Paper Lorenz

What is Paper Lorenz?

This is a simplified paper version of the Lorenz SZ40/42 cipher attachment used during World War 2 which can be used to help understand how the Lorenz SZ40 generated cipher codes.

The Lorenz SZ40 was used to send top secret message between the German High Command and the front-line generals. The Lorenz had twelve cipher wheels which generate two separate key letters. These two letters were then added (using modulo-2 or XOR) to each letter of the plain text to make the cipher message.

Using Paper Lorenz, you can encipher and decipher short messages just like the real machine. Each of the Psi and Chi wheels represent five of the real cipher wheels on Lorenz while the Mu wheel represents the other two


Click the button below to download the PDF files - you will need to print at least the first two pages to assemble Paper Lorenz.

Paper Lorenz v1.0


You will need

  • To download the PDF pages listed above and to print them.
  • Scissors
  • 3 round head paper fasteners
  • Optionally – if you have a laminator available, laminate pages 1 and 2 before assembly to make them more durable.

To assemble

  1. Cut out each of the three wheels on the second page. (It’s fine to just cut around the outer circle)
  2. Line up each wheel with the same marked space on the backing page.
  3. Carefully, push a paper fastener through the cross marked at the centre of each wheel and bend the metal legs apart to hold in place.
  4. Check you can rotate each of the three wheels.

Using Paper Lorenz

  1. Turn the Psi, Chi and Mu wheels to a random starting position or use the Ablesetafel on the other sheet to get your QEP code.
  2. Using the Teleprinter Addition Square, add together the first letter of your message with the letter shown on the Psi wheel.
  3. Add that letter to the one shown on the Chi wheel
  4. Write down the new letter, this is your cipher text
  5. If Mu wheel is currently showing x, step the Psi wheel one letter clockwise otherwise continue.
  6. Turn the Mu wheel one letter clockwise.
  7. Turn the Chi wheel one letter clockwise.
  8. Repeat from step 2 for each letter of your message.
  9. To decrypt, repeat as above but using the cipher text instead of the plaintext message

Example: encrypting a message

In this example, we will encrypt the message "SECRET" using the start positions Psi 1, Mu 1, Chi 1

  • Turn all three wheels so that the black arrows are pointing to number 1 (as on the picture above)
    This is our starting position for this message, a different starting position will give a different result.
  • The first letter of our message is the letter S and we are going to "add" this to the two key letters generated by the Lorenz (the Psi and Chi).
    Adding the letters normally involves converting them to their ITA2 5 bit codes then adding those together using modulo-2 (or XOR) to give another letter. If you want more detail on how this was done for real, follow this link: The Lorenz Machine.
  • Look at the Psi wheel, the current letter the wheel is encoding is P
  • Find the sheet showing the Teleprinter Addition Square - this is a table which gives the resulting letter when adding together any two letters.
  • To add together S + P find the letter S on one axis and the letter P on the other axis and check which letter is shown where the two meet.
    Hint: Use a ruler or straight edge of a paper to line up on row S first then read down from P.
    It doesn't matter which way around you choose, adding S + P is the same as adding P + S!
  • The answer should be the letter W
  • Now, using the same method, we want to add the W we just got to the Chi wheel key letter shown N.
  • The final answer, and our first cipher letter, should be the character 8 - write this down!
  • The real Lorenz, after enciphering a letter, would move the wheels in a set pattern so our second letter will be enciphered using a different two key letters. This means that even if our second letter in our message was the same as the first, the cipher message will be two different letters.
    Follow these rules to change the three cipher wheels.
    1. Look at the current value of the Mu (or motor) wheel, it will either be a cross or a dot.
      If it is a cross, turn the Psi wheel one position clockwise.
      If the Mu wheel is pointing to a dot, continue to the next step without moving the Psi wheel.
      In our example, Mu is in position 1 and showing a dot - don't change anything and just move to step 2.
    2. Always move the Mu wheel one position clockwise.
      It should now be set to position 2, another dot.
    3. Always move the Chi wheel one position clockwise.
      It should now be set to position 2, character 4.
  • Now we repeat the above from the second instruction for our next plain text letter E. We want to add E + P + 4 which should give the answer Y.
    Note: the key letter for the Chi wheel has changed for our second letter, while the Psi wheel has remained at the same one.
  • Move the wheels on as per the rules above.
    Mu is a dot, no change to Psi. Move Mu and Chi one position clockwise.
  • The third letter of our message is C, so the next sum should be adding C + P + L which equals R
  • Again move the wheels as per the rules above.
    Mu is a cross this time, move the Psi wheel on one position to the number 2, character N and also move the Mu and Chi wheels.
  • Fourth letter is R and this time, both Psi and Chi are different key letters. R + N + N = R
  • Fifth letter E + N + F = /
  • Sixth letter T + T + U = U
  • Therefore, our final cipher text is 8YRR/U


Note that in the table above, the Chi wheels change letter for each letter of the message, while the Psi wheels often stay the same. This "stagger" was meant to create a larger pattern of key string which would not repeat very often, but actually was one of the main flaws that allowed the codebreakers at Bletchley Park to break this cipher. See First Break for more detail on how this was done and how the world's first computer was invented!

Deciphering a message is exactly the same - just set the start positions of all three wheels to the same as you enciphered the message then add the first letter of the cipher text to the Psi and Chi rather than the plain text. 8 + P + N = S. Repeat the same steps as listed above to retrieve your message.

Comparing Paper Lorenz to Virtual Lorenz

The three wheels on Paper Lorenz are simplified versions of the twelve actual Lorenz wheels but the letters that are generated are the same as the ones for a Lorenz SZ40 using the Bream pin settings that are available on Virtual Lorenz. This means we can actually use Virtual Lorenz to check a message enciphered using Paper Lorenz (you should be able to get up to around 23 characters, assuming you start at Chi setting 1, before it is no longer accurate).

The Chi wheel on Paper Lorenz represents all five wheels on the right hand side of Virtual Lorenz. If you look on the Paper Lorenz Chi wheel, you will see listed against each position the ITA2 encoded 5-bit code. This is a series of dots and crosses (on and offs) which together represent a character. The table below gives the possible characters for ITA2 code shown as holes punched into paper tape. A cross is represented by a hole in the tape and a dot by a solid piece of tape without a hole.

Adding together two letters involves taking each of the 5 impulses for both characters in turn and using the following rules:
If the symbols are the same, return dot
If the symbols are different, return cross.


Testing our message on Virtual Lorenz

To enciper our message "SECRET" on Virtual Lorenz to see if we get the same result, we first need to set Virtual Lorenz to be a model SZ40 (the basic model) and to set the pins to the Bream message settings.

Select the Prefs menu then select Settings

Make sure the top Lorenz Model switch is set to SZ40

Check that the KT-Schalter (Plaintext Switch) is set to Aus (off).

Select the BREAM pattern button to set the pins on the wheels

Now click Close (at the bottom of the settings menu).

Our start wheel setting is where all of the wheels are set to position 01. If you have moved any wheels, just rotate them all back to 01.

Make sure the Lorenz is switch on - the red switch in the bottom center of the machine should be set to Ein.

To encipher our message, use the keyboard on screen or just type our message SECRET

If all is set correctly, you should find the lower Send/Recv tape shows 8YRR/U - exactly as we generated using Paper Lorenz!

Note: The Lorenz wheels actually turn backwards with respect to the numbers on the wheels - the last Chi wheel (12) goes from position 1 to 23..22.21 etc rather than counting upwards so the letters generated for Paper Lorenz do not match with the positions on the actual wheels but they do run in order!

For example, if you wanted to start Paper Lorenz with a start position of Psi=1, Mu=2, Chi=3 and then wanted to set Virtual Lorenz to start on the same position we can do the following.

The Psi wheels are already set to position 1 so leave the first five wheels all set to 01

Both of the Mu wheels need to move one position on. These are the two wheels marked 6 and 7 on Lorenz. Turn them one position down each (6 should show 37, 7 should show 61).

All of the Chi wheels need to move on two positions. Press the down arrow on all of the last five wheels (8-12) twice. Final settings (40,30,28,25,22).

This should now be set to the same settings as Paper Lorenz.

Important Note: Once any of the wheels on Paper Lorenz get past their highest setting (23 for Chi, 13 for Mu or 19 for Psi), our limited wheels on paper will no longer match up with the real Lorenz, so if you are brave enough to try adding more than a few characters or so and want it to be deciphered on Virtual Lorenz, make sure you start with a low starting position on the wheels.

The Answers

Click this button to download the answers for the challenge decipher page if you're really stuck.

Paper Lorenz Challenge Answers

Paper Engima

If you've enjoyed making a Paper Lorenz, how about making a Paper Enigma as well!

Paper Enigma is a fantastic idea created by Franklin Heath Ltd which allows you to create a working Enigma cipher machine in 3D using a few print outs and a crisp tube.

Paper Enigma


Creative Commons Licence
Paper Lorenz by Virtual Colossus is licensed under a Creative Commons Attribution 4.0 International License.

Please feel free to use Paper Lorenz to learn about the Lorenz SZ40/42 machine and how it enciphered messages - please do let me know if you find it useful or if you have any feedback or ideas for making it easier to use or understand.