Simple encryption algorithm for homework. not getting decryption working properly - algorithm

Question

This is a homework question that I can't get my head around at all
Its a very simple encryption algorithm. You start with a string of characters as your alphabet:
ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789!, .
Then ask the user to enter there own string that will act as a map such as:
0987654321! .,POIUYTREWQASDFGHJKLMNBVCXZ
Then the program uses this to make a map and allows you to enter text that gets encrypted.
For example MY NAME IS JOSEPH would be encrypted as .AX,0.6X2YX1PY6O3
This is all very easy, however he said that its a one to one mapping and thus implied that if I enter .AX,0.6X2YX1PY6O3 back into the program I will get out MY NAME IS JOSEPH
This doesn't happen, because .AX,0.6X2YX1PY6O3 becomes Z0QCDZQGAQFOALDH
The mapping only works to decrypt when you go backwards but the question implies that the program just loops and runs the one algorithm every time.
Even if some could say that it is possible I would be happy, I have pages and pages of paper filled up with possible workings, but I came up with nothing, the only solution to run the algorithm backwards back I don't think we are allowed to do that.
Any ideas?
Edit:
Unfortunately I can't get this to work (Using the orbit computation idea) What am I doing wrong?
//import scanner class
import java.util.Scanner;
public class Encryption {
static Scanner inputString = new Scanner(System.in);
//define alphabet
private static String alpha = "ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789!, .";
private static String map;
private static int[] encryptionMap = new int[40];//mapping int array
private static boolean exit = false;
private static boolean valid = true;
public static void main(String[] args) {
String encrypt, userInput;
userInput = new String();
System.out.println("This program takes a large reordered string");
System.out.println("and uses it to encrypt your data");
System.out.println("Please enter a mapping string of 40 length and the same characters as below but in different order:");
System.out.println(alpha);
//getMap();//don't get user input for map, for testing!
map=".ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789!, ";//forced input for testing only!
do{
if (valid == true){
System.out.println("Enter Q to quit, otherwise enter a string:");
userInput = getInput();
if (userInput.charAt(0) != 'Q' ){//&& userInput.length()<2){
encrypt = encrypt(userInput);
for (int x=0; x<39; x++){//here I am trying to get the orbit computation going
encrypt = encrypt(encrypt);
}
System.out.println("You entered: "+userInput);
System.out.println("Encrypted Version: "+encrypt);
}else if (userInput.charAt(0) == 'Q'){//&& userInput.length()<2){
exit = true;
}
}
else if (valid == false){
System.out.println("Error, your string for mapping is incorrect");
valid = true;//reset condition to repeat
}
}while(exit == false);
System.out.println("Good bye");
}
static String encrypt(String userInput){
//use mapping array to encypt data
String encrypt;
StringBuffer tmp = new StringBuffer();
char current;
int alphaPosition;
int temp;
//run through the user string
for (int x=0; x<userInput.length(); x++){
//get character
current = userInput.charAt(x);
//get location of current character in alphabet
alphaPosition = alpha.indexOf(current);
//encryptionMap.charAt(alphaPosition)
tmp.append(map.charAt(alphaPosition));
}
encrypt = tmp.toString();
return(encrypt);
}
static void getMap(){
//get a mapping string and validate from the user
map = getInput();
//validate code
if (map.length() != 40){
valid = false;
}
else{
for (int x=0; x<40; x++){
if (map.indexOf(alpha.charAt(x)) == -1){
valid = false;
}
}
}
if (valid == true){
for (int x=0; x<40; x++){
int a = (int)(alpha.charAt(x));
int y = (int)( map.charAt(x));
//create encryption map
encryptionMap[x]=(a-y);
}
}
}
static String getInput(){
//get input(this repeats)
String input = inputString.nextLine();
input = input.toUpperCase();
if ("QUIT".equals(input) || "END".equals(input) || "NO".equals(input) || "N".equals(input)){
StringBuffer tmp = new StringBuffer();
tmp.append('Q');
input = tmp.toString();
}
return(input);
}
}

Answer 1

You will (probably) not get your original string back if you apply that substitution again. I say probably because you can construct such inputs (they all do things like if A->B then B->A). But most inputs won't do that. You would have to construct the reverse map to decrypt.
However, there is a trick you can do if you're only allowed to go forward. Keep applying the mapping and you'll eventually return to your original input. The number of times you'll have to do that depends on your input. To figure out how many times, compute the orbit of each character, and take the least common multiple of all the orbit sizes. For your input the orbits are size 1 (T->T, W->W), 2 (B->9->B H->3->H U->R->U P->O->P), 4 (C->8->N->,->C), 9 (A->...->Y->A), and 17 (E->...->V->E). The LCM of all those is 612, so 611 forward mappings applied to the ciphertext will return you to the plaintext.

Answer 2

Well, you can get your string back this way only if you do reverse mapping. One to one mapping means that a single letter of your default alphabet maps to only one letter of your new alphabet and vice versa. I.e. you can't map ABCD to ABBA. It doesn't imply that you can get your initial string by doing a second round of encryption.
The thing you have described can be achieved if you use a finite alphabet and a displacement to encode your string. You can choose the displacement in such a way that after a number of rounds of encryption totalDisplacement mod alphabetSize == 0 Than you will get your string back going only forward.