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Annotation of OpenXM/doc/sci-semi2001/factorb.tex, Revision 1.2

1.2     ! noro        1: % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.1 2001/07/23 06:46:51 noro Exp $
1.1       noro        2:
1.2     ! 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}