DR. JAMES GRIME: This is such an
inspiring story, because it shows how mathematicians
can save lives. We're talking about one of
the most famous cipher machines of all time. It's a code machine called the
Enigma machine used by Nazi Germany in World War II to send
secret, coded messages. And we've got one over here. This is not a copy. This is not a replica. This is an original Enigma
machine actually used in World War II. This was made in 1936. It's an army Enigma machine. Let's see how it works. So this belongs to a man
called Simon Singh. He's an author. He writes popular
science books. And he lends it to
the University of Cambridge, where I work. This was found in a French
field, I was told, by an American cryptographer
after the war. And he took it home. So I guess he took it home
as his souvenir. Yoink, that's mine. The guy who found it died
about 12 years ago. And when he died, that's
when Simon Singh bought the machine. We're going to send a message. Now, we were talking about what
to send before, so we send Numberphile. Let's turn this into
Enigma code. So I'm going to type
in N to begin with. And if you can see there,
the letter Y lights up. So N becomes Y. Your
code lights up. Let's write that down. Let's do the next letter, U. U
becomes T. Let's do M. And I've got H this time. B. And E is W. Notice here, we had
Y turn up twice. And they turned up for two
different letters. So N became Y, and
then later on, E became Y. That's unusual. The other unusual thing you may
notice is that the two E's in Numberphile have turned into
two different letters. So here, E became Y. But
the second E became W. Now, this is why the Germans
thought they had an unbreakable code. Old-fashioned code in the past,
pen-and-paper codes that they used to use, if you had
the same letter, it would become the same letter
in the code. Enigma is different. Probably, each time you did it,
you would get a completely different code. Now, if we can break this code--
and by we, I mean the Polish, the British,
and the Americans-- if we can break this code in
World War II, we'll be able to read those German
secret messages, which is what we did. So let me show you how
this machine works. Let me open up the machine. So we have three things
here at the top. These things are
called rotors. And inside those rotors, try
and imagine lots of wiring. And it's all crisscrossed
wiring as well. Now, if I press a letter, have
a look what happens. The rotors move. When this rotor does a full
turn, it'll kick the next rotor one place. And then, the right-hand
rotor keeps going. Eventually, the middle rotor
does a full turn, and it will kick the left-hand
rotor one place. So a fast rotor, a middle
rotor, and a slow rotor. Imagine it like it's hands on
a clock, like you've got a minute hand and an hour hand
and a second hand. That's the idea. Brilliant machine though
this is, all it is, really, is just a circuit. Here is a battery. You can see that's a modern
battery in there. We've converted this. And this battery is connected
to a bulb. And it lights up. It's the most simple thing you
can make-- a battery and a bulb, the bulb lights up. And that's all it is. The clever bit is the
wires of the circuit are inside the rotors. So when the wires turn, the
battery will connect to a different bulb. Let's try and see that happen. So I'm going to turn the
wires, and the battery connects to this bulb. Do it again, I'm going to turn
the wires, and the battery connects to this bulb. It changes. Do it again, turn the wires,
and the battery connects to this bulb. And that's why it changes each
time, because it has moving parts inside. But otherwise, it's just
a simple circuit. I think we need to know
how to decode. This would be no good if we
can't decode our message, so let's do that. First of all, we're going to
take our code here, we need to know what the setting was. Now, if you notice, in
the little windows, we have three numbers. It's like a combination lock,
like a bike lock. Now, I took a note of what
those three numbers were. Those three numbers, when we
started writing Numberphile, were a 13, 9, and 21. So the idea is, you would type
in your message, and you would get a code. The machine itself doesn't
transmit, so you would have to write that code on
a piece of paper. So that piece of paper would be
given to the radio operator who would then transmit
the message by radio by Morse code. That would travel miles away. So now imagine a second German
officer, maybe on a ship somewhere in the ocean,
now he's tuning into that radio signal. He can hear the code. It's coming in, and
he writes it down. Now, this second officer
will have an Enigma machine as well. And his Enigma machine
is exactly the same as the first one. So there's something
I have to do. I now need to set these rotors
to the correct position. So we've got this code over
here, Y-T-H-M-Y and whatever. Let's type that into the
machine this time and see what happens. OK, so we start with Y. Y
becomes N. T becomes U. H becomes M. M, in the code, is a
B. Y. I. F. And W, finally, is E. So each rotor has
26 starting places. In fact, these rotors come
out and swap over. In fact, again, they had five
rotors to pick from. They had a box of five rotors. You would pick three
from a box of five. And already, we've got thousands
of settings. And each one will produce
a different code. So we're going to work out
those combinations. So the first thing I said was,
we pick three rotors from a box of five. In the first slot, you would
have five rotors to pick from. Once you've picked that, in the
second place, you would have four rotors to pick from. And in the final place, the
third place, you would have three rotors left. And you multiply those
numbers together. So it's 5 times 4 times 3. So there are 60 ways that you
can put in three rotors from a choice of five. We have 26 starting positions
for each rotor. So we have 26 choices for the
first one, 26 choices for the second one, and 26 choices
for the third one. And if you do that, you get
26 cubed, which is 17,576. It gets worse, because the
military, the Army, Air Force, Navy had something extra. The commercial machines-- if you were a bank or a
business, you could buy an Enigma machine to use yourself
to send your own secrets. But the military had
an extra bit. They had this thing at the
front of the machine. Now, this is called
a plugboard. And it's like an old-fashioned
telephone switchboard, like an old patchboard. We have 10 of these wires. And each of these wires
connect two letters into a pair. So in this case, you might see,
the letter Q is going to connect with the letter E. Now
you make 10 of those pairs. Two letters in a pair
will swap over. So if Q is connected to E, then
Q and E would swap over. That's an extra level of
scrambling only available to the military. Now, this had the most number
of combinations. Now, this calculation is going
to be the hardest calculation, but we can do it. So there are 26 letters
in the alphabet. How many ways to arrange
26 letters? Well, it's 26 times 25 times
24 all the way down to 1. That's 26 factorial. But we don't want every
combination of 26 letters. We only want to make 10 pairs. So that means there are 6
letters left over, which we don't care about. Because we don't care about
them, that means we're allowed to divide. And we're going to divide
by 6 factorial. Now there are 10 pairs. We don't care what order
those 10 pairs are in. Because we don't care
about it, we can divide by 10 factorial. And the last thing to divide
by, 2 letters in a pair. Well, if I swap them
over, that would still be the same pair. If I had A and B, that's the
same as B and A. That's still the same pair. So I can divide by 2. And I do that for each pair. There's 10 pairs, so I'm going
to divide by 2 10 times. Just 2 to power 10. And that is how many ways that
you can connect 20 letters into 10 pairs on the front
of the machine. How big is that number? Shall we do that? 150 trillion, 738 billion,
274 million, 937,250. So the total, 158 quintillion,
962 quadrillion, 555 trillion, 217 billion, 826 million,
360,000 flat. That is the total number of
ways that you can set the Enigma machine. This would be an army Enigma
machine around about 1939. -You are telling me that the
Germans would send each other messages like Numberphile
or send-- DR. JAMES GRIME: Yeah, they
did that all the time. -Or send the U-boat to this
position or whatever. But how were they telling each
other their plug settings and their starting rotor numbers? DR. JAMES GRIME: So this
is very important. So these two people, who are
miles apart, need to have the same setting. Now, the setting was written
down for you on a piece of paper. What they would have is a sheet
of paper like that. And it would be a big
sheet of paper for each day of the month. So it was a monthly sheet. For each day of the month, they
told you how to set the machine for that day. If you didn't have this sheet
of paper, you wouldn't know what the setting was that you
had to use for that day. And nice, little story, the Navy
would write these code books in soluble ink. So if you get sunk, if you get
caught, if you throw the code book into water, that's how
you keep the secret. -It sounds to me that all you
need is an Enigma machine and a copy of that book,
and you know everything they're saying. DR. JAMES GRIME: You would do. So if you had the machine and
you had that code sheet, you would be able to decode
all the messages. Fantastic. Great. But we had the machine. And once we've had the machine,
we can pull it apart and find out how it works. Great. But it was getting those
code sheets. That was difficult. They were monthly. They would change every month. If you did capture them, which
we did occasionally, you could use it until it runs out. But without a code sheet, you
would have to break the code. And you would have to do
that with mathematics. -What was the key? What was wrong with Enigma? What was its weakness? DR. JAMES GRIME: Let's have a
look at what the flaw is in the Enigma machine. So the Germans thought
it was unbreakable, but there is a flaw. If I press a letter K--
WHAT KIND OF SOPRANO EDITING IS THIS WHAT IS THE FLAW I MUST KNOW !! ΰ² _ΰ²
Where is part 2? It ended mid sentence just as he was going to show the flaw in the Enigma machine.
Wow, that was fascinating. I watched its entirety, got excited, and then subsequent "flaw" video isn't available yet... :(
http://enigmaco.de/enigma/enigma.swf
Rotors: I,II,III, Start positions: H,D,X Steckers: --
MKGFDIMSXBWHZKYBMHSDRLWN
I did my History Day project on this in 8th grade. It's a little depressing that literally this entire video was news to me...
That's Numberwang!
Having just read Cryptonomicon, this is interesting as fuck.
germans invented the most amazing stuff during war. lets hope we dont get another one.
If you like cryptography, I really recommend the series on cryptography made by Khan Academy. https://www.khanacademy.org/math/applied-math/crypt/v/intro-to-cryptography?v=Kf9KjCKmDcU It consists of several videos, some of them pretty long... but you get the general idea of how asymmetric encryption works (see RSA).
They also mention the Enigma Machine as a study case, so you can see what they mention in this video too :)