Consider The Number 1.2 3, How Many Combinations Are There?
Introduction
Consider the numbers 1 2 3, how many combinations are there? 1x2x3=6, so there is 6 combinations, but how about using combinations and permutations to another level? How does Permutations actually affect our lives? One different combination would result in many other results such as the phone number. Imagine if a particular person’s phone number is 92719071, if i change the 1 at the end into 4, could you still call him? Or would you call another person instead?
One time, there was a new movie that came out 1 year ago which I was really interested in, called the Imitation Game. It was a movie showing how the Enigma works during World War 2 when the British were using it to decipher the german codes. For this investigation, I would like to explain the math and also find the relationship between Combinations behind the use of the enigma machine, where it was first used by the British and was made by Arthur Scherbius. The British used it to decipher the signal traffics in Germany during World War 2. By acheving this, I would have to tell how the Enigma works, what parts …show more content…
is the machine made out of, and also deciphering an Enigma.
A little history background of the the Enigma
Arthur Scherbius, a German engineer, developed his Enigma machine, which is capable of transcribing coded information, in the hope of interesting commercial companies in secure communications. In 1923 he set up his Chiffriermaschinen Aktiengesellschaft in Berlin to manufacture his product. Within three years the German navy was producing its own version, followed by the rest of the military by 1933. The enigma machine was then used in war to decipher and send secret messages to german comrades.
How does the Enigma work?
The enigma machine is an electro mechanical device with electric signals that pass through wires and various mechanical parts. Enigma machine is just one big circuit. Each time a key is pressed at least one rotor is rotated which changes the circuitry of the machine thus lighting a different letter faced light bulb.The light bulb is the bulb that is under the lightboard that shows the light under the plugbaord letter. Now, What is a rotor you may ask? The rotor is like the heart of the Enigma machine, without it, The machine would not work. Inside these rotors are full of mixed up wires.These rotors can only move and each of them has 26 steps, 1 for each alphabet.Each time you press the letter, the rotor would rotate, and if you rotate one full circle, the rotor in the middle will move 1 time, since the wires move, and the battery would get connected to a different bulb. For example, when you press A, you get u, then you press A again, you get t, and you just get other random letters everytime you press the same key “A”. This is why the Germans thought it was an unbreakable code since it changes all the time. The top right corner in the image below is the battery. All the wires and lightboards are connected via the battery itself.
For example, Ill use “hi” for example. When I press H on the keyboard, it shows a “T” on the lightboard, and when I press “I” , the enigma machine shows a “W”. The message is then transmitted through morse code on the radio. The person who recieves it then uses the enigma machine, move 2 rotors backward to reset the machine, then the reciever would type t and w. It would then show the message “HI”, since H and I were T and W respectively in the lightboard from the user, which then concludes that the machine is a code and decode machine. Here’s an example.
But how do they know to set the 2 enigma machines the same from such a far distance?
The setting of the Enigma machine
The following picture above us is the settings of the enigma machine settings, the first column shows us the date of the setting, since the date could be changed every day, month or week.
The second column with the roman numbers are the rotors. Imagine there are 3 rotors in the enigma machine and you can swap order. Since it is combinations, there are 6 combinations since
3x2x1=6.
The third column is the ring setting, Each rotor has 26 letters, a b c d until z. Since there are 3 rotors, each with 26 letters, 26x26x26= 17576.
The fourth column makes pairs. These pairs are the intructions to setting up the plugboard. The total number of ways you can make 6 pairs of the letters out of the alphabet is 100391791500, which is near a hundred billion.
26! (this is how many ways to order 26 alphabets) divided by 14! (divided by the number of ways to order the letters that are not swapped) then divide by 6! (divided by the number of ways to order the pairs) lastly, divide 2^6 , since the other does not matter.
= 100391791500
The final column, which is called the ground setting, which is the rotor starting position. Each rotor has 26 starting places, the first one has 26, the second one has 26 and the third one has 26, the total number of starting positions is the same, which is 17576.
26x26x26=17576
Then the total number of ways to set up the enigma is the product of all 4 numbers, which is around 10000 trillion.
100391791500 x 17576 x 17576 x 6 = 1.06 x 10^16
Sending a secret setting
There are hundreds of enigmas sent everyday during World War 2, then if they were all sent with the same key and security, it would be bad security.
So each message has its own secret starting position. For example, set the machine’s ground setting to abc. The operator could then pick its own secret starting position, lets say its xyz. The operator would send his own secret message in code to the reciever at the beginning of the message. The enigma machine would then be used to encode the three letters xyz. The reciever would then type xyz two times, which xyzxyz would turn into 6 letters of code, for example, JTEQFG. The reciever would then put JTEQFG at the beginning of the message. On the other side, the operator would then use the enigma machine to decode the 6 letters JTEQFG, and he would get XYZXYZ. The operator would then know it is the secret setting for the rest of the
message.
Conclusion
I conclude that the Enigma machine was a huge success and was a great piece of engineering since there was a lot of combinations to set up the enigma machine, which is 10000 trillion. It was a successful piece of engineering since the Enigma machine was the machine that could actually use combinations and permutations to be used for sending messages, and secret messages. It was the great Alan Turing, who was British that actually deciphered the first German message. For my further investigation, I would like to know how the battery keeps on changing to the bulb in the lightboard and the math behind it.
Consider the numbers 1 2 3, how many combinations are there? 1x2x3=6, so there is 6 combinations, but how about using combinations and permutations to another level? How does Permutations actually affect our lives? One different combination would result in many other results such as the phone number. Imagine if a particular person’s phone number is 92719071, if i change the 1 at the end into 4, could you still call him? Or would you call another person instead?
One time, there was a new movie that came out 1 year ago which I was really interested in, called the Imitation Game. It was a movie showing how the Enigma works during World War 2 when the British were using it to decipher the german codes. For this investigation, I would like to explain the math and also find the relationship between Combinations behind the use of the enigma machine, where it was first used by the British and was made by Arthur Scherbius. The British used it to decipher the signal traffics in Germany during World War 2. By acheving this, I would have to tell how the Enigma works, what parts …show more content…
is the machine made out of, and also deciphering an Enigma.
A little history background of the the Enigma
Arthur Scherbius, a German engineer, developed his Enigma machine, which is capable of transcribing coded information, in the hope of interesting commercial companies in secure communications. In 1923 he set up his Chiffriermaschinen Aktiengesellschaft in Berlin to manufacture his product. Within three years the German navy was producing its own version, followed by the rest of the military by 1933. The enigma machine was then used in war to decipher and send secret messages to german comrades.
How does the Enigma work?
The enigma machine is an electro mechanical device with electric signals that pass through wires and various mechanical parts. Enigma machine is just one big circuit. Each time a key is pressed at least one rotor is rotated which changes the circuitry of the machine thus lighting a different letter faced light bulb.The light bulb is the bulb that is under the lightboard that shows the light under the plugbaord letter. Now, What is a rotor you may ask? The rotor is like the heart of the Enigma machine, without it, The machine would not work. Inside these rotors are full of mixed up wires.These rotors can only move and each of them has 26 steps, 1 for each alphabet.Each time you press the letter, the rotor would rotate, and if you rotate one full circle, the rotor in the middle will move 1 time, since the wires move, and the battery would get connected to a different bulb. For example, when you press A, you get u, then you press A again, you get t, and you just get other random letters everytime you press the same key “A”. This is why the Germans thought it was an unbreakable code since it changes all the time. The top right corner in the image below is the battery. All the wires and lightboards are connected via the battery itself.
For example, Ill use “hi” for example. When I press H on the keyboard, it shows a “T” on the lightboard, and when I press “I” , the enigma machine shows a “W”. The message is then transmitted through morse code on the radio. The person who recieves it then uses the enigma machine, move 2 rotors backward to reset the machine, then the reciever would type t and w. It would then show the message “HI”, since H and I were T and W respectively in the lightboard from the user, which then concludes that the machine is a code and decode machine. Here’s an example.
But how do they know to set the 2 enigma machines the same from such a far distance?
The setting of the Enigma machine
The following picture above us is the settings of the enigma machine settings, the first column shows us the date of the setting, since the date could be changed every day, month or week.
The second column with the roman numbers are the rotors. Imagine there are 3 rotors in the enigma machine and you can swap order. Since it is combinations, there are 6 combinations since
3x2x1=6.
The third column is the ring setting, Each rotor has 26 letters, a b c d until z. Since there are 3 rotors, each with 26 letters, 26x26x26= 17576.
The fourth column makes pairs. These pairs are the intructions to setting up the plugboard. The total number of ways you can make 6 pairs of the letters out of the alphabet is 100391791500, which is near a hundred billion.
26! (this is how many ways to order 26 alphabets) divided by 14! (divided by the number of ways to order the letters that are not swapped) then divide by 6! (divided by the number of ways to order the pairs) lastly, divide 2^6 , since the other does not matter.
= 100391791500
The final column, which is called the ground setting, which is the rotor starting position. Each rotor has 26 starting places, the first one has 26, the second one has 26 and the third one has 26, the total number of starting positions is the same, which is 17576.
26x26x26=17576
Then the total number of ways to set up the enigma is the product of all 4 numbers, which is around 10000 trillion.
100391791500 x 17576 x 17576 x 6 = 1.06 x 10^16
Sending a secret setting
There are hundreds of enigmas sent everyday during World War 2, then if they were all sent with the same key and security, it would be bad security.
So each message has its own secret starting position. For example, set the machine’s ground setting to abc. The operator could then pick its own secret starting position, lets say its xyz. The operator would send his own secret message in code to the reciever at the beginning of the message. The enigma machine would then be used to encode the three letters xyz. The reciever would then type xyz two times, which xyzxyz would turn into 6 letters of code, for example, JTEQFG. The reciever would then put JTEQFG at the beginning of the message. On the other side, the operator would then use the enigma machine to decode the 6 letters JTEQFG, and he would get XYZXYZ. The operator would then know it is the secret setting for the rest of the
message.
Conclusion
I conclude that the Enigma machine was a huge success and was a great piece of engineering since there was a lot of combinations to set up the enigma machine, which is 10000 trillion. It was a successful piece of engineering since the Enigma machine was the machine that could actually use combinations and permutations to be used for sending messages, and secret messages. It was the great Alan Turing, who was British that actually deciphered the first German message. For my further investigation, I would like to know how the battery keeps on changing to the bulb in the lightboard and the math behind it.