+9235x+47282$\\ \hline
470: 11&$x^2+7821x+89347$&$x^2+9235x+165380$\\ \hline
1.2 noro 471: 12&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
472: 13&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
1.1 noro 473: \end{tabular}}
474: \end{slide}
475:
476: \begin{slide}{}
1.2 noro 477: \underline{\uc $\bmod 3^k$ $B$G$N0x;R$+$i??$N0x;R$X(B}
1.1 noro 478:
1.2 noro 479: $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}
1.1 noro 480:
1.4 noro 481: $\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0}
1.1 noro 482:
1.2 noro 483: {\eec
1.4 noro 484: $f(x) =$
1.1 noro 485:
1.4 noro 486: $ (x^2+7821x+89347)(x^2+9235x+342527)$}
1.1 noro 487:
1.2 noro 488: \underline{\uc $B<B:]$K$O(B...}
1.1 noro 489:
490: \begin{itemize}
491: \item $BIi$N78?t$N>l9g$r07$&$?$a$N9)IW$,I,MW(B
492:
493: \item $B<:GT$N2DG=@-$b$"$k$N$G(B, $k$ $B$r$I$3$^$G>e$2$l$P$$$$$+$N>e8B$,I,MW(B
494: \end{itemize}
495: \end{slide}
496:
497: \begin{slide}{}
1.4 noro 498: \underline{\uc $\bmod p$ $B$G$NJ,2r$,LdBj(B}
1.1 noro 499:
1.2 noro 500: $B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B
1.1 noro 501:
502: \begin{itemize}
1.2 noro 503: \item {\eec $k > 1$}
1.1 noro 504:
1.2 noro 505: $BO"N)0l<!J}Dx<0(B ($B<B:]$K$O9gF1<0(B)
1.1 noro 506:
1.2 noro 507: \item {\eec $k = 1$}
1.1 noro 508:
1.2 noro 509: $B0l<!J}Dx<0$G$J$$(B
510:
511: $\Rightarrow$ $B$7$i$_$D$V$7$G2r$/$N$O$"$^$j$K8zN((B
1.1 noro 512: $B$,$o$k$$(B ($B$$$/$i%3%s%T%e!<%?$G$b(B)
513: \end{itemize}
514: \end{slide}
515:
516: \begin{slide}{}
1.2 noro 517: \fbox{\sc 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }
518:
519: $p$ $B$,(B{\ec $BAG?t(B}$B$N$H$-(B,
1.1 noro 520:
1.2 noro 521: {\eec $GF(p) = \{0,1,\cdots,p-1\}$} $B$K(B, $+$, $-$, $\times$ $B$r(B
522: {\eec $B!V7k2L$r(B $p$ $B$G(B $B3d$C$?M>$j!W(B}$B$GDj5A$9$k$H(B
1.1 noro 523:
524: \begin{enumerate}
525: \item $B2C8:>h;;$GJD$8$F$$$k(B.
1.2 noro 526: \item {\eec 0 $B0J30$N85$G3d;;$,$G$-$k(B. }
1.1 noro 527:
1.2 noro 528: $B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$J$k(B $b$ $B$,$"$k!W(B
1.1 noro 529: \end{enumerate}
530:
1.2 noro 531: $B$9$J$o$A(B, {\eec $GF(p)$ $B$OBN(B($B%?%$(B)}
1.1 noro 532:
1.2 noro 533: $B85$N8D?t$,M-8B8D(B ($p$ $B8D(B)$B$J$N$G(B {\ec $BM-8BBN(B} $B$H$h$V(B.
1.1 noro 534:
535: \end{slide}
536:
537: \begin{slide}{}
1.2 noro 538: \underline{\uc $k=1$ $\Rightarrow$ $BM-8BBN>e$G$N0x?tJ,2r(B}
1.1 noro 539:
540: $a_0 \equiv f \bmod p$ $B$r(B $GF(p)$ $B78?tB?9`<0$H$_$k(B.
541:
542: $\Rightarrow$ $a_0 \equiv b_0c_0 \bmod p$ $B$H$J$k(B $b_0$, $c_0$ $B$r(B
543: $B5a$a$k$3$H$O(B, $GF(p)$ $B>e$G$N0x?tJ,2r$KAjEv(B
544:
1.2 noro 545: $\Rightarrow$ {\eec $B<B$O$h$$%"%k%4%j%:%`$,$"$k(B}
546:
547: \vskip 1cm
1.1 noro 548:
1.2 noro 549: \underline{\uc $k > 1$ $\Rightarrow$ $BM-8BBN>e$G$NO"N)0l<!J}Dx<05a2r(B}
1.1 noro 550:
551: $B<B:]$K$O(B, $k=1$ $B$N7k2L$+$i5!3#E*$K7W;;$G$-$k(B.
552: \end{slide}
553:
554: \begin{slide}{}
1.2 noro 555: \underline{\uc $B0x?tJ,2r$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}
1.1 noro 556:
557: \begin{enumerate}
1.2 noro 558: \item {\eec $B$h$$AG?t(B $p$ $B$rA*$s$G(B $f \bmod p$ $B$r0x?tJ,2r(B}
1.1 noro 559:
1.2 noro 560: {\eec $B!V$h$$!W(B} $B$H$O(B
1.1 noro 561:
1.2 noro 562: \begin{itemize}
563: \item $f$ $B$N:G9b<!78?t$r3d$i$J$$(B
564:
565: \item $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.
566: \end{itemize}
1.1 noro 567:
1.2 noro 568: \item {\eec $B<!$r7+$jJV$7(B}
1.1 noro 569:
570: \begin{enumerate}
571: \item $GF(p)$ $B>e$N0x;R$rFsAH$KJ,$1$k(B
572:
573: \item $B3FAH$N@Q$r(B $g_1$, $h_1$ $B$H$9$k(B.
574:
575: \item $f \equiv g_kh_k \bmod p^k$ $B$J$k(B $g_k$, $h_k$ $B$r:n$k(B
576:
577: \item $B78?t$N@5Ii$rD4@a$7$F;n$73d$j(B
578: \end{enumerate}
579:
580: \end{enumerate}
581: \end{slide}
582:
583: \begin{slide}{}
1.2 noro 584: \underline{\uc $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}
1.1 noro 585:
586: \begin{itemize}
1.2 noro 587: \item {\eec $BBN$N>e$G$N0x?tJ,2r$N0l0U@-(B}
1.1 noro 588:
589: $BBN>e$NB?9`<04D$N@-<A(B
590:
1.2 noro 591: \item {\eec $BM-8BBN>e$G$N0x?tJ,2r%"%k%4%j%:%`(B}
1.1 noro 592:
593: Berlekamp $B%"%k%4%j%:%`(B
594:
1.2 noro 595: \item {\eec $\bmod p$ $B$+$i(B $\bmod p^k$ $B$X$N;}$A>e$2(B}
1.1 noro 596:
597: Euclid $B$N8_=|K!(B, Hensel $B$NJdBj(B
598: \end{itemize}
599:
1.2 noro 600: $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
601:
602: {\ec $B?t3X$r$&$^$/;H$C$?%"%k%4%j%:%`@_7W$,I,MW(B}
1.1 noro 603:
604: \end{slide}
605:
606: \begin{slide}{}
1.2 noro 607: \fbox{\sc 6. $BM-8BBN$N1~MQNc(B : $B0E9f(B}
1.1 noro 608:
1.2 noro 609: \underline{\uc $B%M%C%H%o!<%/>e$G$NDL?.$O4pK\E*$KE{H4$1(B}
1.1 noro 610:
1.2 noro 611: $B<+J,$N?H$O<+J,$G<i$k(B $\Rightarrow$ $BDL?.FbMF$r(B{\ec $B0E9f(B}$B2=(B
1.1 noro 612:
1.2 noro 613: \underline{\uc $B0E9f2=DL?.$N0lNc(B}
1.1 noro 614:
615: \begin{enumerate}
1.2 noro 616: \item $B0E9f2=(B/$BI|9f2=(B{\ec $B80(B}$B$r(B{\ec $B6&M-(B}$B$9$k(B.
1.1 noro 617:
618: \item $BAw?.B&(B : $B80$G0E9f2=(B $\Rightarrow$ $B<u?.B&(B : $B80$GI|9f2=(B
619: \end{enumerate}
620:
1.2 noro 621: \underline{\uc $BLdBj(B : $BDL?.O)$,E{H4$1$N$H$-$K(B,}
622:
623: \underline{\uc $B$I$&$d$C$F80$r6&M-(B?}
1.1 noro 624: \end{slide}
625:
626: \begin{slide}{}
1.2 noro 627: \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}
1.1 noro 628:
629: \begin{itemize}
1.2 noro 630: \item {\eec $B8x3+>pJs(B}
1.1 noro 631:
632: $BBg$-$$AG?t(B $p$, $0 < g < p$ $B$J$k@0?t(B $g$
633:
1.2 noro 634: \item {\eec A $B$5$s$N;E;v(B}
1.1 noro 635:
636: \begin{enumerate}
1.2 noro 637: \item $0 < s_A < p$ $B$J$k@0?t(B {\eec $s_A$} ($BHkL)(B) $B$r:n$k(B.
638: \item $w_A =$ {\eec $g^{s_A} \bmod p$} $B$r(B B $B$5$s$KAw$k(B.
1.4 noro 639: \item $B<u$1<h$C$?(B $w_B$ $B$+$i(B $s =$ {\eec $w_B^{s_A} \bmod p$} $B$r:n$k(B.
1.1 noro 640: \end{enumerate}
641:
1.2 noro 642: \item {\eec B $B$5$s$N;E;v(B}
1.1 noro 643:
644: \begin{enumerate}
1.2 noro 645: \item $0 < s_B < p$ $B$J$k@0?t(B {\eec $s_B$} ($BHkL)(B) $B$r:n$k(B.
646: \item $w_B =$ {\eec $g^{s_B} \bmod p$} $B$r(B A $B$5$s$KAw$k(B.
1.4 noro 647: \item $B<u$1<h$C$?(B $w_A$ $B$+$i(B $s =$ {\eec $w_A^{s_B} \bmod p$} $B$r:n$k(B.
1.1 noro 648: \end{enumerate}
649:
1.6 ! noro 650: \end{itemize}
1.1 noro 651: \end{slide}
652:
653: \begin{slide}{}
1.2 noro 654: \underline{\uc $BBg;v$JE@(B}
1.1 noro 655:
656: \begin{itemize}
1.2 noro 657: \item {\eec $w_B^{s_A} = w_A^{s_B} \bmod p$}
1.1 noro 658:
659: $B$3$l$G80$,6&M-$G$-$?(B
660:
1.4 noro 661: \item {\eec $w_A$, $w_B$ $B$O0E9f2=$NI,MW$J$7(B}
1.1 noro 662:
663: $g^{s_A} \bmod p$ $B$+$i(B $s_A$ $B$r5a$a$k$N$OFq$7$$(B
664:
1.2 noro 665: {\ec ($BM-8BBN$N>hK!72$K$*$1$kN%;6BP?tLdBj(B)}
1.1 noro 666:
1.2 noro 667: \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 668:
669: $\overline{a^{100}} = \overline{(\overline{a^{50}})^2}$,
670: $\overline{a^{50}} = \overline{(\overline{a^{25}})^2}$,
671: $\overline{a^{25}} = \overline{\overline{(\overline{a^{12}})^2} \times \overline{a}}$,
672: $\overline{a^{12}} = \overline{(\overline{a^{6}})^2}$,
673: $\overline{a^{6}} = \overline{(\overline{a^{3}})^2}$,
674: $\overline{a^{3}} = \overline{\overline{(\overline{a})^2} \times \overline{a}}$
675:
676: \end{itemize}
677: \end{slide}
678:
679: \begin{slide}{}
1.2 noro 680: \underline{\uc $BB>$K$b$$$m$$$m$"$k(B}
1.1 noro 681:
682: \begin{itemize}
1.2 noro 683: \item {\eec RSA $B0E9f(B}
1.1 noro 684:
1.2 noro 685: {\eec $BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5(B}$B$rMxMQ(B
1.1 noro 686:
1.2 noro 687: \item {\eec $BBJ1_6J@~0E9f(B}
1.1 noro 688:
689: $BM-8BBN>e$G(B $y^2=x^3+ax+b$ $B$N2r(B $P=(x,y)$ $B$r9M$($k$H(B,
690: $k$ $BG\;;(B $kP$ $B$,Dj5A$G$-$k(B.
691:
1.2 noro 692: {\eec $kP$ $B$+$i(B $k$ $B$r5a$a$k7W;;$NFq$7$5(B}$B$rMxMQ(B
1.1 noro 693: \end{itemize}
694:
1.2 noro 695: $\Rightarrow$ {\ec $BD>@\(B, $B4V@\$K@0?t$N>jM>1i;;$,4XM?(B}
1.1 noro 696: \end{slide}
697:
698: \begin{slide}{}
1.2 noro 699: \fbox{\sc 7. $B$^$H$a(B}
1.1 noro 700:
701: \begin{enumerate}
1.2 noro 702: \item {\eec $BB?9`<00x?tJ,2rDxEY$G$b(B, $B8zN($h$$<B8=$OBgJQ(B}
703:
704: $B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$K(B{\ec $BM-8BBN(B}
705:
706: \item {\eec $B$G$bM-8BBN$J$s$FB>$K2?$NLr$KN)$D$N(B?}
707:
708: $B<B$O(B IT $B<R2q$rN"$G;Y$($F$$$?$j$9$k(B.
1.1 noro 709:
1.2 noro 710: \item {\eec $B?t3X$N2{$N?<$5(B}
1.1 noro 711:
1.2 noro 712: $B8e$K$J$C$F$H$s$G$b$J$$$H$3$m$K1~MQ$5$l$k2DG=@-(B
1.1 noro 713:
1.2 noro 714: $B7W;;$NFq$7$5$,Lr$KN)$D$3$H$b$"$k$H$$$&IT;W5D(B
1.1 noro 715:
716:
717: \end{enumerate}
718:
719: \end{slide}
720:
721: %\begin{slide}{}
722: %\fbox{\bf}
723: %\end{slide}
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