COMP 547: Assignment 1 Solutions

square root. In other words, we would like to be able to let msqrt know
n’s factorisation n = pq, as this would allow for an efficient search for
a square root modulo n, using the techniques seen in class. However,
since msqrt does not allow us to do this, we must write our own function,
pqsqrt, which works exactly like msqrt, except that instead of providing
the function with n, we provide it with n’s prime factorisation n = pq.
9. To pick my two primes p, q starting by 1 followed by 669955887 and ending
by 01 or 03 of a 100 bits :
genpq := proc()
local p, q;
randomize();
p := 0;
q := 0;
while not isprime(p) do
p := 1669955887*10^90 + rand(0..10^88-1)()*10^2 + 1;
end do;
while not isprime(q) do
q := 1669955887*10^90 + rand(0..10^88-1)()*10^2 + 3;
end do;
return(p, q);
end proc;
This generated the following p, q pair:
p := 1669955887263026818036412887272431375849039042767749903\
121235706543134605608754758160586255799694301;
q := 1669955887003361797744279814925958377127037938538442466\
825381252778399643783166093655256052781392403;
Computing n = pq yielded the following:
n := 1587706207511861907832769125027465150069895085728428488\
006051557553392116910484620485578137364900762730201415942\
132714983409510118890246103424460950790756546532252049282\
409606090137788255468523795303;
10. To pick my two random quadratic non-residues y and z,here is the following
Maple function:
genyz := proc(n, p, q)
local a, b, y, z;
randomize();
with(numtheory);
12