Annotation of OpenXM/doc/sci-semi2001/factorb.tex, Revision 1.4
1.4 ! noro 1: % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.3 2001/07/24 09:35:54 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:
1.4 ! noro 190: \underline{\uc $B%3%s%T%e!<%?$K9g$C$?J}K!$O(B?}
! 191:
! 192: \begin{itemize}
! 193: \item {\eec $B!V6a;w!W(B}$B$r$&$^$/;H$&(B
1.1 noro 194:
1.2 noro 195: {\eec $BCf4VCM$NDjM}(B} = {\eec $B<B?t$K$*$1$k6a;w(B} $B$NMxMQ(B
196:
197: $BJL$N6a;w(B $\Rightarrow$ {\ec $B3d$C$?M>$j(B}$B$KCmL\(B
1.4 ! noro 198:
! 199: \item $B%3%s%T%e!<%?$O(B{\eec $B7+$jJV$7(B}$B$,F@0U(B
! 200:
! 201: $B6a;w$r7+$jJV$7$F@:EY$r>e$2$k(B
! 202: \end{itemize}
1.1 noro 203: \end{slide}
204:
205: \begin{slide}{}
1.2 noro 206: \fbox{\sc 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}
207: {\Large\parskip 0pt
1.1 noro 208:
1.2 noro 209: \underline{\uc $B86M}(B} : {\eec $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$
1.1 noro 210:
1.2 noro 211: {\eec $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k(B}
1.1 noro 212:
1.2 noro 213: ({\eec $m$ $B$O==J,Bg$-$$@0?t$G3d$j@Z$l$k(B})
1.1 noro 214:
215: $B$?$H$($P(B,
216:
217: \begin{enumerate}
218: \item $B:G=i(B, $f(x)-g_1(x)h_1(x)$ $B$N78?t$,@0?t(B $p$ $B$G3d$j@Z$l$k$h$&$J(B $g_1$,
219: $h_1$ $B$r8+$D$1$k(B.
220:
221: \item $f(x)-g_k(x)h_k(x)$ $B$N78?t$,(B $p^k$ $B$G3d$j@Z$l$k$h$&$K(B $g_k$, $h_k$ $B$r(B
1.4 ! noro 222: $B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$)
1.1 noro 223:
1.2 noro 224: \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.
225: \end{enumerate}}
1.1 noro 226: \end{slide}
227:
228: \begin{slide}{}
1.2 noro 229: \underline{\uc $B8@$$$+$($l$P(B...}
230:
231: $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 232:
1.2 noro 233: {\eec $f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$}
1.1 noro 234:
1.2 noro 235: $B$H!V$Y$-5i?tE83+!W(B ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
1.1 noro 236:
1.2 noro 237: {\ec $g(x) = b_0(x)+p\cdot b_1(x)+p^2\cdot b_2(x)+\cdots$}
1.1 noro 238:
1.2 noro 239: {\ec $h(x) = c_0(x)+p\cdot c_1(x)+p^2\cdot c_2(x)+\cdots$}
1.1 noro 240:
1.2 noro 241: ($b_i$, $c_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
1.1 noro 242:
243: $B$H$*$$$F(B $f(x)-g(x)h(x)=0$ $B$+$i(B $b_i$, $c_i$ $B$r7h$a$k(B.
244: \end{slide}
245:
246: \begin{slide}{}
1.2 noro 247: \underline{\uc $B5-9f(B $a \equiv b \bmod M$}
1.1 noro 248:
1.4 ! noro 249: $M$ $B$r@0?t$H$9$k(B.
1.1 noro 250:
251: \begin{itemize}
252: \item $a,b$ $B$,@0?t$N$H$-(B,
253:
1.4 ! noro 254: {\eec $a \equiv b \bmod M$} $\Leftrightarrow$
! 255: {\eec $a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k(B}
1.1 noro 256:
257: \item $a,b$ $B$,@0?t78?tB?9`<0$N$H$-(B
258:
1.4 ! noro 259: {\eec $a \equiv b \bmod M$} $\Leftrightarrow$
! 260: {\eec $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k(B}
! 261:
! 262: \item {\eec $a$ $B$r(B $M$ $B$G3d$C$?M>$j(B} $B$b(B {\eec $a \bmod M$} $B$H=q$/(B
! 263:
! 264: \item $\equiv$ $B$G7k$P$l$?<0(B : {\eec $BEy<0$HF1MM$K07$($k(B}
1.1 noro 265: \end{itemize}
266:
267: \end{slide}
268:
269: \begin{slide}{}
1.2 noro 270: \underline{\uc $b_0(x)$, $c_0(x)$ $B$+$i%9%?!<%H(B}
271:
272: $f-gh$
273:
274: $\quad = a_0-${\ec $b_0c_0$} + ($p$$B$G3d$j@Z$l$kB?9`<0(B)
1.1 noro 275:
1.2 noro 276: $B$@$+$i(B, $f=gh$ $B$J$i(B
1.1 noro 277:
1.2 noro 278: $a_0 \equiv$ {\ec $b_0c_0$} $\bmod p$ $B$N$O$:(B
1.1 noro 279:
1.2 noro 280: \underline{\uc $BNc(B}
1.1 noro 281:
1.2 noro 282: {\eec
1.1 noro 283: \begin{tabbing}
284: $f(x)=$ \= $x^4+17056x^3+72658809x^2$ \\
285: \> $+3504023212x+30603759869$
1.2 noro 286: \end{tabbing}}
1.1 noro 287:
288: $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
289: \end{slide}
290:
291: \begin{slide}{}
1.2 noro 292: \underline{\uc $B0l<!0x;R$,$"$k$+(B?}
1.1 noro 293:
1.2 noro 294: {\ec $b_0(x) = x+q$},
295: {\ec $c_0(x) = x^3+rx^2+sx+t$} $B$H$*$/(B.
1.1 noro 296:
297: $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
298:
1.2 noro 299: {\ec
1.1 noro 300: $\left\{
301: \parbox[c]{6in}{
1.2 noro 302: $q+r \equiv 1 \bmod 3$ \\
303: $qr+s \equiv 0 \bmod 3$ \\
304: $qs+t \equiv 1 \bmod 3$ \\
305: $qt \equiv 2 \bmod 3$}
306: \right.$\\}
1.1 noro 307:
1.2 noro 308: $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$r$I$&F~$l$F$b%@%a(B.
1.1 noro 309:
1.2 noro 310: $B$h$C$F(B, {\eec $B0l<!0x;R$O$J$$(B}.
1.1 noro 311:
312: \end{slide}
313:
314: \begin{slide}{}
1.2 noro 315: \underline{\uc $BFs<!0x;R$O$"$k$+(B? --- $B$^$:(B $b_0$, $c_0$ $B$rC5$9(B}
1.1 noro 316:
1.2 noro 317: {\ec $b_0(x) = x^2+qx+r$},
318: {\ec $c_0(x) = x^2+sx+t$}
1.1 noro 319:
1.2 noro 320: $B$H$*$/$H(B, $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
1.1 noro 321:
1.2 noro 322: {\ec
1.1 noro 323: $\left\{
324: \parbox[c]{6in}{
1.2 noro 325: $q+s \equiv 1 \bmod 3$ \\
326: $qs+r+t \equiv 0 \bmod 3$ \\
327: $qt+rs \equiv 1 \bmod 3$ \\
328: $tr \equiv 2 \bmod 3$}
329: \right.$\\}
1.1 noro 330:
1.2 noro 331: $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$l$P(B
1.1 noro 332:
1.2 noro 333: {\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}
1.1 noro 334:
1.4 ! noro 335: ($b_0$,$c_0$) $B$N%Z%"$H$7$F$O$3$l$i$OF1$8$b$N(B
1.1 noro 336: \end{slide}
337:
338: \begin{slide}{}
1.4 ! noro 339: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9>r7o(B}
1.1 noro 340:
1.2 noro 341: {\Large\parskip 0pt
342: {\eec $b_0 = x^2+1$},
343: {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B
1.1 noro 344:
1.4 ! noro 345: \centerline{\eec $f \equiv b_0c_0 \bmod 3$}
! 346:
! 347: $gh \equiv (b_0+3${\ec $b_1$}$)(c_0+3${\ec$c_1$}$) \bmod 3^2$ $B$h$j(B
1.1 noro 348:
1.4 ! noro 349: $f-gh \equiv a_0-b_0c_0+3(a_1-$
1.2 noro 350: $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3^2$
1.1 noro 351:
1.4 ! noro 352: $BN>JU$r(B 3 $B$G3d$C$F(B
1.1 noro 353:
1.2 noro 354: ${{f-gh}\over 3} \equiv {{a_0-b_0c_0}\over 3}+(a_1-$
355: $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3$
1.1 noro 356:
357: $B:8JU$O(B $3$ $B$G2?2s$G$b3d$l$k(B $\Rightarrow$ $B1&JU$O(B $3$ $B$G3d$l$k(B
358:
1.2 noro 359: $BJd@59`(B {\ec $b_1$}, {\ec $c_1$} : $x^2$ $B$N78?t$O(B 0 $B$H$7$F$h$$(B}
1.1 noro 360:
361: \end{slide}
362:
363: \begin{slide}{}
1.2 noro 364: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}
1.1 noro 365:
1.2 noro 366: {\Large\parskip 0pt
367: {\ec $b_1 = qx+r$},
368: {\ec $c_1 = sx+t$} $B$H$*$/(B.
1.1 noro 369:
370: \begin{tabbing}
1.2 noro 371: $B1&JU(B = \= {\ec $-(q+s)x^3-(q+r+t+1)x^2$}\\
372: \> {\ec $-(2q+r+s-1)x-(2r+t)$}
1.1 noro 373: \end{tabbing}
374:
375: $$B1&JU(B \equiv 0 \bmod 3$ $B$h$j(B\\
376:
1.2 noro 377: {\ec
1.1 noro 378: $\left\{
379: \parbox[c]{6in}{
1.2 noro 380: $q+s \equiv 0 \bmod 3$ \\
381: $q+r+t+1 \equiv 0 \bmod 3$ \\
382: $2q+r+s-1 \equiv 0 \bmod 3$ \\
383: $2r+t \equiv 0 \bmod 3$}
384: \right.$\\}
1.1 noro 385:
386: $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B
387:
1.2 noro 388: {\eec $(q,r,s,t) = (0,1,0,1)$} $B$9$J$o$A(B {\eec $b_1 = 1$}, {\eec $c_1 = 1$}}
1.1 noro 389:
390: \end{slide}
391:
392: \begin{slide}{}
1.2 noro 393: \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}
1.1 noro 394:
1.2 noro 395: {\Large\parskip 0pt
1.1 noro 396: $B$3$l$G(B,
397:
1.2 noro 398: \centerline{\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
1.1 noro 399:
400: $B0J2<F1MM$K(B,
401:
1.4 ! noro 402: \centerline{\ec $b_i = qx+r, c_i = sx+t$}
1.1 noro 403:
1.4 ! noro 404: ($i=2,3,\ldots$) $B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r=g<!(B
! 405: $B2r$$$F$$$1$P(B
1.1 noro 406:
1.2 noro 407: \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 408:
409: $B$9$J$o$A(B
410:
1.2 noro 411: \centerline{\eec $f \equiv g_kh_k \bmod 3^k$}
1.1 noro 412:
1.2 noro 413: $B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B. }
1.1 noro 414:
415: \end{slide}
416:
417: \begin{slide}{}
1.2 noro 418: \underline{\uc $(g_k, h_k)$ $B$NI=(B}
1.1 noro 419:
420: {\large
421: \begin{tabular} { c | c c }
422: $k$ & $g_k$ & $h_k$ \\ \hline
423: 1&$x^2+1$&$x^2+x+2$\\ \hline
424: 2&$x^2+4$&$x^2+x+5$\\ \hline
425: 3&$x^2+18x+4$&$x^2+x+5$\\ \hline
426: 4&$x^2+45x+4$&$x^2+x+59$\\ \hline
427: 5&$x^2+45x+166$&$x^2+x+140$\\ \hline
428: 6&$x^2+531x+409$&$x^2+487x+626$\\ \hline
429: 7&$x^2+1260x+1867$&$x^2+487x+1355$\\ \hline
430: 8&$x^2+1260x+4054$&$x^2+2674x+1355$\\ \hline
431: 9&$x^2+7821x+10615$&$x^2+9235x+7916$\\ \hline
432: 10&$x^2+7821x+30298$&$x^2+9235x+47282$\\ \hline
433: 11&$x^2+7821x+89347$&$x^2+9235x+165380$\\ \hline
1.2 noro 434: 12&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
435: 13&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
1.1 noro 436: \end{tabular}}
437: \end{slide}
438:
439: \begin{slide}{}
1.2 noro 440: \underline{\uc $\bmod 3^k$ $B$G$N0x;R$+$i??$N0x;R$X(B}
1.1 noro 441:
1.2 noro 442: $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}
1.1 noro 443:
1.4 ! noro 444: $\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0}
1.1 noro 445:
1.2 noro 446: {\eec
1.4 ! noro 447: $f(x) =$
1.1 noro 448:
1.4 ! noro 449: $ (x^2+7821x+89347)(x^2+9235x+342527)$}
1.1 noro 450:
1.2 noro 451: \underline{\uc $B<B:]$K$O(B...}
1.1 noro 452:
453: \begin{itemize}
454: \item $BIi$N78?t$N>l9g$r07$&$?$a$N9)IW$,I,MW(B
455:
456: \item $B<:GT$N2DG=@-$b$"$k$N$G(B, $k$ $B$r$I$3$^$G>e$2$l$P$$$$$+$N>e8B$,I,MW(B
457: \end{itemize}
458: \end{slide}
459:
460: \begin{slide}{}
1.4 ! noro 461: \underline{\uc $\bmod p$ $B$G$NJ,2r$,LdBj(B}
1.1 noro 462:
1.2 noro 463: $B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B
1.1 noro 464:
465: \begin{itemize}
1.2 noro 466: \item {\eec $k > 1$}
1.1 noro 467:
1.2 noro 468: $BO"N)0l<!J}Dx<0(B ($B<B:]$K$O9gF1<0(B)
1.1 noro 469:
1.2 noro 470: \item {\eec $k = 1$}
1.1 noro 471:
1.2 noro 472: $B0l<!J}Dx<0$G$J$$(B
473:
474: $\Rightarrow$ $B$7$i$_$D$V$7$G2r$/$N$O$"$^$j$K8zN((B
1.1 noro 475: $B$,$o$k$$(B ($B$$$/$i%3%s%T%e!<%?$G$b(B)
476: \end{itemize}
477: \end{slide}
478:
479: \begin{slide}{}
1.2 noro 480: \fbox{\sc 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }
481:
482: $p$ $B$,(B{\ec $BAG?t(B}$B$N$H$-(B,
1.1 noro 483:
1.2 noro 484: {\eec $GF(p) = \{0,1,\cdots,p-1\}$} $B$K(B, $+$, $-$, $\times$ $B$r(B
485: {\eec $B!V7k2L$r(B $p$ $B$G(B $B3d$C$?M>$j!W(B}$B$GDj5A$9$k$H(B
1.1 noro 486:
487: \begin{enumerate}
488: \item $B2C8:>h;;$GJD$8$F$$$k(B.
1.2 noro 489: \item {\eec 0 $B0J30$N85$G3d;;$,$G$-$k(B. }
1.1 noro 490:
1.2 noro 491: $B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$J$k(B $b$ $B$,$"$k!W(B
1.1 noro 492: \end{enumerate}
493:
1.2 noro 494: $B$9$J$o$A(B, {\eec $GF(p)$ $B$OBN(B($B%?%$(B)}
1.1 noro 495:
1.2 noro 496: $B85$N8D?t$,M-8B8D(B ($p$ $B8D(B)$B$J$N$G(B {\ec $BM-8BBN(B} $B$H$h$V(B.
1.1 noro 497:
498: \end{slide}
499:
500: \begin{slide}{}
1.2 noro 501: \underline{\uc $k=1$ $\Rightarrow$ $BM-8BBN>e$G$N0x?tJ,2r(B}
1.1 noro 502:
503: $a_0 \equiv f \bmod p$ $B$r(B $GF(p)$ $B78?tB?9`<0$H$_$k(B.
504:
505: $\Rightarrow$ $a_0 \equiv b_0c_0 \bmod p$ $B$H$J$k(B $b_0$, $c_0$ $B$r(B
506: $B5a$a$k$3$H$O(B, $GF(p)$ $B>e$G$N0x?tJ,2r$KAjEv(B
507:
1.2 noro 508: $\Rightarrow$ {\eec $B<B$O$h$$%"%k%4%j%:%`$,$"$k(B}
509:
510: \vskip 1cm
1.1 noro 511:
1.2 noro 512: \underline{\uc $k > 1$ $\Rightarrow$ $BM-8BBN>e$G$NO"N)0l<!J}Dx<05a2r(B}
1.1 noro 513:
514: $B<B:]$K$O(B, $k=1$ $B$N7k2L$+$i5!3#E*$K7W;;$G$-$k(B.
515: \end{slide}
516:
517: \begin{slide}{}
1.2 noro 518: \underline{\uc $B0x?tJ,2r$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}
1.1 noro 519:
520: \begin{enumerate}
1.2 noro 521: \item {\eec $B$h$$AG?t(B $p$ $B$rA*$s$G(B $f \bmod p$ $B$r0x?tJ,2r(B}
1.1 noro 522:
1.2 noro 523: {\eec $B!V$h$$!W(B} $B$H$O(B
1.1 noro 524:
1.2 noro 525: \begin{itemize}
526: \item $f$ $B$N:G9b<!78?t$r3d$i$J$$(B
527:
528: \item $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.
529: \end{itemize}
1.1 noro 530:
1.2 noro 531: \item {\eec $B<!$r7+$jJV$7(B}
1.1 noro 532:
533: \begin{enumerate}
534: \item $GF(p)$ $B>e$N0x;R$rFsAH$KJ,$1$k(B
535:
536: \item $B3FAH$N@Q$r(B $g_1$, $h_1$ $B$H$9$k(B.
537:
538: \item $f \equiv g_kh_k \bmod p^k$ $B$J$k(B $g_k$, $h_k$ $B$r:n$k(B
539:
540: \item $B78?t$N@5Ii$rD4@a$7$F;n$73d$j(B
541: \end{enumerate}
542:
543: \end{enumerate}
544: \end{slide}
545:
546: \begin{slide}{}
1.2 noro 547: \underline{\uc $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}
1.1 noro 548:
549: \begin{itemize}
1.2 noro 550: \item {\eec $BBN$N>e$G$N0x?tJ,2r$N0l0U@-(B}
1.1 noro 551:
552: $BBN>e$NB?9`<04D$N@-<A(B
553:
1.2 noro 554: \item {\eec $BM-8BBN>e$G$N0x?tJ,2r%"%k%4%j%:%`(B}
1.1 noro 555:
556: Berlekamp $B%"%k%4%j%:%`(B
557:
1.2 noro 558: \item {\eec $\bmod p$ $B$+$i(B $\bmod p^k$ $B$X$N;}$A>e$2(B}
1.1 noro 559:
560: Euclid $B$N8_=|K!(B, Hensel $B$NJdBj(B
561: \end{itemize}
562:
1.2 noro 563: $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
564:
565: {\ec $B?t3X$r$&$^$/;H$C$?%"%k%4%j%:%`@_7W$,I,MW(B}
1.1 noro 566:
567: \end{slide}
568:
569: \begin{slide}{}
1.2 noro 570: \fbox{\sc 6. $BM-8BBN$N1~MQNc(B : $B0E9f(B}
1.1 noro 571:
1.2 noro 572: \underline{\uc $B%M%C%H%o!<%/>e$G$NDL?.$O4pK\E*$KE{H4$1(B}
1.1 noro 573:
1.2 noro 574: $B<+J,$N?H$O<+J,$G<i$k(B $\Rightarrow$ $BDL?.FbMF$r(B{\ec $B0E9f(B}$B2=(B
1.1 noro 575:
1.2 noro 576: \underline{\uc $B0E9f2=DL?.$N0lNc(B}
1.1 noro 577:
578: \begin{enumerate}
1.2 noro 579: \item $B0E9f2=(B/$BI|9f2=(B{\ec $B80(B}$B$r(B{\ec $B6&M-(B}$B$9$k(B.
1.1 noro 580:
581: \item $BAw?.B&(B : $B80$G0E9f2=(B $\Rightarrow$ $B<u?.B&(B : $B80$GI|9f2=(B
582: \end{enumerate}
583:
1.2 noro 584: \underline{\uc $BLdBj(B : $BDL?.O)$,E{H4$1$N$H$-$K(B,}
585:
586: \underline{\uc $B$I$&$d$C$F80$r6&M-(B?}
1.1 noro 587: \end{slide}
588:
589: \begin{slide}{}
1.2 noro 590: {\Large\parskip 0pt
591: \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}
1.1 noro 592:
593: \begin{itemize}
1.2 noro 594: \item {\eec $B8x3+>pJs(B}
1.1 noro 595:
596: $BBg$-$$AG?t(B $p$, $0 < g < p$ $B$J$k@0?t(B $g$
597:
1.2 noro 598: \item {\eec A $B$5$s$N;E;v(B}
1.1 noro 599:
600: \begin{enumerate}
1.2 noro 601: \item $0 < s_A < p$ $B$J$k@0?t(B {\eec $s_A$} ($BHkL)(B) $B$r:n$k(B.
602: \item $w_A =$ {\eec $g^{s_A} \bmod p$} $B$r(B B $B$5$s$KAw$k(B.
1.4 ! noro 603: \item $B<u$1<h$C$?(B $w_B$ $B$+$i(B $s =$ {\eec $w_B^{s_A} \bmod p$} $B$r:n$k(B.
1.1 noro 604: \end{enumerate}
605:
1.2 noro 606: \item {\eec B $B$5$s$N;E;v(B}
1.1 noro 607:
608: \begin{enumerate}
1.2 noro 609: \item $0 < s_B < p$ $B$J$k@0?t(B {\eec $s_B$} ($BHkL)(B) $B$r:n$k(B.
610: \item $w_B =$ {\eec $g^{s_B} \bmod p$} $B$r(B A $B$5$s$KAw$k(B.
1.4 ! noro 611: \item $B<u$1<h$C$?(B $w_A$ $B$+$i(B $s =$ {\eec $w_A^{s_B} \bmod p$} $B$r:n$k(B.
1.1 noro 612: \end{enumerate}
613:
1.2 noro 614: \end{itemize}}
1.1 noro 615: \end{slide}
616:
617: \begin{slide}{}
1.2 noro 618: \underline{\uc $BBg;v$JE@(B}
1.1 noro 619:
620: \begin{itemize}
1.2 noro 621: \item {\eec $w_B^{s_A} = w_A^{s_B} \bmod p$}
1.1 noro 622:
623: $B$3$l$G80$,6&M-$G$-$?(B
624:
1.4 ! noro 625: \item {\eec $w_A$, $w_B$ $B$O0E9f2=$NI,MW$J$7(B}
1.1 noro 626:
627: $g^{s_A} \bmod p$ $B$+$i(B $s_A$ $B$r5a$a$k$N$OFq$7$$(B
628:
1.2 noro 629: {\ec ($BM-8BBN$N>hK!72$K$*$1$kN%;6BP?tLdBj(B)}
1.1 noro 630:
1.2 noro 631: \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 632:
633: $\overline{a^{100}} = \overline{(\overline{a^{50}})^2}$,
634: $\overline{a^{50}} = \overline{(\overline{a^{25}})^2}$,
635: $\overline{a^{25}} = \overline{\overline{(\overline{a^{12}})^2} \times \overline{a}}$,
636: $\overline{a^{12}} = \overline{(\overline{a^{6}})^2}$,
637: $\overline{a^{6}} = \overline{(\overline{a^{3}})^2}$,
638: $\overline{a^{3}} = \overline{\overline{(\overline{a})^2} \times \overline{a}}$
639:
640: \end{itemize}
641: \end{slide}
642:
643: \begin{slide}{}
1.2 noro 644: \underline{\uc $BB>$K$b$$$m$$$m$"$k(B}
1.1 noro 645:
646: \begin{itemize}
1.2 noro 647: \item {\eec RSA $B0E9f(B}
1.1 noro 648:
1.2 noro 649: {\eec $BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5(B}$B$rMxMQ(B
1.1 noro 650:
1.2 noro 651: \item {\eec $BBJ1_6J@~0E9f(B}
1.1 noro 652:
653: $BM-8BBN>e$G(B $y^2=x^3+ax+b$ $B$N2r(B $P=(x,y)$ $B$r9M$($k$H(B,
654: $k$ $BG\;;(B $kP$ $B$,Dj5A$G$-$k(B.
655:
1.2 noro 656: {\eec $kP$ $B$+$i(B $k$ $B$r5a$a$k7W;;$NFq$7$5(B}$B$rMxMQ(B
1.1 noro 657: \end{itemize}
658:
1.2 noro 659: $\Rightarrow$ {\ec $BD>@\(B, $B4V@\$K@0?t$N>jM>1i;;$,4XM?(B}
1.1 noro 660: \end{slide}
661:
662: \begin{slide}{}
1.2 noro 663: \fbox{\sc 7. $B$^$H$a(B}
1.1 noro 664:
665: \begin{enumerate}
1.2 noro 666: \item {\eec $BB?9`<00x?tJ,2rDxEY$G$b(B, $B8zN($h$$<B8=$OBgJQ(B}
667:
668: $B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$K(B{\ec $BM-8BBN(B}
669:
670: \item {\eec $B$G$bM-8BBN$J$s$FB>$K2?$NLr$KN)$D$N(B?}
671:
672: $B<B$O(B IT $B<R2q$rN"$G;Y$($F$$$?$j$9$k(B.
1.1 noro 673:
1.2 noro 674: \item {\eec $B?t3X$N2{$N?<$5(B}
1.1 noro 675:
1.2 noro 676: $B8e$K$J$C$F$H$s$G$b$J$$$H$3$m$K1~MQ$5$l$k2DG=@-(B
1.1 noro 677:
1.2 noro 678: $B7W;;$NFq$7$5$,Lr$KN)$D$3$H$b$"$k$H$$$&IT;W5D(B
1.1 noro 679:
680:
681: \end{enumerate}
682:
683: \end{slide}
684:
685: %\begin{slide}{}
686: %\fbox{\bf}
687: %\end{slide}
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