COMP 547: Assignment 1 Solutions

all functions I wrote for this assignment, I used the randomize function
to set the seed used by functions such as rand to a number that is based
on the system clock. Although my random numbers are still deterministically
computed, they are, in a sense, more random and more difficult to
randomly reproduce.
But there is worst, the random generator Randpoly is not very good and
loops around quite quickly. You could not, in fact generate the entire field.
2. To build the field F 2 1000 using P , one must use Maple’s GF function:
F21000 := GF(2, 1000, P);
This generated the following output:
F21000 := module () export +, -, *, /, ^, input, output,
inverse, extension, variable, factors, norm, trace, order,
random, size, isPrimitiveElement, PrimitiveElement,
ConvertIn, ConvertOut, zero, one, init; end module
3. myFindPrim := proc(Field)
local factors, m, f, g, i;
randomize();
factors := [31, 601, 4710883168879506001, 269089806001,
1801, 3, 11, 251, 229668251, 5519485418336288303251,
4051, 5, 94291866932171243501, 268501, 28001, 96001,
41, 101, 47970133603445383501, 3775501, 7001, 8101, 17,
61681, 401, 3173389601, 2787601, 340801, 8877945148742\
945001146041439025147034098690503591013177336356694416\
517527310181938001, 4001, 1074001, 2020001, 22624001,
1481124532001];
m := [seq((2^1000 - 1)/factors[i], i = 1..34)];
f := 0;
while f = 0 do
f := 1;
g := Field[input](rand(0..2^1000-1)());
for i from 1 to 34 do
if Field[‘^‘](g, m[i]) = Field[one] then
f := 0;
break;
end if
end do;
end do;
return(g);
end proc;
This generated the following primitive element g:
4