#Homework Monday, Jan 22, 2024
#Ok to post

Help := proc(): print(`Beginner(), EulerThe_(n), MyIfactor(n)`) : end:
with(NumberTheory):

# Textbook examples with modified changes
#------------------------------------------
Beginner := proc() local k,z:
  Sum(i^3,i=1..5);
  print(value(%));

  z := exp(x^2*y)(sqrt(x^3*y^2+2*y+x^2));
  diff(z,x);
end:

# Euler's Theorem
#------------------------------------------
EulerThe_ := proc(n) local A, i, a, phi, status:
  A := {}:
  for i from 1 to (n-1) do:
    if AreCoprime(i, n) then:
      A := A union {i}:
    fi:
  od:

  phi := nops(A):
  status := "SUCCESS":
  for a in A do:
    if (a^phi mod n) <> 1 then:
      return 1;
    fi:
  od:
  
  return 0;

end:

#Checking to see if my implementation of Euler's Theorem works

status := SUCCESS:
for i from 1 to 1200 do
 if EulerThe_(i) = 1 then
  status := FAILURE:
  break;
 fi:
od:
print(status);


# RSA Encryption
#____________________________________________

p := nextprime(rand(10^5..10^6-1)()):
q := nextprime(rand(10^5..10^6-1)()):

n := p * q:

do
  e := nextprime(rand(10^5..10^6-1)()):
until AreCoprime(e, n):

phi_ := Totient(n):

d := e&^(-1) mod phi_;

do
  cipher_message := rand(0..n)():
until AreCoprime(cipher_message, n):

message := cipher_message&^(d) mod n;

encrypted_message := message&^(e) mod n;

print("RSA Encryption is complete");
  

# Procedure for MyIfactor(n)
#-------------------------------------------
MyIfactor := proc(n) local p, prime_factors, i, temp:
  temp := n:
  prime_factors := []:
  while (temp <> 1) do
    p := 0:
    while (p = 0 or temp mod p <> 0) do
      p := nextprime(p):
    od:
    prime_factors := [op(prime_factors), p]:
    temp := temp / p:
  od:
  return prime_factors;
end:  

# Time Comparison between MyIfactor(n) and ifactor(n)

myifactor_time := 0: ifactor_time := 0:

for i from 1 to 100000 do
  myifactor_time := myifactor_time + time(MyIfactor(i)):
  ifactor_time := ifactor_time + time(ifactor(i)):
od:

printf("My time: %f\nMaple time: %f\n", myifactor_time, ifactor_time);