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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 36 "Modular Arithmetic and Fa
st Powering" }}{PARA 19 "" 0 "" {TEXT -1 14 "Jeremy Johnson" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Modula
r Arithmetic" }}{PARA 0 "" 0 "" {TEXT -1 101 "Use mod function after e
very arithmetic operation.  By default the positive representatives ar
e used." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "3*3 mod 5;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "1/2  mod 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "For the symmetric form use mo
ds." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "mods(4,5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
256 16 "Computing Powers" }}{PARA 0 "" 0 "" {TEXT -1 77 "Several recur
sive and iterative functions for computing powers are presented." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "Powerit := proc(M,e)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "  local
 i,nu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  if (e = 0) then return \+
1 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "  nu := M;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "  for i from 1 to e-1 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "    nu := nu*M;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "
  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  return nu;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Pow
eritGf*6$%\"MG%\"eG6$%\"iG%#nuG6\"F,C&@$/9%\"\"!O\"\"\">8%9$?(8$F3F3,&
F0F3F3!\"\"%%trueG>F5*&F5F3F6F3OF5F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "Powerit(2,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Powerrec1 := proc(M,e
)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  if e = 0 then return 1 fi;  \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  return M*Powerrec1(M,e-1);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%*Powerrec1Gf*6$%\"MG%\"eG6\"F)F)C$@$/9%\"\"!O\"\"\"O*&9$F0-F$6
$F3,&F-F0F0!\"\"F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"Powerrec1(2,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Powerrec := proc(M,e)" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "  local P;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "  if e = 0 then return 1 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"  P := Powerrec(M,floor(e/2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
  P := P * P;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "  if type(e,odd) t
hen P := P*M; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  return P;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%)PowerrecGf*6$%\"MG%\"eG6#%\"PG6\"F+C'@$/9%\"\"!O\"\"\">8$-F$6
$9$-%&floorG6#,$*&#F2\"\"#F2F/F2F2>F4*&F4F2F4F2@$-%%typeG6$F/%$oddG>F4
*&F4F2F7F2OF4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Powe
rrec(2,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%C5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "Powerrec(3,100) mod 100;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "Powerrec(3,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"Q,?_2@qs7il(
H6YO5L6?t?vP:&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Powerrecm
od := proc(M,e,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "  local P;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  if e = 0 then return 1 fi;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "  P := Powerrecmod(M,floor(e/2),n);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "  P := P * P mod n;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 41 "  if type(e,odd) then P := P*M mod n; fi;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  return P;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,Powerrecmod
Gf*6%%\"MG%\"eG%\"nG6#%\"PG6\"F,C'@$/9%\"\"!O\"\"\">8$-F$6%9$-%&floorG
6#,$*&#F3\"\"#F3F0F3F39&>F5-%$modG6$*&F5F3F5F3F@@$-%%typeG6$F0%$oddG>F
5-FC6$*&F5F3F8F3F@OF5F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "Powerrecmod(3,5,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Powerrec(3,5) mod 7;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "time(Powerrec(3,1000000) mod 1023);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"$^&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "time(Powerrecmod(3,1000000,1023));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "In Maple, powe
rs are computed using the ^ notation.  If you want to incorporate the \+
mod function into the computation of powers you need to use the inert \+
function Power." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "3^100;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"Q,?_2@qs7il(H6YO5L6?t?vP:&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Power(3,100) mod 100;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 27 "Analysis of Binary Powering" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "PowerRec := proc(a,e)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "  local y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  if (e = 0) \+
then return 1; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  if (e = 1) \+
then return a; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  y := PowerR
ec(a,floor(e/2)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  y := y*y;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  if (e mod 2) = 1 then y := y*a;
 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "  return y;  " }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Pow
erRecGf*6$%\"aG%\"eG6#%\"yG6\"F+C(@$/9%\"\"!O\"\"\"@$/F/F2O9$>8$-F$6$F
6-%&floorG6#,$*&#F2\"\"#F2F/F2F2>F8*&F8F2F8F2@$/-%$modG6$F/FAF2>F8*&F8
F2F6F2OF8F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "PowerRec
(2,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%C5" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 15 "PowerRec(2,15);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"&oF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "PowerRecCou
nt := proc(a,e)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  local count;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  if (e = 0) then return 0; fi;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  if (e = 1) then return 0; fi;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  count := PowerRecCount(a,floor(
e/2)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "  count := count + 1;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "  if (e mod 2) = 1 then count := c
ount + 1; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "  return count;  \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%.PowerRecCountGf*6$%\"aG%\"eG6#%&countG6\"F+C(@$/9%\"
\"!OF0@$/F/\"\"\"OF0>8$-F$6$9$-%&floorG6#,$*&#F4\"\"#F4F/F4F4>F7,&F7F4
F4F4@$/-%$modG6$F/FAF4>F7FCOF7F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "PowerRecCount(a,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "PowerRecCount(a,
9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "data := [seq([e,PowerRecCount(a,e)],e=0..128)]:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 
7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin
ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pointplot(data,styl
e=LINE);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6$-%'C
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F.7$$\"\"&F)F17$$\"\"'F)F17$$\"\"(F)F47$$\"\")F)F17$$\"\"*F)F47$$\"#5F
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7$$\"##*F)FC7$$\"#$*F)FF7$$\"#%*F)FF7$$\"#&*F)FI7$$\"#'*F)F=7$$\"#(*F)
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$.\"F)FF7$$\"$/\"F)F@7$$\"$0\"F)FC7$$\"$1\"F)FC7$$\"$2\"F)FF7$$\"$3\"F
)FC7$$\"$4\"F)FF7$$\"$5\"F)FF7$$\"$6\"F)FI7$$\"$7\"F)F@7$$\"$8\"F)FC7$
$\"$9\"F)FC7$$\"$:\"F)FF7$$\"$;\"F)FC7$$\"$<\"F)FF7$$\"$=\"F)FF7$$\"$>
\"F)FI7$$\"$?\"F)FC7$$\"$@\"F)FF7$$\"$A\"F)FF7$$\"$B\"F)FI7$$\"$C\"F)F
F7$$\"$D\"F)FI7$$\"$E\"F)FI7$$\"$F\"F)FL7$$\"$G\"F)F=-%&STYLEG6#%%LINE
G" 1 6 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "counts := seq(PowerRecCou
nt(a,e),e=8..16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'countsG6+\"\"$
\"\"%F'\"\"&F'F(F(\"\"'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "seq(convert(e,binary),e=8..16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
+\"%+5\"%,5\"%55\"%65\"%+6\"%,6\"%56\"%66\"&++\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 18 "numbits := proc(n)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "  local count, nb;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "  count := 0;  nb := n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "  wh
ile (nb <> 0) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "    if nb mod 2
 = 1 then count := count + 1; fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "    nb := floor(nb/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "  return count;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(numb
itsGf*6#%\"nG6$%&countG%#nbG6\"F+C&>8$\"\"!>8%9$?(F+\"\"\"F4F+0F1F/C$@
$/-%$modG6$F1\"\"#F4>F.,&F.F4F4F4>F1-%&floorG6#,$*&#F4F<F4F1F4F4OF.F+F
+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(numbits(e),e=8..
16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\"\"\"\"\"#F$\"\"$F$F%F%\"\"%F
#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "counts := seq(PowerRec
Count(a,e),e=8..16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'countsG6+\"
\"$\"\"%F'\"\"&F'F(F(\"\"'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 64 "counts2 := seq(floor(simplify(log[2](e)))+numbits(e)-1,e=8..16);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(counts2G6+\"\"$\"\"%F'\"\"&F'F(
F(\"\"'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "seq(PowerRecCo
unt(a,e)-(floor(simplify(log[2](e)))+numbits(e)-1),e=1..128);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6\\s\"\"!F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F
#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F
#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F
#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#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 0 "" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
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