Jump to content

Do You have any Interesting but, Not Very Useful Facts


Recommended Posts

Factorial 26 x 12. (twelve rotors on a Lorenz)

'factorial'  is written " ! " so 26 x 25 x 24 x 23 and so on,  x 12.     Or 26! x 12 = 5 x 10^27 (27 noughts after the '5')   but that is out by many, many orders of magnitude.

Number of atoms in the universe is about 10^80 (80 noughts after the 10)

Link to post
Share on other sites

The Lorenz didn't use the 26 letter alphabet John -  it used 35 characters in 5bit Baudet code since it was an extension of a teleprinter.  It was also far more complicated than just a twelve-wheel Enigma machine for which your calculation might be valid.  The Enigma relied on combination substitution of letters but the Lorenz uses 'exclusive OR" addition for each bit with a pseudo-random stream generated by the movement of the rotors, which is a rather different process.

to quote from this link:

http://www.quadibloc.com/crypto/te0301.htm

"It had twelve pinwheels, all of which could have all their pins set by the user. Ten of these pinwheels formed two groups of five, and one wheel from each group inverted its corresponding plaintext bit when a pin was active on it.

The wheels of the first group had sizes 41, 31, 29, 26, and 23. Those of the second group had sizes 43, 47, 51, 53, and 59.

Two additional wheels were of size 37 and 61. The wheels of the first group, and the wheel with 61 positions, advanced one position with every letter enciphered. When the current pin on the 61-position wheel was active, the wheel with 37 positions advanced one space. When the current pin on the 37-position wheel was active, then the wheels of the second group advanced one space."

Difficult to understand but you get the gist. You will note the use of prime numbers for the pins on some of the wheels. 

Link to post
Share on other sites
On 3/27/2020 at 7:38 PM, RogerH said:

The old Ford slogan - you can have any colour as long as it is Black.

Is all to do with the production line. It traveled at a constant speed. 

Black paint dries quicker than any other colour and fitted into the production line speed.

 

Roger 

Just to add another misconception, black isn’t a colour, neither is white. So if your asked what the highest scoring coloured ball is in snooker, it’s the Pink.

Link to post
Share on other sites

All the numbers of pins (which gives the number of start positions) on the wheels are co-primes. Picking up on John’s calculation the number of start positions comes to about 1.6x10^16. It’s not a factorial but as Rob points out it derives from the number of pins or cams on each wheel. The total complexity multiplies that fairly big number by an even bigger one. I'll leave that hanging for a bit. Incidentally, the Baudot-Murray 5 bit code has 32 binary states  and this has certain consequences.

When this madness is over I’ll set up another National Museum of Computing visit. Then I can explain things with the real props to hand including Colossus. 

Edited by peejay4A
Link to post
Share on other sites
49 minutes ago, Cameron said:

Just to add another misconception, black isn’t a colour, neither is white. So if your asked what the highest scoring coloured ball is in snooker, it’s the Pink.

Neither is pink a colour, its not part of ROYGBIV

Link to post
Share on other sites

Here is a link to a virtual Lorenz machine

 

http://www.lorenz.virtualcolossus.co.uk/LorenzSZ/

 

https://www.theregister.co.uk/2017/05/15/virtual_lorenz_tnmoc/

Edited by 4Mal
Link to post
Share on other sites
1 hour ago, RobH said:

The Lorenz didn't use the 26 letter alphabet John -  it used 35 characters in 5bit Baudet code since it was an extension of a teleprinter.  It was also far more complicated than just a twelve-wheel Enigma machine for which your calculation might be valid.  The Enigma relied on combination substitution of letters but the Lorenz uses 'exclusive OR" addition for each bit with a pseudo-random stream generated by the movement of the rotors, which is a rather different process.

to quote from this link:

http://www.quadibloc.com/crypto/te0301.htm

"It had twelve pinwheels, all of which could have all their pins set by the user. Ten of these pinwheels formed two groups of five, and one wheel from each group inverted its corresponding plaintext bit when a pin was active on it.

The wheels of the first group had sizes 41, 31, 29, 26, and 23. Those of the second group had sizes 43, 47, 51, 53, 59. 

Two additional wheels were of size 37 and 61. The wheels of the first group, and the wheel with 61 positions, advanced one position with every letter enciphered. When the current pin on the 61-position wheel was active, the wheel with 37 positions advanced one space. When the current pin on the 37-position wheel was active, then the wheels of the second group advanced one space."

Difficult to understand but you get the gist. You will note the use of prime numbers for the pins on some of the wheels. 

Evenn ignoring the complexities of Either/Or, the first quintet of wheels gives 2.5 x 10^163 possibilities!     Combining only the next four gets 1 x10^248 (Google can't cut any more!) , so in excess of 10^400 possibilities.   That would do it! 

PS 4Mal, no wonder we won the war - the Nazis must have been deafened!

Edited by john.r.davies
Link to post
Share on other sites
38 minutes ago, 4Mal said:

Martin Gillow, creator of the “Virtual” suite of emulations, is a frequent visitor to TNMOC and a very smart guy. There’s a lot of misconceptions and conflations in the responses to the Register post. For the accurate picture come to TNMOC. 

Edited by peejay4A
Link to post
Share on other sites
2 hours ago, peejay4A said:

I’d be interested to see how you worked that out John. 

Same as when I did it as for an Enigma.  Factorial 41, 31, 29, 26, and 23, and 43, 47, 51, 53, 59, as per RobH's description of the size of each wheel.   I didn't get as far as adding the two extra wheels!  Even Google's computing power ran out!

The two sets came to 2.5 x 10^163 and 1 x 10^248 (the '^' symbol is used to indicate a power, like 2 squared, 2^2, in typescript, where resetting to a superscript character isn't possible)     And because it's a power, adding them together is the same as multiplying.     OK, one set of digits was 2.5 and the other 1, but for a number as large as this, the digits are insignificant!

 

Link to post
Share on other sites

A bigish number then?

Link to post
Share on other sites

An immense number John, but of course the really interesting thing is that despite this the encryption was cracked because the Germans used the same setting twice for a similar message.  in fact the basic operation of the machine was worked out from first principles using little more than paper and pencil by Bill Tutte and others.  Very, very clever people. 

 

Link to post
Share on other sites

Factorials aren’t the right approach to deriving the theoretical number of settings on a Lorenz. What you’re looking for is twofold. The start positions of al 12 wheels for each message (should be different for every message),  and the disposition of all 501 pins or cams (changed monthly, with some caveats, at the outset but daily in 1945). 

To work out the number of start positions it’s the product of all the possible states of all 12 wheels. 43×47×51×53×59×37×61×41×31×29×26×23 which is about 1.6 x 10^16. 

The possible combinations of the pins is 2^501, there are 501 pins each of which can be in a binary 1 or 0 state. Not all combinations were allowed for reasons I won’t go into here but...

The total theoretical settings is 2^501 times 1.6x10^16  About 10^170, which is why I said above that it’s the complexity of the universe squared  

The ciphering principle, a Vernam cipher, was well understood but the internal design of the Lorenz was not until Bill Tutte exposed it. He then spotted a weakness in the design which led to his double delta attack. This was a statistical, zero knowledge attack which allowed the disposition of all 10^170 possible settings for each message  to be derived by processing ciphertext alone.  That’s a lot of processing which led to Colossus. Tutte’s work was the single most significant intellectual achievement if the Second World War in my opinion.

For the Full Monty explanation get down to The National Museum of Computing or get on one of my TR tours  

 

Link to post
Share on other sites

Pete, I just found this whilst looking for something else. This is the machine we spoke to you about some time ago now. To be found in Coronation Park aka Millbrook Park Jersey.

A7A81C3C-57E2-4939-879F-30D3C00F21A7.png

Edited by SuzanneH
Link to post
Share on other sites

158,962,555,217,826,360,000 according to the Numberphile presenter. See about 6:15 onwards, or just watch the whole thing - it's interesting.

... and for the explanation of the flaw in Enigma,

Enjoy!

Cheers, Richard

PS: My niece's husband is a Numberphile presenter (not this one) - all this is far too clever for me!

Edited by Spit_2.5PI
Link to post
Share on other sites
1 hour ago, peejay4A said:

Factorials aren’t the right approach to deriving the theoretical number of settings on a Lorenz. What you’re looking for is twofold. The start positions of al 12 wheels for each message (should be different for every message),  and the disposition of all 501 pins or cams (changed monthly, with some caveats, at the outset but daily in 1945). 

To work out the number of start positions it’s the product of all the possible states of all 12 wheels. 43×47×51×53×59×37×61×41×31×29×26×23 which is about 1.6 x 10^16. 

The possible combinations of the pins is 2^501, there are 501 pins each of which can be in a binary 1 or 0 state. Not all combinations were allowed for reasons I won’t go into here but...

The total theoretical settings is 2^501 times 1.6x10^16  About 10^170, which is why I said above that it’s the complexity of the universe squared  

The ciphering principle, a Vernam cipher, was well understood but the internal design of the Lorenz was not until Bill Tutte exposed it. He then spotted a weakness in the design which led to his double delta attack. This was a statistical, zero knowledge attack which allowed the disposition of all 10^170 possible settings for each message  to be derived by processing ciphertext alone.  That’s a lot of processing which led to Colossus. Tutte’s work was the single most significant intellectual achievement if the Second World War in my opinion.

For the Full Monty explanation get down to The National Museum of Computing or get on one of my TR tours  

 

 

Bletchley Park is somewhere I really have to go.

No1 grandson went and came back buzzing

Link to post
Share on other sites
47 minutes ago, SuzanneH said:

Pete, I just found this whilst looking for something else. This is the machine we spoke to you about some time ago now. To be found in Coronation Park aka Millbrook Park Jersey.

A7A81C3C-57E2-4939-879F-30D3C00F21A7.png

Presumably a relic from wartime occupation Suzanne. Probably an M3 Enigma. 

Link to post
Share on other sites

Hi Pete,

you would have been proud of me. Shortly after the Forum visit to the computer museum and your presentation we were in the Jersey tunnels and an old boy was telling his chums what the typewriter was. 

I couldn't resist it and stepped in and gave him everything I took in at Bletchley..

I could see they were flagging a little so I excited them with my take on the Lorenz. Great day. 

They were so excited they couldn't get out to the sunshine quick enough.

 

Roger

Link to post
Share on other sites
1 hour ago, Spit_2.5PI said:

158,962,555,217,826,360,000 according to the Numberphile presenter. See about 6:15 onwards, or just watch the whole thing - it's interesting.

... and for the explanation of the flaw in Enigma,

Enjoy!

Cheers, Richard

PS: My niece's husband is a Numberphile presenter (not this one) - all this is far too clever for me!

There’s quite a few inaccuracies in those videos but the general thrust is OK. A couple of things;

“Weather report” wasn’t the only crib (possible plaintext). Indeed if it was the only one you relied on  it would probably only get you into one network and there were around 60 discrete networks all with their own key of the day. You didn’t get a result in 30 minutes from a bombe. You’d have used multiple bombes to get one key of the day. There’s a lot of back room effort involved over and above bombe time. Usually the key of the day was found by breakfast time, the key having changed at midnight Berlin time. 
 

159 million million million is a theoretical number. Poor operating procedures reduced that by quite a large amount. 
 

Without cribs (or the daily coding sheets for a network) no Enigma messages would have been read. 

PS. You can see the Bombe replica in action at TNMOC. 

Edited by peejay4A
Link to post
Share on other sites

It does John. The Lorenz could have been practically unbreakable in wartime but for an overcomplicated design which looked good on paper.  We have Bill Tutte to thank although he always downplayed his contribution.  He’s my hero. 

Link to post
Share on other sites
  • 3 weeks later...

This is a bit interesting..... and possibly even topical.....

During the Black Death, incoming ships were forced to wait for 40 days to prevent possible infection.

The Italian word for 40 “Quaranta”, is where we get the word “Quarantine” from.......

 

Link to post
Share on other sites
On 3/30/2020 at 2:18 PM, SuzanneH said:

Pete, I just found this whilst looking for something else. This is the machine we spoke to you about some time ago now. To be found in Coronation Park aka Millbrook Park Jersey.

A7A81C3C-57E2-4939-879F-30D3C00F21A7.png

It's interesting that the Z and the Y have been transposed - presumably because it is a German keyboard. Oh and the P and L have been moved to the bottom row.

Rgds ian

Link to post
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Restore formatting

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
×
×
  • Create New...

Important Information

Please familiarise yourself with our Terms and Conditions. By using this site, you agree to the following: Terms of Use.