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Numberphile - The Fatal Flaw of the Enigma Code Machine

radx says...

Edit: Oh boy, wall of text crits for 10k.

His explanation was rather short and somewhat misleading. Maybe they thought a proper explanation would have been too dry or too lengthy to be of any interest for a sufficient number of their viewers.

tl:dr

If all rotor settings are indicated to be correct, a feedback loop within the circuit indicated a subset of correct connections on the plugboard, even if the initially assumed connection turned out to be wrong. It didn't show all connections, but enough to run it through a modified Enigma to determine if it's a false positive or in fact the correct setting. If it was correct, the rest could be done by hand.

----------------------- Long version -----------------------

Apologies in advance. We had to recreate parts of the Bombe as a simulation, but a) it's been a while and b) it was in German. I'll try to explain the concept behind it, hopefully without screwing it up entirely.

The combination of clear message and code snippet (2:25) is called a crib. This can be used to create a graph, wherein letters are the vertices and connections together with their numerical positions are the edges.

For example, at position 1, "A" corresponds to "W". So you'd create an edge between "A" and "W" and mark that edge as "1". At position 4, "B" corresponds to "T", so there's the edge marked as "4". All edges are bidirectional, the transformation at a specific position can go either way.

Once your graph is finished, you check for loops. These are essential. Without loops, you're boned. In this case, one loop can be found at positions 2,3,5 in form of "T->E->Q->T".

Here the Bombe comes into play. It uses scramblers, each combining all three rotors plus reflector of an enigma into one segment. This way, one Enigma setting is functionally equal to a single scrambler.

Now you can use those scramblers to create an electrical circuit that corresponds to your graph -- scrambler = edge. All scramblers are set to the same initial configuration. The first scramber remains at in the inital configuration, while the second and third get configurations in relation to their edge's numerical value. Configuration in this case means the value of their internal three rotors, so there are 26*26*26 possible settings within each scrambler.

It's basically a sequence of three encryptions.

Example: in our little TEQ triangle, the first scrambler (TE, 2) gets a random starting position. The second scrambler (QE, 5) gets turned three notches, the third scrambler (QT, 3) gets turned one notch. The initial configuration might be wrong, but only the relation between the scramblers matters. A wrong result simply tells you to turn all scramblers another notch, until you get it right.

You have a possibly correct setting when the output matches the input. Specifically, a voltage is applied to the wire of letter "T", leading into the first scrambler. And on a test register attached to the last scrambler, the wire of letter "T" should have a voltage on it as well. If the setting is incorrect, a different letter will light up. Similarly, all incorrect inputs for this particular setup will always light up a different letter at the the end, never the same (thanks to the reflector). If output equals input, you're golden. And if several loops are used, all with the same input/output letter, each of their outputs must equal the input.

To reduce the number of false positives, you need as many connected loops within the crib as possible.

So far, that's an Enigma without a plugboard. To account for that, they introduced feedback loops into the circuit. In our small scale case, the output of the third scrambler would be coupled back into the input of the first scrambler. The number of loops determines the number of possible outcomes with each specific setting. All of these are fed back into the first scrambler of each loop.

The plugboard, however, changed the input into the system of rotors. Instead of a "T" in our example, it might be a "Z", if those two letters were connected on the board.

A random hypothesis is made and fed into the machine. If the scramblers are set incorrectly, a different letter comes out at the end of each loop and is in return fed back into the first scramblers. Result: (almost) everything lights up. If you start with a good graph, everything will light up.

-----
A key element for this was the "diagonal board", which represented a) all possible connections on the plugboard and b) the bidirectional nature of those connections (AB = BA). Maybe it can be explained without pictures, but I sure as hell can't, so "a grid of all possible connections between scramblers and letters + forced reciprocity" will have to suffice.
-----

If, however, the setting was correct, a wrong hypothesis for the input connection merely meant that everything except the right connections was lit up.

Let's say the fix point of the loops in our graph is the letter "T". We assume that it's connected to the letter "Z" on the plugboard. A voltage is applied to "Z" on the test register, and thereby inserted into the circuit at the first scrambler. Loop #1 applies voltage to the letter "A" on the test register, #2 lights up "B", #3 lights up "F". These three outputs are now fed back into the first scrambler, so now the scrambler has voltage on ZABF, which in return lights up ZABF+GEK on the test register.
This goes on until everything except "U" is lit up on the test register. That means three things: a) the settings are correct, b) the hypothesis is wrong, c) "T" is connected to "U".

Reasons:
a) if the settings were incorrect, the entire register would be alive
b) if the hypothesis was correct, only the letter "Z" would be alive on the register
c) due to the feedback loop, the only way for the output to be "U" is if the input was also "U", and the reciprocity within the system makes it impossible for any other input to generate the output "U". Since "T" was the fix point for our loops, "T" is connected to "U".

Similarly, if the initial hypothesis is correct, everything on the test register except "U" stays dead.

The diagonal board provides registers for every single letter and allows the user to pick one as a test register. During operation, all the other registers serve as visual representations of the deductions based on the initial hypothesis. So you actually get to see more than just the initial connection, all based on the same concept.

rychan said:

I do not understand at all why finding one contradictory plug setting, e.g. (t a) and (t g), means that every other plug setting you found during that trial was wrong. That cannot possibly be true. The space of possible plug connections (on the order of 26*25) is too small. You've probably got millions of trials that end in conflicting plug settings. You would end up invalidating all of them. I must be misunderstanding what he was trying to say.

WTF: Ukrainian Plant Radio

Helicopter Rescue Of An RC Plane In the Tree Tops

First Person View Of A Lion Encounter

Porksandwich says...

Race car drivers, sky divers, scuba divers, high voltage line repair guys, anyone who works around machinery, tree trimmers, .......etc. Lots of jobs out there can kill you in freak ways that no matter how much you prepare you'll still die. And none of the stuff you're working on will protect you either when those things happen.

At least with an animal, if you're in a pack of them one going nuts might end up being the pack of them on you or the others protecting you. A lot like groups of people...once you get enough people together, one guy doing something stupid go either way. More people joining in or people quashing it.......

Think it'd be cool, especially if you raised them....you could probably tell more easily WTF is up with a big cat than WTF up with your teenage daughter.

ReverendTed (Member Profile)

GeeSussFreeK says...

Safe nuclear refers to many different new gen4 reactor units that rely on passive safety instead of engineered safety. The real difference comes with a slight bit of understanding of how nuclear tech works now, and why that isn't optimal.

Let us first consider this, even with current nuclear technology, the amount of people that have died as a direct and indirect result of nuclear is very low per unit energy produced. The only rival is big hydro, even wind and solar have a great deal of risk compared to nuclear as we do it and have done it for years. The main difference is when a nuclear plant fails, everyone hears about it...but when a oil pipeline explodes and kills dozens, or solar panel installers fall off a roof or get electrocuted and dies...it just isn't as interesting.

Pound per pound nuclear is already statistically very safe, but that isn't really what we are talking about, we are talking about what makes them more unsafe compared to new nuclear techs. Well, that has to do with how normal nukes work. So, firstly, normal reactor tech uses solid fuel rods. It isn't a "metal" either, it is uranium dioxide, has the same physical characteristics as ceramic pots you buy in a store. When the fuel fissions, the uranium is transmuted into other, lighter, elements some of which are gases. Over time, these non-fissile elements damage the fuel rod to the point where it can no longer sustain fission and need to be replaced. At this point, they have only burned about 4% of the uranium content, but they are all "used up". So while there are some highly radioactive fission products contained in the fuel rods, the vast majority is just normal uranium, and that isn't very radioactive (you could eat it and not really suffer any radiation effects, now chemical toxicity is a different matter). The vast majority of nuclear waste, as a result of this way of burning uranium, generates huge volumes of waste products that aren't really waste products, just normal uranium.

But this isn't what makes light water reactors unsafe compared to other designs. It is all about the water. Normal reactors use water to both cool the core, extract the heat, and moderate the neutrons to sustain the fission reaction. Water boils at 100c which is far to low a temperature to run a thermal reactor on, you need much higher temps to get power. As a result, nuclear reactors use highly pressurized water to keep it liquid. The pressure is an amazingly high 2200psi or so! This is where the real problem comes in. If pressure is lost catastrophically, the chance to release radioactivity into the environment increases. This is further complicated by the lack of water then cooling the core. Without water, the fission chain reaction that generates the main source of heat in the reactor shuts down, however, the radioactive fission products contained in the fuel rods are very unstable and generate lots of heat. So much heat over time, they end up causing the rods to melt if they aren't supplied with water. This is the "melt down" you always hear about. If you start then spraying water on them after they melt down, it caries away some of those highly radioactive fission products with the steam. This is what happened in Chernobyl, there was also a human element that overdid all their safety equipment, but that just goes to show you the worst case.

The same thing didn't happen in Fukushima. What happened in Fukushima is that coolant was lost to the core and they started to melt down. The tubes which contain the uranium are made from zirconium. At high temps, water and zirconium react to form hydrogen gas. Now modern reactor buildings are designed to trap gases, usually steam, in the event of a reactor breach. In the case of hydrogen, that gas builds up till a spark of some kind happens and causes an explosion. These are the explosions that occurred at Fukushima. Both of the major failures and dangers of current reactors deal with the high pressure water; but water isn't needed to make a reactor run, just this type of reactor.

The fact that reactors have radioactive materials in them isn't really unsafe itself. What is unsafe is reactor designs that create a pressure to push that radioactivity into other areas. A electroplating plant, for example, uses concentrated acids along with high voltage electricity in their fabrication processes. It "sounds" dangerous, and it is in a certain sense, but it is a manageable danger that will most likely only have very localized effects in the event of a catastrophic event. This is due mainly to the fact that there are no forces driving those toxic chemical elements into the surrounding areas...they are just acid baths. The same goes for nuclear materials, they aren't more or less dangerus than gasoline (gas go boom!), if handled properly.

I think one of the best reactor designs in terms of both safety and efficiency are the molten salt reactors. They don't use water as a coolant, and as a result operate at normal preasures. The fuel and coolant is a liquid lithium, fluoride, and beryllium salt instead of water, and the initial fuel is thorium instead of uranium. Since it is a liquid instead of a solid, you can do all sorts of neat things with it, most notably, in case of an emergency, you can just dump all the fuel into a storage tank that is passively cooled then pump it back to the reactor once the issue is resolved. It is a safety feature that doesn't require much engineering, you are just using the ever constant force of gravity. This is what is known as passive safety, it isn't something you have to do, it is something that happens automatically. So in many cases, what they designed is a freeze plug that is being cooled. If that fails for any reason, and you desire a shutdown, the freeze plug melts and the entire contents of the reactor are drained into the tanks and fission stops (fission needs a certain geometry to happen).

So while the reactor will still be as dangerous as any other industrial machine would be...like a blast furnace, it wouldn't pose any threat to the surrounding area. This is boosted by the fact that even if you lost containment AND you had a ruptured emergency storage tank, these liquid salts solidify at temps below 400c, so while they are liquid in the reactor, they quickly solidify outside of it. And another great benefit is they are remarkably stable. Air and water don't really leach anything from them, fluoride and lithium are just so happy binding with things, they don't let go!

The fuel burn up is also really great. You burn up 90% of what you put in, and if you try hard, you can burn up to 99%. So, comparing them to "clean coal" doesn't really give new reactor tech its fair shake. The tech we use was actually sort of denounced by the person who made them, Alvin Weinberg, and he advocated the molten salt reactor instead. I could babble on about this for ages, but I think Kirk Sorensen explains that better than I could...hell most likely the bulk of what I said is said better by him



http://www.youtube.com/watch?v=N2vzotsvvkw

But the real question is why. Why use nuclear and not solar, for instance?

http://en.wikipedia.org/wiki/Energy_density

This is the answer. The power of the atom is a MILLION times more dense that fossil fuels...a million! It is a number that is beyond what we can normal grasp as people. Right now, current reactors harness less that 1% of that power because of their reactor design and fuel choice.

And unfortunately, renewables just cost to darn much for how much energy they contribute. In that, they also use WAY more resources to make per unit energy produced. So wind, for example, uses 10x more steal per unit energy contributed than other technologies. It is because renewables is more like energy farming.

http://videosift.com/video/TEDxWarwick-Physics-Constrain-Sustainable-Energy-Options


This is a really great video on that maths behind what makes renewables less than attractive for many countries. But to rap it up, finally, the real benefit is that cheap, clean power is what helps makes nations great. There is an inexorable link with access to energy and financial well being. Poor nations burn coal to try and bridge that gap, but that has a huge health toll. Renewables are way to costly for them per unit energy, they really need other answers. New nuclear could be just that, because it can be made nearly completely safe, very cheap to operate, and easier to manufacture (this means very cheap compared to today's reactors as they are basically huge pressure vessels). If you watch a couple of videos from Kirk and have more questions or problems, let me know, as you can see, I love talking about this stuff Sorry if I gabbed your ear off, but this is the stuff I am going back to school for because I do believe it will change the world. It is the closest thing to free energy we are going to get in the next 20 years.

In reply to this comment by ReverendTed:
Just stumbled onto your profile page and noticed an exchange you had with dag a few months back.
What constitutes "safe nuclear"? Is that a specific type or category of nuclear power?
Without context (which I'm sure I could obtain elsewise with a simple Google search, but I'd rather just ask), it sounds like "clean coal".

Last jump from high voltage line

Tree Branch on Powerlines - High Voltage Wicked Effect

Payback says...

>> ^ForgedReality:

Surprising how it didn't trip any transformer fuses. Around here, whenever a squirrel tries to go from one live wire to another while touching both, he gets fried, and everyone on that circuit loses power until they can replace the fuse. It doesn't keep pumping power through it.


Probably 3 phase industrial lines.

Tree Branch on Powerlines - High Voltage Wicked Effect

Tree Branch on Powerlines - High Voltage Wicked Effect

Tree Branch on Powerlines - High Voltage Wicked Effect

xxovercastxx (Member Profile)

Tree Branch on Powerlines - High Voltage Wicked Effect

Tree Branch on Powerlines - High Voltage Wicked Effect

There is a secret message on your digital music player (Blog Entry by dystopianfuturetoday)

Boise_Lib (Member Profile)

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