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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "beta := n^sigma; # b
locking factor" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG)%\"nG%&sigm
aG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "s := n^tau; # number \+
of giant steps" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG)%\"nG%$tauG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "r := simplify( (n/beta) /
 s); # number of baby steps" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG)
%\"nG,(\"\"\"F(%&sigmaG!\"\"%$tauGF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 79 "Standard matrix arithmetic, \+
quadratic B/M, Chinese remainder integer arithmetic" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 18 "Step 2.1: Compute " }{XPPEDIT 18 0 "B^j y" "6#*
&)%\"BG%\"jG\"\"\"%\"yGF'" }{TEXT -1 2 ", " }{TEXT 256 1 "j" }{TEXT 
-1 9 " = 0,...," }{TEXT 257 1 "r" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "substep1 := simplify( r * beta * n^2 * r );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)substep1G)%\"nG,(\"\"%\"\"\"%&sigmaG!\"\"
*&\"\"#F)%$tauGF)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Step 2.2: \+
Compute " }{XPPEDIT 18 0 "Z = B^r" "6#/%\"ZG)%\"BG%\"rG" }{TEXT -1 21 
" by repeated squaring" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s
ubstep2 := simplify( n^3 * r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)s
ubstep2G)%\"nG,(\"\"%\"\"\"%&sigmaG!\"\"%$tauGF+" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 18 "Step 2.3: Compute " }{XPPEDIT 18 0 "x^Tr*Z^k;" "6#
*&)%\"xG%#TrG\"\"\")%\"ZG%\"kGF'" }{TEXT -1 2 ", " }{TEXT 258 1 "k" }
{TEXT -1 9 " = 0,...," }{TEXT 259 2 "s " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "substep3 := simplify( s * beta * n^2 * n/beta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)substep3G)%\"nG,&%$tauG\"\"\"\"\"$F
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Step 2.4: Compute (" }
{XPPEDIT 18 0 "x^Tr*Z^k;" "6#*&)%\"xG%#TrG\"\"\")%\"ZG%\"kGF'" }{TEXT 
-1 3 ") (" }{XPPEDIT 18 0 "B^j y" "6#*&)%\"BG%\"jG\"\"\"%\"yGF'" }
{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "substep4 :
= simplify( r * s * beta^2 * n * n/beta );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)substep4G*$)%\"nG\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 38 "Step 3: Blocked Berlekamp/Massey for  " }{TEXT 270 
1 "n" }{TEXT -1 7 " moduli" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "step2 := simplify( (n/beta)^2 * beta^3 * n );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&step2G)%\"nG,&\"\"$\"\"\"%&sigmaGF)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "Step 4: Determinant of generator matrix p
olynomial for " }{TEXT 271 1 "n" }{TEXT -1 7 " moduli" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "step3 := simplify( beta^3 * n * n )
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&step3G)%\"nG,&%&sigmaG\"\"$\"
\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Overall bit complexi
ty" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "eval([substep1, subst
ep2, substep3, substep4, step2, step3], \{sigma=1/3, tau=1/3\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7(*$)%\"nG\"\"$\"\"\"*$)F&#\"#5F'F(F)F
$F)F$" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 2 "``" }{TEXT 267 54 "The asymptotically best algorithms freq
uently turn out" }}{PARA 259 "" 0 "" {TEXT 268 52 "to be worst on all \+
problems for which they are used." }{TEXT -1 2 "''" }}{PARA 260 "" 0 "
" {TEXT -1 4 "--- " }{TEXT 269 37 "D. G. CANTOR and H. ZASSENHAUS (198
1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 29 "F
ast matrix multiplication O(" }{XPPEDIT 18 0 "n^omega" "6#)%\"nG%&omeg
aG" }{TEXT -1 40 "), linear B/M, linear integer arithmetic" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 18 "Step 2.1: Compute " }{XPPEDIT 18 0 "B^j y
" "6#*&)%\"BG%\"jG\"\"\"%\"yGF'" }{TEXT -1 2 ", " }{TEXT 260 1 "j" }
{TEXT -1 9 " = 0,...," }{TEXT 261 2 "r " }{TEXT -1 3 "by " }{XPPEDIT 
18 0 "B^(2^i)" "6#)%\"BG)\"\"#%\"iG" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "
B^0 y" "6#*&%\"BG\"\"!%\"yG\"\"\"" }{TEXT -1 9 " | ... | " }{XPPEDIT 
18 0 "B^(2^i - 1)" "6#)%\"BG,&)\"\"#%\"iG\"\"\"F)!\"\"" }{TEXT -1 3 " \+
y]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fastsubstep1 := simpl
ify( n^omega * r );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-fastsubstep1
G)%\"nG,*%&omegaG\"\"\"F)F)%&sigmaG!\"\"%$tauGF+" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 18 "Step 2.2: Compute " }{XPPEDIT 18 0 "Z = B^r" "6#/%
\"ZG)%\"BG%\"rG" }{TEXT -1 21 " by repeated squaring" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "fastsubstep2 := simplify( n^omega * r);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-fastsubstep2G)%\"nG,*%&omegaG\"\"
\"F)F)%&sigmaG!\"\"%$tauGF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "St
ep 2.3: Compute " }{XPPEDIT 18 0 "x^Tr*Z^k;" "6#*&)%\"xG%#TrG\"\"\")%
\"ZG%\"kGF'" }{TEXT -1 2 ", " }{TEXT 262 1 "k" }{TEXT -1 9 " = 0,...,
" }{TEXT 263 2 "s " }{TEXT -1 3 "as " }{TEXT 265 3 "fat" }{TEXT -1 17 
" vectors times a " }{TEXT 266 4 "thin" }{TEXT -1 8 " matrix." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "fastsubstep3 := simplify( s \+
* (n/(beta*s))^2 * (beta*s)^omega * r, 'power', 'symbolic' );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-fastsubstep3G)%\"nG,*%$tauG!\"#\"\"
$\"\"\"*&F*F+%&sigmaGF+!\"\"*&,&F-F+F(F+F+%&omegaGF+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 18 "Step 2.3: Compute " }{XPPEDIT 18 0 "x^Tr*
Z^k;" "6#*&)%\"xG%#TrG\"\"\")%\"ZG%\"kGF'" }{TEXT -1 2 ", " }{TEXT 
273 1 "k" }{TEXT -1 9 " = 0,...," }{TEXT 274 2 "s " }{TEXT -1 23 "as r
ectangular product:" }}{PARA 0 "" 0 "" {TEXT -1 101 "       s parts of
 x^Tr to make them thin leads to beta*s by n times n (namely Z) produc
t of lenght r." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "fastsubst
ep3 := simplify( s * n^(omega-gamma) * (beta*s)^gamma * r, 'power', 's
ymbolic');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-fastsubstep3G)%\"nG,,
%&omegaG\"\"\"%&gammaG!\"\"*&,&%&sigmaGF)%$tauGF)F)F*F)F)F)F)F.F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Step 2.3: using Huang/Pan n by b t
imes n by n^r where r = sigma+tau" }}{PARA 0 "" 0 "" {TEXT -1 15 "(sam
e as above)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "# fastsubst
ep3 := simplify( s * n^( (2*(1-sigma-tau) + (sigma+tau-alpha)*omega)/(
1-alpha) ) * r, 'power','symbolic');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Step 2.4: Compute (" }
{XPPEDIT 18 0 "x^Tr*Z^k;" "6#*&)%\"xG%#TrG\"\"\")%\"ZG%\"kGF'" }{TEXT 
-1 3 ") (" }{XPPEDIT 18 0 "B^j y" "6#*&)%\"BG%\"jG\"\"\"%\"yGF'" }
{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "fastsubste
p4 := simplify( r^2/s * (beta*s)^omega * n/beta, 'power', 'symbolic' )
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-fastsubstep4G)%\"nG,*\"\"$\"\"
\"*&F(F)%&sigmaGF)!\"\"*&F(F)%$tauGF)F,*&,&F+F)F.F)F)%&omegaGF)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Step 3: (Displacement complexity) \+
Blocked Berlekamp/Massey for " }{TEXT 264 2 "n " }{TEXT -1 6 "moduli" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "faststep2 := simplify( be
ta^2 * n * n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*faststep2G)%\"nG,
&%&sigmaG\"\"#F)\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Step 3:
 (Gilles) Matrix half GCD for polys of degree " }{XPPEDIT 18 0 "n" "6#
%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "faststep2 := simpl
ify(beta^omega * n/beta * n, 'power', 'symbolic');" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%*faststep2G)%\"nG,(*&%&sigmaG\"\"\"%&omegaGF*F*\"\"
#F*F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Step 4: Leading and
 constant coefficient of generator matrix polynomial for " }{TEXT 272 
1 "n" }{TEXT -1 7 " moduli" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "faststep3 := simplify( beta^omega * n, 'power', 'symbolic' );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*faststep3G)%\"nG,&*&%&sigmaG\"\"\"%
&omegaGF*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Overall bit co
mplexity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "total := eval([
fastsubstep1, fastsubstep2, fastsubstep3, fastsubstep4, faststep2, fas
tstep3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&totalG7()%\"nG,*%&omeg
aG\"\"\"F*F*%&sigmaG!\"\"%$tauGF,F&)F',,F)F*%&gammaGF,*&,&F+F*F-F*F*F0
F*F*F*F*F+F,)F',*\"\"$F**&F5F*F+F*F,*&F5F*F-F*F,*&F2F*F)F*F*)F',(*&F+F
*F)F*F*\"\"#F*F+F,)F',&F;F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 "expos:=simplify(map(x -> log[n](x), total), 'symbolic');" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&exposG7(,*%&omegaG\"\"\"F(F(%&sigma
G!\"\"%$tauGF*F&,.F'F(%&gammaGF**&F-F(F)F(F(*&F-F(F+F(F(F(F(F)F*,,\"\"
$F(*&F1F(F)F(F**&F1F(F+F(F**&F)F(F'F(F(*&F'F(F+F(F(,(F4F(\"\"#F(F)F*,&
F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "expos:=subs(gam
ma=(omega-2)/(1-alpha), expos);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&
exposG7(,*%&omegaG\"\"\"F(F(%&sigmaG!\"\"%$tauGF*F&,.F'F(*&,&F'F(\"\"#
F*F(,&F(F(%&alphaGF*F*F**&*&F.F(F)F(F(F0F*F(*&*&F.F(F+F(F(F0F*F(F(F(F)
F*,,\"\"$F(*&F7F(F)F(F**&F7F(F+F(F**&F)F(F'F(F(*&F'F(F+F(F(,(F:F(F/F(F
)F*,&F:F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "expos:=sub
s(alpha=0.2946289, expos);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&expos
G7(,*%&omegaG\"\"\"F(F(%&sigmaG!\"\"%$tauGF*F&,,F'$!*nMp<%!\"*$\"+MpQN
QF/F(*($\"+nMp<9F/F(,&F'F(\"\"#F*F(F)F(F(*(F3F(F5F(F+F(F(F)F*,,\"\"$F(
*&F9F(F)F(F**&F9F(F+F(F**&F)F(F'F(F(*&F'F(F+F(F(,(F<F(F6F(F)F*,&F<F(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "numexpos:=eval(expos,
 omega=2.375477);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)numexposG7(,($
\"(xaP$!\"'\"\"\"%&sigmaG!\"\"%$tauGF,F&,($\"+5d;VG!\"*F**&$\"+,r)on%!
#5F*F+F*F,*&$\"+**G6B`F5F*F-F*F*,($\"\"$\"\"!F**&$\"'BXiF)F*F+F*F,*&$F
?F)F*F-F*F,,&F+$\"(xaP\"F)\"\"#F*,&F+$\"(xaP#F)F*F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 82 "minexpo := solve(\{numexpos[2]=numexpos[3
], numexpos[2]=numexpos[5]\}, \{sigma,tau\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(minexpoG<$/%&sigmaG$\"+1CCp]!#5/%$tauG$\"+e7!Hr\"F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval(numexpos, minexpo)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7($\"+LEE(p#!\"*F$$\"+MEE(p#F&$\"
+H&Rkd#F&F'$\"+uo=/AF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 276 24 "Division free char poly:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 66 "Step 4: Char poly: Determinant of generator matrix polyno
mial for " }{TEXT 275 1 "n" }{TEXT -1 7 " moduli" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 64 "faststep3 := simplify( beta^omega * n *n, 'po
wer', 'symbolic' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*faststep3G)%
\"nG,&*&%&sigmaG\"\"\"%&omegaGF*F*\"\"#F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "Overall bit complexity" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "total := eval([fastsubstep1, fastsubstep2, fastsubste
p3, fastsubstep4, faststep2, faststep3]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&totalG7()%\"nG,*%&omegaG\"\"\"F*F*%&sigmaG!\"\"%$tau
GF,F&)F',,F)F*%&gammaGF,*&,&F+F*F-F*F*F0F*F*F*F*F+F,)F',*\"\"$F**&F5F*
F+F*F,*&F5F*F-F*F,*&F2F*F)F*F*)F',(*&F+F*F)F*F*\"\"#F*F+F,)F',&F;F*F<F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "expos:=simplify(map(x \+
-> log[n](x), total), 'symbolic');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>%&exposG7(,*%&omegaG\"\"\"F(F(%&sigmaG!\"\"%$tauGF*F&,.F'F(%&gammaGF*
*&F-F(F)F(F(*&F-F(F+F(F(F(F(F)F*,,\"\"$F(*&F1F(F)F(F**&F1F(F+F(F**&F)F
(F'F(F(*&F'F(F+F(F(,(F4F(\"\"#F(F)F*,&F4F(F7F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "expos:=subs(gamma=(omega-2)/(1-alpha), expos);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&exposG7(,*%&omegaG\"\"\"F(F(%&s
igmaG!\"\"%$tauGF*F&,.F'F(*&,&F'F(\"\"#F*F(,&F(F(%&alphaGF*F*F**&*&F.F
(F)F(F(F0F*F(*&*&F.F(F+F(F(F0F*F(F(F(F)F*,,\"\"$F(*&F7F(F)F(F**&F7F(F+
F(F**&F)F(F'F(F(*&F'F(F+F(F(,(F:F(F/F(F)F*,&F:F(F/F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "expos:=subs(alpha=0.2946289, expos);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&exposG7(,*%&omegaG\"\"\"F(F(%&sigma
G!\"\"%$tauGF*F&,,F'$!*nMp<%!\"*$\"+MpQNQF/F(*($\"+nMp<9F/F(,&F'F(\"\"
#F*F(F)F(F(*(F3F(F5F(F+F(F(F)F*,,\"\"$F(*&F9F(F)F(F**&F9F(F+F(F**&F)F(
F'F(F(*&F'F(F+F(F(,(F<F(F6F(F)F*,&F<F(F6F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "numexpos:=eval(expos, omega=2.375477);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%)numexposG7(,($\"(xaP$!\"'\"\"\"%&sigmaG!\"\"
%$tauGF,F&,($\"+5d;VG!\"*F**&$\"+,r)on%!#5F*F+F*F,*&$\"+**G6B`F5F*F-F*
F*,($\"\"$\"\"!F**&$\"'BXiF)F*F+F*F,*&$F?F)F*F-F*F,,&F+$\"(xaP\"F)\"\"
#F*,&F+$\"(xaP#F)FEF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "mi
nexpo := solve(\{numexpos[2]=numexpos[3], numexpos[2]=numexpos[6]\}, \+
\{sigma,tau\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(minexpoG<$/%$tau
G$\"+R*fWH#!#5/%&sigmaG$\"+vv;&R$F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "eval(numexpos, minexpo);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7($\"+[U^1G!\"*F$$\"+\\U^1GF&$\"+w'pYk#F&$\"+\"\\(*pY#F
&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "56 1" 0 
}{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }