Flaw in the Enigma Code - Numberphile

The flaw which allowed the Allies to break the Nazi Enigma code.
More links & stuff in full description below ↓↓↓
First video explaining Enigma: • 158,962,555,217,826,36...
Extra footage: • Video
Brown papers on ebay: bit.ly/brownpapers
Periodic Videos: / periodicvideos
This video features Dr James Grime discussing Enigma, the Bombe and Alan Turing.
James' "day job" is touring with the Enigma machine - he could even visit you - see more at enigma.maths.org/content/proje...
The maths of breaking the Enigma by James Grime enigma.maths.org/content/sites...
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[@chaslington]

The fact that the messages were in German was top level encryption in itself.

[@jfloresmac]

German is an easy language to learn. It is very regular. You should try it.

[@Leon_der_Luftige]

Edigy Maybe on paper. In reality, it really isnt't.

[@ali-azizimayer-peters6686]

Ja lern Deutsch ! Das ist echt eine wundervolle Sprache, du musst dich nur an die tollen Endlos-Koffer-Wörter gewöhnen. :D Viel Spaß dabei !

[@elonmush4793]

English and German are fairly close relatives. It's not like learning Japanese or something like that.

[@TotalImmort7l]

@@jfloresmac i prefer google translate and russian over Deutsch without an `o`.

[@xXFluffers]

I love how the germans and british built these complex encryption machines, but the US just plopped two Navajo indians on two ends of a radio line and no one could figure out what they were saying because no one could speak Navajo, and the only way to get someone who could speak Navajo would be to kidnap a Navajo indian, lol

[@Geographus666]

That is "security by obscurity", which is something you do not want in cryptography.

[@daniellbondad6670]

But Navajo is an extremely difficult to learn language once you learned another one(definitely English and/or Japanese).

[@daniellbondad6670]

Different grammar system and phonology system.

[@valante7]

Furball: That is so true. I think that in the movie called "Windtalkers" they showed how this was done, but the US armed forces also had a big job of protecting the Navajo Indians from being kidnapped.

[@dumbcowgomoo8923]

I've read a book based on this and yea, they did use a code on top of the language. If I remember correctly they used many words to describe stuff. For example, frog would be for an amphibious vehicle since tadpoles turn into frogs.

[@Quasihamster]

"The Germans sent a weather report. It was the same every day." -An Englishman.

[@appleslover]

?

[@Jajdjejwi28]

@@appleslover reference to the blitz

[@moomoomachines7193]

@@appleslover England has the same weather every day.

[@DenyLoneWolf]

I've seen what u did there! AHAHAHA Nice one, nice one.

[@birdy_]

Gutes Wetter, ab in den Kampf!

[@yungee3921]

I've invented an improvement on the Type X machine where a letter ALWAYS becomes itself! ;-)

[@pinkponyofprey1965]

It existed even before the WW I and was known as a typewriter.

[@arkivelikesmilk6046]

This why using 100% of your brain is dangerous

[@pholzman2918]

:-} There must be a political Enigma machine, politicians never say exactly what they are thinking!

[@nukehunterlp1371]

Pathetic!

[@aniruddhalimaye2616]

Ha ha ha great idea !

[@danielharrington8691]

How dare you make me enjoy Maths.

[@chachnaq7337]

‍ ‍ so true

[@JorgetePanete]

Daniel Harrington you forgot the question mark

[@davenn7597]

‍ ‍ Numberphile is fun Not math itself

[@jrk1666]

Better enjoy math than meth

[@jfloresmac]

Math can be your friend. Teachers are the enemy (the great majority of them) Just remember, find the little x at the end of the rainbow using the given formulas.

[@skeetersorenson4909]

It's so sad what happened to Alan Turing after the war.

[@frozenfeet4534]

?

[@skeetersorenson4909]

Garen Crownguard He was found to be homosexual, and was forced to take hormonal medications or something. He committed suicide.

[@MadMargaretGaming]

Skeeter Sorenson It's not known whether it was suicide or accidental poisoning, but he did die from cyanide poisoning.

[@Audiack]

Imitation Game = Dramatized Version. It is said that Alan was quite happy with his life.

[@MadMargaretGaming]

Audiack Alan was probably quite happy before he was forced to ruin his own life.

[@Kredroth]

This has to be one of the simplest ways I’ve seen something so complex being explained. Great video.

[@Bri_bees]

My mom worked in the Weather office at bletchley park. One of the keys to successful code braking was nothing to to with codes or math but hard work and filing. The ladies endlessly filled out card's , cross-referencing every operator . This allowed them to get a feel for operators who would say use there mom's name each day as a test message and give them a starting point.

[@eugenio5774]

ohh, this sounds like typing "signature" with the morse code! morse operators have a typing style, and you can actually recognise individual operators by their rythm and speed. I remember reading somewhere that the english could pinpoint where Rommel was because they knew the style of his morse typist, so once they pinpointed where he was, bam, by extension they knew where rommel was.

[@lozoft9]

This is something that the movie The Imitation Game (terrible name) depicted. Women were crucial to cryptanalysis b/c they could more readily pick up on social and speech patterns. In other words, they were the first social engineers.

[@pauls3075]

I think your mum should be sent to prison for divulging state secrets. However you are a liar because you are American and your 'mom' wouldnt be working at Bletchley Park! Also your prom was in 2020 so your 'mom' would have been at least 75 when you were born! I believe you call this 'Stolen valour' you are a nasty person.

[@johnbennett757]

@@lozoft9 Not the first time that women's contribution went unappreciated.

[@Xehemoth]

Even considering the flaw of the Enigma, it is an incredible machine even to this day. It was probably one of the most innovative machines of that time.

[@KipIngram]

It really is absolutely brilliant.

[@IchWillNichtMehr853]

German quality

[@thebanjo7023]

Right up until the point a better machine cracked it

[@LS-Moto]

@@thebanjo7023 There will always be better machines as time goes on. You could crack the type x machine today as well with brute force computer software. It would take about the time of making a cup of coffee to the duration of a comfortable shower. What makes the cracking of Enigme so unique is the fact, that it was done by hands and a manual machine. At that time, this was a huge breakthrough.

[@legendgames128]

And to think that now we could write code (as in C++ or Python) to mimic that same machine or better.

[@therealzilch]

It's still a bit surprising that the engineers who developed the Enigma, a very sophisticated bit of cryptology, didn't see the flaw of not allowing a letter to represent itself, which seems pretty obvious in hindsight. But I guess people do make mistakes. Thanks, James, for this very clear explanation. Lunch is on me if you're ever in Vienna, Hitler's favorite city.

[@therealzilch]

jonesgerard While I agree with you, and the Bible, about the dangers of declaring oneself wise, I doubt that Social Darwinism was the cause of this blunder.

[@therealzilch]

jonesgerard Yeah, I basically agree, but it's still a puzzling mistake when the Germans were probably the best engineers in the world at that time. Their bad military decisions- invading Russia in winter was another one- are a different sort of error, caused by pride and conceit. Pride and conceit don't necessarily make you susceptible to mistakes in formal systems of logic like math (including cryptography), though.

[@therealzilch]

jonesgerard Can you give me an example?

[@MultiAlxndr]

+Scott Wallace aka the titanic

[@therealzilch]

MultiAlxndr But the problem with the Titanic was not a simple mistake in logic, as with the Enigma, but rather a very general underestimation of what an iceberg could do. To correct the problem with the Enigma would have merely required a very minor change in one tiny part of the machine, very easy to define and accomplish. In contrast- the Titanic would need to have been redesigned in very complex ways, and it still would have required a captain who would at least sometimes avoid icebergs- or is it possible to make a ship that can never be sunk? cheers from rainy Vienna, Scott

[@mghyy2846]

As used in practice, the Enigma encryption was broken from 1932 by cryptanalytic attacks from the Polish Cipher Bureau, which passed its techniques to their French and British allies in 1939. Subsequently, a dedicated decryption centre was established by the United Kingdom at Bletchley Park as part of the Ultra program for the rest of the war.

[@blipco5]

The British should have called it the X-Box instead of the X-Machine, they would have made a fortune.

[@PADARM]

X-Bomb

[@jfloresmac]

Used exclusively by the X-Men. They would have won the war in months and not years...

[@Pulsonar]

blipco5 Then perhaps Microsoft would have used X-machine for their game console name.

[@btnt5209]

Not necessarily since they are in 2 separate industries

[@ISO-Certified-pimp]

And whenever a kid wanted a xbox their parents accidently bought them an enigma decoder

[@manueltrinidad9970]

Welp, the Enigma had a little flaw, but the worst flaw was made by te german by ending each message with the word Hitler...

[@EngineersAnon]

Also, just don't bother trying any secure message via Enigma on Hitler's birthday. Since sending him birthday wishes was essentially non-discretionary, there were plenty of known-plaintext messages that day.

[@acousticviking7499]

No. It wasn't quite that simple. That you got from "The Imitation Game". As James mentioned they used weather reports, and other Crib Words. The German Navy used even 4 Letter Codes to encode 150 or so phrases, eg. AABB = "I attack convoy" etc.. you shouldn't take movies too serously ;)

[@Doriandotslash]

Agreed @fjordweingeist . I found that comment funny as well. Movies are not history lol

[@ianmoseley9910]

According to a booklet I got from the Station X museum, one remote German post usually sent the message "nothing to report" which helped crib one of the daily code settings.

[@51WCDodge]

The British also used Frequency analysis. Which station sent a message and who jumped at the other end. That allows you to surmise that Station A reports to Station C that reports to Station B. Then when Station B transmits A & C jump ie B is the lead. The reason for Meterology reports being important, is the information cannot be changed, it has to be taken from certain points at a set time to be any use. As the only source of such information was either U Boats or Long range aircraft by Enigma, and through HF/DF the Allies knew the area from which the report was made. So Allied vessels and aircraft were recording the same information, wind ,cloud , humidity and barometric pressure. So The Allies already knew what should be in the report. Then when the message reached a German station it was retransmitted to the end user, mostly the Luftwaffe, by Enigma. So you had the same info being transmitted by two stations, at roughly the same time on a regular basis, otherwise the Met info was useless. Add to that long range telegraphy was carrier wave, better known as Morse. Every Morse operator develops a rythum, know as the Fist, it is very distinctive. The British Y Service operators who did the actual . interception of the transmisions became familiar with the operator's fist and also the habits that each operator developed. The Enigma require an intial random setting of the rotas. Think of modern day passwords, how many people actually use a random password for very site? If all else failed then the British would provoke a message. A bombing raid would be carried out, or the guns at Dover would lob a few shells over the Channel. The local garrison would then be likley to report, air attack at certain hour or shells falling in an area in their routine reports.

[@johncgibson4720]

This episode is the most important episode of the numberphile series. And they almost omitted it. They made this episode by accident due to popular comments for another video!

[@joeblow8593]

His explanation of the Enigma code machine is the best I've heard yet from anyone. Kudos

[@christopherpeery7436]

"The inner machinations of my mind are an enigma" ~ alan turing

[@davidbaird1090]

I absolutely love this guy, he seems so genuinely excited to tell us about this machine! His other videos are all the same way, excellent content!

[@konstanty8094]

Additional weakness is the Germans have very long words, which makes it easier to guess if the word fits.

[@Droggelbecherbot]

not if you leave out the spaces between the words, which would be a no brainer. would be surprised if they didnt do that.

[@ricarleite]

+1234bliblablau No space bar in enygma. Thewordswerekepttigether, like this.

[@16dedikodu34]

Aspecially spelling errors like tigether would make it extra difficult to break

[@l3p3]

+16dedi kodu Git point.

[@rub800]

Let'sGeilo Lp kik

[@theturtlepwn]

this guy is so positive and articulate and excited about math!! i've watched this video a few times now because i'm writing a paper on alan turing and it's so difficult for me to understand how he cracked enigma, but this vid is really helpful

[@quietman482]

Been to Bletchley, did the tour, read a couple of books on Alan Turing but never could get my head around the Enigma Code being a non-mathematician. Watched both the videos and now, thanks to you, I have some grasp on the complexity of the problem and how it was solved. Great videos! Many thanks.

[@brianmurray8199]

For anybody wondering, "But wouldn't the plug board allow you to cause a letter to encrypt as itself? Imagine K after the rotors maps to U. Why not just route U to K via the plug board so that pressing K results in K (in this particular button push)?" Here's why not: The plug board is used both directions. So that U->K mapping on the output would also be a K->U mapping on the input. So that K you entered and hoped to get back out (to avoid this flaw) would become a U before entering the rotors. Now, to get K back as the final answer you still need to get U out of the rotors because of that U->K in the plug board. So, you're left with needing U->U coming from the rotors, which is no different than needing K->K coming from the rotors. The plug board doesn't add the ability for a letter to map to itself, and given that it won't happen in the rotors it won't happen via Enigma.

[@KipIngram]

Yep.

[@espadrine]

I don’t understand why they added a reflector though. It single-handedly weakens the whole cipher. If the circuit was just “plug board, rotors, then directly light up the output letter”, you couldn’t rely on those simplifying deduction shenanigans, and you’d be back on pure brute-forcing the key.

[@AyCe]

@@espadrine And counterintuitive stuff like doing it twice actually making it worse is why people keep warning against trying to be clever with inventing your own crypto method in programming. Just use existing algorithms, properly configured, on your plain data. You trying to "improve" them probably just makes them weaker.

[@greeny-dev]

@@espadrine I think the point of the reflector is to allow for single machine to be used both as encryption and decryption. The current that goes from e.g. K to T needs to also go from T to K (given the same configuration of rotors and plugboard).

[@theaccordian9377]

Why can't the rotors route a letter to itself then?

[@jaredstearns970]

It seems to me that the more messages you send encrypted, the more chances you give your opponents to crack them. Regional weather is generally not a mystery, I would think that it could be sent with much lower level of encryption, or even unencrypted. Not to mention that the forecasts were probably reasonably accurate, so you could compare the actual observed weather conditions to the encrypted message fro additional hints.

[@thewackychaps]

It wasn't the number of messages, all they needed was one to break it with the machine

[@acm8559]

@@thewackychaps Incorrect, the amount of messages were able to give it a pattern that gave them the key to cracking the machine in the first place, repeated phrases and standard formats like the video said. If they didn't have multiple messages to find consistent words, they wouldn't be able to have this so called "key" that reduced the possibilities by considerable powers which allowed the machine to crack the code.

[@thewackychaps]

@@acm8559 you only need one key is what I'm saying, heil Hitler was on every one and Turing's machine only had to check every single combination until they matched

[@davidelliott5843]

Germany used Enigma for every message (repeated or otherwise) because they believed it was 100% unbreakable. The belief was so solid that Soviet Russia used the machines until the 1960s. The British kept their code breaking systems totally, secret because they were reading Russian communications. The fact they could conceal the truth for so long is even more amazing than breaking the code itself.

[@sorio99]

The thing about the Nazis is, for every intelligent idea they had, they had about three absolutely idiotic ones. Including the weather messages, and as thewackychaps mentioned, including something like “Heil Hitler” on every possible message.

[@evilpandakillabzonattkoccu4879]

Just something: I don't really care for math usually but the excitement and fascination that Dr Grimes delivers the information with is contagious. I find myself engaged in a manner that doesnt usually happen when math is involved and thats something special. I try to remember that when I'm teaching my own children....that excitement, curiosity and fascination can be inspired in others. A sincere thank you from me to all of you! Very well done! I'm not new to Brady's work or numberphile but it still amazes me how effective these videos are at teaching concepts. I went too long without saying thanks, imho.

[@matihari79]

Marian Rejewski, Jerzy Różycki i Henryk Zygalski, thank you guys for breaking the enigma code in 1932

[@robertgoss4842]

Zowie. That is some serious spelling. My compliments!

[@GunMeat]

And that smiling Englishitman is lyng worse than Goebbels.

[@vincentfreddoyle7555]

@@GunMeat ?

[@kurumi394]

I'll have a stab at pronouncing these names Ma-ree-an Re-yev-skee Ye-zhi Roo-zhits-kee and Hen-rik Zi-gal-skee Are these acceptable Poles?

[@Knukszt]

@@kurumi394 well, hard to tell from writting. English Y means Polish J, and Polish Y doesnt have an English counterpart as far as im aware, so its deffinitely not spoken as "i". If i had to describe it, it sounds like a drunkard would make a caveman sound when he gets mad at you. But from what i see, id say its like, how a typical englishman would pronounce polish words with "broken polish". But thats acceptable especialy if you dont live in poland heh id say.

[@bernardpower5876]

The enigma code was cracked by the polish mathematician Marian Rejewski. He showed this technique to the British and was then sidelined. The computer to achieve the breaking of the code was designed and built (almost single handedly) by Tommy Flowers. The breaking of enigma is largely due to these two who are rarely credited.

[@MomMom4Cubs]

Seeing what happened to the man widely credited (Alan Turing), perhaps that's best.

[@IroAppe]

What did Alan Turing then do? Was he involved at all? Edit: Now I really wanted to know who was responsible for what, and as always, it's a team effort of many people, working at it at different times and locations, until it becomes the one being used. As far as I can see, the mathematician Rejewski indeed figured it out, also there was first the Polish machine Bomba (with an 'a'), that, as numberphile said, was not able to decrypt Navi codes. Then the British came in with the Bombe (with an 'e'), and so far as I have read, Alan Turing designed and produced the prototype, the initial design (whatever they mean by 'produced', I thought that also means to build it, not just design it). Tommy Flowers as far as I can see on his page, did not actually work at all with the Bombe. As far as the information is provided there, Turing wanted him to build a counter for the Bombe (which even on Tommy Flowers page states, that Turing created), but that project was abandoned, and so Tommy Flowers continued to make "Colossus", a machine for decrypting the German Lorenz SZ-40/42 cipher machine, which, and I quote: "was a much more complex system than Enigma". So yes, Tommy Flowers has a significant role, but not quite with the Bombe itself. For the "Colossus", he should be remembered for doing something even more complicated, somehow the world only focuses on the Enigma. So perhaps you confused those two machines?

[@pdwmr]

some facts: On 5 August 2014 the Institute of Electrical and Electronics Engineers (IEEE) honored Rejewski, Różycki, and Zygalski with its prestigious Milestone Award, which recognizes achievements that have changed the world. The uniqueness of the device lay in both the concept of mechanical cipher-breaking and the exceptional mathematical ideas that Polish cryptanalysts employed to crack the supposedly unbreakable encryption mechanism. July 2005 Rejewski's daughter, Janina Sylwestrzak, received on his behalf the War Medal 1939-1945 from the British Chief of the Defence Staff. On 1 August 2012 Marian Rejewski posthumously received the Knowlton Award of the U.S. Military Intelligence Corps Association; his daughter Janina accepted the award at his home town, Bydgoszcz, on 4 September 2012. Rejewski had been nominated for the Award by NATO Allied Command

[@lylaley]

Why sending a weather report encrypted in the first place. Fun fact: After the War the Britons sold Enigma machines to other countries, without telling them that they could decrypt it

[@grayscribe2125]

Just a guess, but I think they sent the weather reports on different frequencies for different parts of the military and different regions. Sending the weather report in the clear would make it easier to know which frequency was used for which region and which part of the military. Given, they could figure that out sooner or later, so on to reason number two. Weather reports give you a time and a place. A weather report about a certain part of the atlantic would indicate that they had ships there or planned to have them there. Just as a weather report for a specific part of england could indicate a bomber raid there. A weather report for Gibraltar could indicate a ship going to try to get through. And so on. The weather report would also indicate the time frame for something going to happen there. Given, you could simply read out the complete weather reports for all of Europe and the Atlantic Ocean, but that might take some time. And some, like submarines, were not always able to wait that long.

[@skalty9868]

Thorsten Lucht having accurate information about the weather patterns of a location, before satellites, would be super valuable.

[@yuxin7440]

The purpose of cryptography are not only for the security of transmission but also guarantee the authenticity of the message because the fact you can decrypt it shows that the sender of the encrypted message are someone who have the password (or settings of the machine in this case). If the weather report is not encrypted, anyone will be able to produce it and thus you cannot verify the sender and the authenticity of the message.

[@doogleticker5183]

A few weather reports allow meteorologists to build an isobar chart...invaluable in predicting wind speed, storms, precipitation, temperature...all useful for being prepared for combat.

[@andrewemery4272]

The weather over German locations would be very useful information for the RAF when planning raids.

[@linkdeminsk]

I am amazed at the piece of editing mastery at 4:13, how the still image comes from and to the video before and after as though it was a still image from a single video, expect DocG keeps talking and his sentence carries over to the end of the still image back into the video... You've done it guys, you broke time.

[@54johnpaul]

You guys are great......I've read widely about Enigma but this is the first time I've seen such a graphic explanation. I understand much more now. Thank you so much.

[@grahammartin9494]

Not being a mathematician I have been struggling with following the Alan Turing story. This has made things much clearer. Thank you so much... will look out for more of your lectures!

[@johnbones261]

The poor Polish guys who really broke the code are completely forgotten. Sad.

[@lhaviland8602]

Muh standing alone. Muh imperialism. Muh abandoning Poland to Stalin to save Greek RAF bases.

[@shazzo3667]

the polish broke it but didnt keep it a secret so the germans made it harder to break and we broke that

[@stevenzhao3414]

Ok to be fair though, the Polish dude who broke it also didn't go on to basically single-handedly start computer science...

[@johnbones261]

@@stevenzhao3414 you've missed the point. Someone got the credit due to some one else.

[@1313tennisman]

@@johnbones261 no the poles broke it and the germas figured it out and made it more complicated and then the british broke the more complicated version turing and co get respect that they deserve

[@foreverkurome]

I remember my teacher showing me this in high school as a sort of motivation for why one might study mathematics, I think only now I appreciate things like this outside of "that's pretty cool bro"

[@maindepth8830]

all of the guests that u btring are always so bright with energy

[@KokkiePiet]

Polish military intelligence broke the enigma initially, Turing automated it, he and others broke the updated versions

[@jaybpl666]

True

[@marekkoacinski500]

Różycki, Rejewski, Zygalski, that was the names of mathematicians, who has broke the code.

[@staliniumprojectile]

8:28

[@aleksanderwierzejski1346]

@@marekkoacinski500 did the author mentioned that?

[@tubemein2007]

@Tweaky Robin If you hate the truth just because it was the poles - polish mathematicians - who did the hard work and NOT Turing himself, well then, there is little anyone can do about it. Just like flat earth and creationists alike they have their own lies built up inside of them and they seem to be willing to get away with it.

[@xkcdstickfigure]

"What could you put in to make it more secure?" An ssl certificate.

[@stargazer7644]

That laughing sound is coming from the NSA

[@alzeNL]

signed by a chinese CA somewhere in the authentication path :D

[@keninswed]

So well done and will most definitely withstand the test of time. This was so exciting to watch, top grade content

[@ayeehdilly]

so glad the decided to do a bigger follow up video, so interesting.

[@samarvora6355]

Absolutely brilliant! Love content like this! Amazing videos, mate! The way he is presenting it, it's pretty clear that he loves the stuff and loves presenting it as well. His energy make sit even better...

[@futhamucka]

What a phenomenal piece of engineering, and a phenomenal team it took to crack it.

[@waheisel]

Fascinating! Great stuff-thank you. The "extra footage" is no longer available as of Jan 2022.

[@chrish.8241]

Great couple of videos, finally explained in a way even I can understand. Many thanks.

[@jbyeats]

Dear Dr Turing , WE want to thank you for your enormous contribution & for your work relating to breaking the German Military codes & to acknowledge your unique input into developing the very first computer. Now -- this won't hurt at all -- Dr Turing. - We just want to CHEMICALLY CASTRATE you. You won't feel a thing.

[@heatherbluelove]

Mentality of 1950's that seems to be stuck to homophobic idiots these days

[@jbyeats]

Unfortunately very much alive today -- just as you say.

[@heatherbluelove]

I live in the middle east Folks are very radical to say the least

[@jbyeats]

For goodness sake -- this was nothing to do with COMMON LAW or ANY LAW. This was the British Establishment deciding that it HAD TO ACT to PROTECT ITSELF because its NO 1 -- COMPUTER SCIENTIST was consorting with young boys for sexual gratification. Turing was followed 24 hrs a day. His phone was tapped. His mail opened. He was a marked man. The British Establishment is ruthless in dealing with any of its KEY PERSONNEL -- " WHO STEP OUT OF LINE " ( They murdered Dr David Kelly -- so that themselves & the Yanks could invade IRAQ .) Turing's problem was that his social behaviour left him open to BLACKMAIL . The Brits were in the middle of a cold war with the Soviets. They knew at that time that their Intelligence Agencies were full of Russian Spies. They could NOT -- UNDER ANY CIRCUMSTANCES -- HAVE TURING BEING USED as a SPY by the SOVIETS. By the way -- your English is Superb. Just look at some of these morons on KZbin -- who know absolutely nothing about GRAMMAR or SYNTAX. -- and ENGLISH is their MOTHER TONGUE.

[@heatherbluelove]

jbyeats hahaha English isn't my mother language yet I am not only decent at the language but I also write poetry in English In any case has anyone seen what is happening in Ukraine?

[@OtakusRUs2]

Such a coincidence that I find these two videos right after I get back from watching The Imitation Game. Wonderful movie, and a wonderful story. I highly recommend it.

[@wagnerrrrr]

「S」 Coincidence, or KZbin spying on you? :)

[@geekymonkey]

+「S」 I came here for Rock Paper Scissors and stumbled upon this not long after watching The Imitation Game. Crazy!

[@LoganCovers91]

+「S」 Not a coincidence, you will notice Facebook does this too, talk about a brand on say Reddit, and surprisingly you'll see an ad for it on Facebook the next day or within the week, it's actually scary and sad

[@mysterious_billionaire]

thanks for the recommendation

[@oldcowbb]

the movie make it way to dramatic, and they make turing a stereotypical nerd

[@bigendianian]

Thanks for the great video. Please make "Extra footage" video available

[@jeffreywickens3379]

Dr. James Grime has such a pleasant, gentle and humble personality.

[@winnerlu2882]

Really helpful with my studying on project!! :)

[@alanthomas8836]

Alan Turin's Bombe machine albeit a substantial help, was really only an extension and development from a device that had been first designed in 1938 in Poland at the Biuro Szyfrów (Cipher Bureau) by cryptologist Marian Rejewski, and known as the "cryptologic bomb" (Polish: bomba kryptologiczna).

[@welshpete12]

Very clear explanation and fascinating thank you for posting !

[@darcynog]

Great video. Clearer explanation than in the books I have read on the matter.

[@ElectricPyroclast]

"Mein Deutsch ist sehr schlecht, enschuldigung." No, James, your German is better than most Americans'.

[@cesaros11]

He's not American though. His German is probably better than most Japanese or Brazilians as well.

[@ElectricPyroclast]

cesaros11 I know he's British. The stereotype for typical Americans is that they are unable to speak anything but American English, and despite that, they make a LOT of spelling, grammar, and pronunciation mistakes.

[@ralusek]

ElectricPyroclast White Americans as opposed to what, Black Americans?

[@ElectricPyroclast]

ralusek I guess my statement wasn't very correct. I guess it's just basically American Americans. Been there, ancestors have also been there, don't care about the rest of the world Americans.

[@ralusek17]

ElectricPyroclast There are a lot of Americans like that, but it's not exactly fair to compare an American only speaking English to a European speaking numerous languages. For one, the United States is pretty expansive, with the majority of Canada and therefore the majority of the entire continent speaking only English. With the much smaller European countries, you're bordering with 2-4 entirely different cultures a few cities away. They have older histories that likely come with their own languages. So on top of the mere size and proximity to different cultures, you have the fact that English is the universal common language of the majority of our media, technology, etc. So from a practical standpoint, there is very little incentive to learn a different language, nor is there any real existing culture in place to be inherited from outside of our relatively young, English speaking colony. It's not an apples to oranges comparison. Even in our education system, learning another language is considered primarily a hobby.

[@gregfaris6959]

I think the most interesting part of the story is the least often told, which is how the Polish managed to crack all but one enhanced variant of Enigma code with a small, desktop machine, while it took Turing and his team years of work, and a machine the size of a Panzer tank do do them one click better!

[@thenerdguy9985]

Also, the fact that poles gave all their findings they had in cracking.

[@dindin3655]

can i learn more about these polish, what's their name?

[@kashmir99scor]

Marian Rejewski, Jerzy Różycki and Henryk Zygalski,

[@E1craZ4life]

8:08

[@88porpoise]

The Polish machines required one machine for every possible rotor order. That was fine when there were three rotors (so six machines required), but became impractical when the Germans went to five rotors requiring 60 bomby (and the Navy soon went to eight, requiring 336 bomby). The Bombes were much larger and more complex than the Bomby, but they were also much more capable. It is also important to note that the biggest accomplishment of Bletchley Park wasn't breaking the codes, it was breaking them fast enough to provide useful information about ongoing operations. The Polish efforts provided the foundation that the British would use, but comparing a bomba to a bombe like that is just silly.

[@johnmiller4859]

The best explanation I've heard. Thank you.

[@normanedwards7220]

I do not understand everything, but I enjoy your presentation, and will continue to replay your videos , thank you for explaining the enigma 😊😊😊

[@nand3kudasai]

the 2nd optimization of using physical electrical connections is really smart (y)

[@justintheoreo]

Praise Benedict Cumb- ... Uh I mean Alan Turing

[@aeriumsoft]

ich liebe kartoffelein

[@parthiancapitalist2733]

Awsomiihill the ch is pronounced /x/ not /k/

[@pppfan103]

It's funny because Cumberbatch looks NOTHING like Turing

[@xandercoulton8541]

Jackson DeStefano he played Alan Turing in the imitation game....

[@MrDaiseymay]

or his brain

[@aucourant9998]

That was really brilliant. So well explained.

[@jsfbr]

Excellent! Send more videos about Enigma, please.

[@OKMX5]

Wow, man would think that some German mathematician would have seen that flaw. Letter never becoming the same letter might see clever to common people but not to someone who knows how ciphers work...

[@xoites877]

Thank you so much for this. The Imitation Game is one of my favorite movies, but understanding the process was beyond its scope. I understand a little bit more now.

[@simonpersonaltrainer5624]

Do some research on the Polish scientists. They managed to crack all but one enhanced variant of Enigma code with a small, desktop machine, while it took Turing and his team years of work, and a machine the size of a Panzer tank do do them one click better!

[@cooldawg2009]

Cant get enough of this subject! Great vid

[@daviddredge1178]

This is an OUTSTANDING video. Thank you.

[@mikosoft]

"Mein Deutsch ist sehr schlecht, Entschuldigung." That was the German bit but with heavy accent. It means "My German is very bad, I apologise".

[@mikosoft]

I scrolled through a couple of comments and didn't see it, so sorry for repeating without knowing.

[@aktan4ik]

+mikosoft Hi (sorry for my bad english)

[@turboapples1233]

mikosoft the accent was confusing couldn't tell if he was apologising or not

[@MrRainierSalu]

when did he say that again?

[@texannationalist5887]

right at the beginning

[@mephostopheles3752]

I want to know how fast our computers can decode Enigma now. Is it faster? Is it instant these days? How far have we come?

[@EbonyWolf.]

By brute force they cannot, not even super computers. By exploiting the flaws they can do it instantly though(also called breaking with cryptanalysis).

[@mhdawod8350]

i think like a 1024 bit

[@EbonyWolf.]

Alkenrinnstet navy enigma has 158,962,555,217,826,360,000 combinations(about 2^67). Our 3ghz computers can at best compute 3*2^30 combinations per second. So it will take 3*2^37 seconds or 13074 years to solve it by brute force. A super computer might break it in several months

[@alkenrinnstet]

Ebony Wolf Putting aside the obvious errors in your arithmetic, you were the one to claim "not even super computers". Also you are using exactly the wrong tool for the job.

[@EbonyWolf.]

Alkenrinnstet Id love to know my arithmetic errors. Also they had one day to break the enigma code. So several months to break a analog coding machine is far far from trivially solvable.

[@mikecobb2466]

Just great! I love this type of stuff, thank you so much for explaining it.

[@mesaplayer9636]

So im a bit confused about what is going on at 5:46 if the first plug board passthrough isn't changing so "t" always turns into "a" why is "a" turning into different letters when i thought you said the rotor isn't changing? I thought the rotor changes once you find a violation. Are the pairs coming from 2:22 because t matches with e,q,b,g? Then my next question would be how much of the message do you have to guess to do this and get a complete code solution because in order to find a violation you would need enough moments in the message where you think t is used in this example.

[@TheEloheim]

@Numberphile: I love the channel! Also, maybe a more overt tribute would not fit the lean format of these episodes, but I think we should be sure to not forget the absolutely reprehensible treatment of Turing by the British government after the war. It may well be different across the pond, but as an American around the age of 30, I've been well aware of Turning's incredible achievements and contributions to the Allied victory in WW2. Alternatively, I had zero clue untill recently that in 1952(!) he was charged and convicted for the crime of homosexuality, and resultingly fired and banned from any future national security work, and forced to undergo chemical castration to leave him impotent. The sum total of this unbelievable public humiliation led him to commit suicide in 1954, at the age of 41, less than 10 years after the end of the war he'd helped win! Speaking personally, as an observer in the 21st century, learning those awful facts of Turing's fate, for me, felt like a punch to the stomach. I know the values of the time may have been different (also Turing was recently pardoned by the Queen), and the point of this post isn't to direct hated toward any persons or institutions, but hopefully to inform some who were not aware, and remind any others, not to forget the human struggles that can lie closer to home than one might rather imagine. So here's to Alan Turing and all those like him, whose stories may not be known.

[@lhaviland8602]

Some people think he may have been murdered to stop him from spilling what he knew to the Soviet. The 50s were wild and yes the UK fight the war for nothing.

[@MJC1124]

Aleph One: I agree with you about the treatment of Turing. However, what happened at Bletchley Park was kept secret until about the 1980s. Those responsible for his trial and criminal conviction would have been totally unaware of the vital role he had in the war effort. As we go though life, some things that were once unacceptable become acceptable and vice versa. I can think of several such examples.

[@nukclear2741]

Lets not forget the polish here…

[@YtubeUserr]

Sorry for my bad England, I'm from English

[@ahmedelshahawy4148]

😂😂😂😂😂

[@zogzog1063]

'snot a problem

[@HolahkuTaigiTWFormosanDiplomat]

NCE

[@jfloresmac]

You dont speak english very goodly, do you?

[@meloniejen8400]

accurate lol

[@deancyrus1]

Thanks for that I've always wanted to know how that works. Now i almost do. Amazing stories behind this machine. Thanks Alan Turing and sorry for the treatment you got after the war.

[@enquiryplay]

Still KZbin's best explanation of the Enigma machine.

[@bardokgokusfather]

I bet one of these views is Benedict Cumberbatch's. To research his role on Alan Turing.

[@kira.b]

Its kinda endearing how excited he is/seems about the machine and the mathematics behind it

[@ivanhornak7644]

I have a question; if all 26 positions are wrong, does that necessary mean that my rotor setting is wrong? Couldn't it mean that I've just tried letter, that is not connected to a pair? Assuming there are 10 pairs, thus 6 letters are left over. Also, when there is a conflict in a deduction are all other deductions wrong just for current rotor setting, or does that apply for all rotor settings?

[@roxanedemunck4883]

26 positions includes mapping (for example) T to itself

[@anyadubois]

Great video! I have a question - If there was an option in letter connection (f.e. like mentioned in the video A-T) that there was no connection of the first letter with another at all, wouldnt that mean that then it would be coded as an initial letter?

[@FifiRX]

please remember that three Polish gus broke enigma code before the WW2 has started, and later during the war they give codes to brithish etc

[@krowa1010]

first the Poles broke it, then the work was handled to be under Alan's Turing's supervision, thats the way it was, shouldnt it therefore be mentioned?

[@LiquidFluorine]

​@Tweaky Robin Being factually correct isn't salty, so stop replying to every thread mentioning the truth as 'being salty'. If somebody's broke the code and then somebody else is basing on that work, you don't say that the later guy broke the code. Turing improved on the ideas brought by Rejewski, Zygalski and Różycki. He did not reinvent the ideas by himself or independently developed the mathematical theory behind it. He had direct access to all the mathematical concepts and created devices and also met and talked to the people who invented them. And that is very important to answering the question who broke the Enigma code. The first Enigma devices were broken as early as in 1932, and that is 6 years before Turing's involvement with Government Code and Cypher School. Turing's work is immensely important in improving the way the Enigma could be decoded as the complexity of the ciphers increased with time but the idea of how it worked did not change much. So no, Turing was not the first to break the code which is what is implied by asking 'who broke the enigma code'.

[@Honeythief_]

I as a german really enjoyed you speaking german :D

[@niaschimnoski882]

at [starting at] 8:41 / 8:42 up until 8:46, what is the source of that background noise? Am the only one picking up on it?

[@57thorns]

One important factor to remember here is what the actual numbers needed to break the enigma was: The huge factor from the reciprocal partial substitution cipher in the switchboard added zero security. Yes, zero. The whole factor is irrelevant. One Bombe had 12 sets of 3 rotors each. You would basically use one set for each letter in the crib. Let's call this a group. You would then need one group for each possible rotor combination (wheel order). This would be 60 for the standard 3 out of 6 wheel machine and 336 for the navy machine. These numbers were met by building more and more of the machines. Expensive (the whole project was about the size of the Apollo or Manhattan projects). The Bombe then would find candidates among the 17567 possible rotor starting positions in about 20 minutes. Those drums really spun fast. And in my opinion, this is one of the main reasons why it was possible to break the Enigma, the people at Bletchley Park figured out a way to solve the three main settings (wheel order, wheel starting position and stecker board setting) independently of each other. That, and the way the Germans kept giving them presents in the form of daily cribs. Any cipher is harder to break if you have no idea about the content.

[@NaderR]

Credits should go to the person who created that machine not only to the one who broke the code...

[@ZER0--]

***** Are you talking to me ?

[@jasonnung2645]

Credits would go to German engineer Arthur Scherbius

[@ZER0--]

Jason Nung Lol. But why should some one be credited for making a code machine that was broken almost immediately.

[@jasonnung2645]

It wasn't. It was created near the end of the First World War, and was originally for commercial purposes. During WWII it was adapted for military use, and extra rotors and a plug board was added to increase the number of code combinations by several thousand times. It took until just before Poland was occupied by Germany before Polish cryptographers were able to solve the 3-rotor version of the Enigma. But it was not until the early 1940s for the 5-rotor version to be solved by Turing. 1941-1918= 29 years of it being the most perplexing code in the Western World, the apex of science and technology at the time. I think that's quite impressive.

[@ZER0--]

Jason Nung It was cracked !! So it is a failure. Churchill said it shortened the war by a couple of years. The war lasted 5 years and we cracked every single machine so wtf are you talking about 29 years. Remember you believe he deserves credit. I repeat we broke every single enigma machine even Hitler's personal machine with 10 rotors. Twist the words all you like but Alan Turing was 6 in 1918 so I doubt he started work on it then. Unless you know some thing I don't.

[@jacquesj.j.soudan4670]

How did the Allies find out/know about this flaw (no letter was ever represented as itself)? From captured machines? Or trial and error as well? Also, how many machines were in use (distributed) at one time? Because it might have been worth the effort to try to intercept those monthly code-sheets? PS A few weeks back I found these sites - very insightful! - www.ellsbury.com/enigmabombe.htm - enigma.louisedade.co.uk/enigma.html

[@TheTck90]

Jacques J.J. Soudan In the first video they mentioned that Allies had captured a Enigma machine, and after pressing for example ''k'' 10 000 times and all the other letters have lighten up few times but never ''k'' itself they could assume that the letter is never itself. They were trying to intercept those code-sheets, but they could be destroyed by any contact with liquid (water) and only high officers had them. So if any of the officers knew they would get caught they simply had to apply some water in the sheet or eat it, making it very hard to get one. Even when they got one, they could only use it until the end of the month. Don't know how many machines were in use, but I would guess that every base of operations and ships had one.

[@jacquesj.j.soudan4670]

Hi TheTck90 - thanks! I missed having them an Enigma machine itself - I figure they would have needed at least one, as how would you start cracking it in the first place? I also realized after posting that they could have obtained a commercial version - although they had no plugboard in the front - but you would have a starting point. The water-destroyable ones were Navy (only) - he mentions it, in case a ship would sink. But those Navy-machines and codes were high on their wishlist for fighting the U-boats - later in the war, with the cracked codes they could hunt them, reducing losses considerably - and making U-boat service a suicide-mission, as 75% of the sailors got killed.

[@TheTck90]

Jacques J.J. Soudan It's amazing how much more power information gives you compared to guns!

[@jacquesj.j.soudan4670]

But guns and bombs sank the U-boats, TheTck90 - it's best to have both, so you have more options.

[@TheTck90]

Jacques J.J. Soudan True that!

[@sakunikajayaweera9800]

I watched plenty of videos on this topic but only this made me understand properly.

[@paulg444]

He is a dynamic and charismatic personality!!

[@ZeroRyoko]

Why has no one mentioned Thomas "Tommy" Harold Flowers, MBE (22 December 1905 - 28 October 1998) The Inventor and creator of the First Digital computer 'Colossus'? This was the machine that allowed the Allies break the Enigma almost instantly and the Lorenz Cypher. Without this man, the D-day landings would have been a Spectacular Failure, We could have lost the War in a very real way without him. With no help from the government, and simply because he was convinced he could help, He bankrupted himself to prove his "Programmable Computer" was viable. This man Is the farther of Modern Computing, he deserves the recognition, thanks to this man we live in the Digital Information Age. Yet most 'Computer Nerds' have never heard of him, its almost criminal in my opinion!

[@jameswalker199]

Because Colossus never cracked enigma codes, they were used solely for tunny codes.

[@NuisanceMan]

Grime's head: red on the outside, abstract on the inside

[@Albrecht8000]

You are an ABSOLUTE BRILLIANT teacher!!! :-) Perfect explained. Greetings from germany

[@NottsBobUK]

Great work. Thank you for showing us.

[@moeaftab]

The fact that math helped take down one of the most misguided and evil regimes in history is truly amazing. I now officially love math... these are words I thought I would never utter before today. Thanks, Numberphile.

[@NGCgalaxy]

arguably ... war develops us as much as it destroys us

[@DarkPaladinDE1]

Yeah, but it's technically not *needed*. If we'd all work as hard as if we were in war and push technology as hard the progress would be even bigger, because nothing gets destroyed.

[@josephcope2737]

The reason why war is such a stimulus for progress is that the necessity of victory to preserve a nations way of life concentrates its citizens' efforts. Someone once said that "nothing concentrates a man's thinking like the knowledge he's going to be hanged in the morning." Maybe it was Oscar Wilde. Unfortunately, during times of peace nations tend to get lazy and frivolous.

[@amojak]

suffering is a requirement of human nature to evolve. Without it people invent causes to fight for as we have a lot of now

[@DakotaGraftt]

DakrPaladinDE1 That's called capitalism.

[@terryhigson434]

War is why we have the technology we have... FULL STOP.

[@martadavies995]

Wow! What a brainiac. I'm impressed and in admiration of how easy you made it all seem even to me who, as a creative, am not gifted with a box of math or science skills. Thank you!

[@Bagunka]

I am going to college soon, will be majoring in Computer Science. Gosh, I really hope that we will learn at some point about how exactly did Alan Turing come up with his machine for breaking the code, which algorithm he used and so on.

[@anthonywallis2058]

I now understand The imitation Game

[@victariongreyjoy7261]

KygerGames24 Same

[@gohhoekiat4828]

deep

[@yahliamir8388]

Just watched it today before watching this video, quite useful indeed

[@pepecohetes492]

Not a great version of the events.

[@bananian]

Actually the movie doesn't really make sense. They hadn't figured out how to find the corresponding encrypted letters; yet the Bomba was already being built. How would turing know he's got the right settings if he doesn't have a deceypted message to compare to?

[@NoneN1nordy123]

It took one Swedish guy two weeks to break the T52 (Enigmas big brother, used by the German embassies and navy) using a pen and paper. He never explained exactly how he’d figured it out. But this has been overshadowed by well-known Enigma Machine story.

[@JohnSmith-eo5sp]

The Achilles Heal of the Enigma Cipher was the Umkerwalze itself. The reciprocal letter arrangement was a starting point for a process of elimination in a mass analysis

[@codyvivian6347]

I never in my life thought I would be watching video after video about math, bro yall are awesome!

[@novat9731]

The navy sent the rotation position in a different code because the navy would regularly be at sea for months at a time, and it would not be safe to produce codes months in advance. In addition, if a single submarine was captures, all the others could not communicate with the outside world. But since the code position was not predetermined, and was sent in a different code, a single captured submarine was not an issue to communications for the entire fleet.

[@b-chroniumproductions3177]

Admittedly it's somewhat difficult to capture a submarine, especially with an intact codebook (they'd probably throw it into the water before surrendering)

[@norbertfleck812]

@@b-chroniumproductions3177 Actually a codebook was captured in the 1940ies ...

[@JanPBtest]

This is all nice but what he says at 3:00 was not how the Enigma was broken. First of all, it wasn't Alan Turing who did this (although he was instrumental later), it was Marian Rejewski in 1933 in Poland. His method was something called characteristics method involving cycles of letters, not exclusion if the same letters. Likewise, the "bomb" was also invented in Poland around 1935 (the name derives from an "ice cream bomb" dessert served at the Hotel Europejski cafe in Warsaw across the Saxon Palace where Polish Cipher Bureau was located - Rejewski and his coworkers would go there for lunch now and then. Apparently they hit on the key idea of the "bomb" while having that dessert). Turing later massively improved on these ideas, he also had much greater resources. One weird aspect of it all was that when the British were setting up Bletchley Park, they did NOT invite Rejewski! And he was in fact IN London at the time! That's truly bizarre: to have a task of such magnitude, one upon which so many LIVES depended on, and to have the very Enigma BREAKER RIGHT THERE, in London, and NOT to have him work at Bletchley. Incomprehensible! It's a mystery which AFAIK has not been explained to this day. ("A mystery wrapped in an enigma", no?) At best a case of Brobdingnagian incompetence on the British side. Can you imagine?? In the end the Brits got _extremely_ lucky that Alan Turing did in fact turn out to be an absolutely brilliant cryptographer but of course they had no way of knowing this in advance (it's well-known that the talent for mathematics and the talent for codebreaking do not correlate as well as one might expect).

[@KurtRichterCISSP]

This is mentioned in the video. Well, not the ice cream etymology, but the rest.

[@mrtrololorific]

JanPBtest you should've watched the entire video before commenting

[@LakrimaProject]

mrtrololorific You should read whole comment before commenting too, there are flaws in the video that JanPBtest mention.

[@MissesWitch]

Great, I'd like that Icecream now~

[@Bucefal76]

It was worse, poles had crack down the addditional rotors used by germans, they gived blue papers to brits but british goverment didnt pass this plans (itentionally) to Turing. They pass this plans when Turing could not do any prpgress on his own...

[@simonbeasley989]

Brilliant explanation!

[@sammyspaniel6054]

Super thumbs up. Great explanation.

[@kemp10]

What about the letter ß (Esszett)?

[@PawelDzierzega]

Hey! The first person who decoded Engima was the polish mathematicians: M. Rajewski, J. Różycki and H. Zygaleski. They had done it (in 1932) before IIWW started.

[@PopeLando]

At the end of this video click the Hidden Extras link. In it James explains how the Poles had cracked Enigma before the war and had held a special meeting with the French and the British to pass on the secrets they had learned. It ought to go without saying that though the work done by the Polish workers was invaluable, that did not mean that the British didn't have anything to do or that they didn't have very difficult decryption tasks that had to be completed once the war got started. For one thing, the Germans did change the way they used the machine and did increase its security, in the case of at least one service, by adding two new rotors. But there seem to be many commenters who think that the Poles cracked Enigma, that was that and the British took credit for things they didn't do. They didn't. Alan Turing did things that the Poles couldn't have done, the Americans and Russians couldn't have done and indeed anybody else at Bletchley Park couldn't have done.

[@TheRobbex]

The Polish cryptographers were acclaimed at the time the breaking of the Enigma was made public in the U.K. in 1975. The importance of the Polish contribution (which cannot be over estimated) was to show how codes could be attacked using advanced mathematics. The version of the Enigma system they solved so brilliantly was much less complex, having fewer rotors and much less frequent rotor setting changes. Turing was responsible for attacking the German U boat code system unknown to the Poles. Then the German Navy changed their Enigma machine in early 1942, introducing an extra rotor (1+3) and a much more complex setting system that defeated previous techniques for finding solutions. This new system was unbreakable until a Royal Navy team boarded a sinking submarine 'blown to the surface' at night off Haifa in late 1942. Two men, Lt. Anthony Fasson and AB Colin Grazier died when the submarine finally sank with them still on board. The code books recovered enabled the U boat messages to be read once more but with a longer decipherment time due to their much greater complexity. Dr Turing's other war work is still largely secret. The fiendishly difficult Geheimschreiber system was broken at Bletchley Park during the war using an electronic computer.

[@applecounty]

Anton Deque " Then the German Navy changed their Enigma machine in early 1942, introducing an extra rotor (1+3) and a much more complex setting system that defeated previous techniques for finding solutions." Presumably you are describing the desperate efforts to get back into SHARK?

[@KaitainCPS]

The Poles solved an earlier, much easier version of Enigma. The one used by the Kriegsmarine during the war had a state space that was orders of magnitude larger. All the breakthroughs at Bletchley Park were specifically to address this vastly larger state space. This required both new computing machines and the invention of ingenious new heuristics for reducing the size of the state space. The Polish work was just the starting point for ULTRA. By their own admission, the Poles did not have the resources to solve the later incarnations of Enigma. But their contribution was of course important: they had already done the basic analysis of the machine, how it worked and the nature of the problem domain.

[@KaitainCPS]

PopeLando Correct.

[@honortruth5227]

I really appreciate your videos. ⭐️

[@RichardDominguezTheMagicIsReal]

i said this before, never been a numbers person but I can't express how amazing and interesting I find your videos, almost wish I was young enough to start a road down the math path, ty for sharing