Project 4: Constructing Colossi

Deliverables: Bring in to class on Wednesday, 16 March a stapled turn-in containing your written answers to all the questions.


  1. Get practice with material that will be on the Blue Belt Exam including: recursive definitions, procedures, lists, asymptotic notation, and analyzing procedures.

  2. Learn about cryptography and the first important problems that were solved by computers.


Before doing this project, you are expected to have read Chapter 7: Cost of the coursebook, and completed Lesson 5: How Programs Run of Udacity CS101.

Collaboration Policy

This problem set is intended to help you prepare the Blue Belt Exam. You may work on it by yourself or with any number of other students of your choice, but you will need to do the exam on your own. If you work with others, you should turn in one assignment with everyone's name on it, but it is required that everyone understand everything in the submission you turn in.

Regardless of whether you work alone or with a partner, you may discuss the assignment with anyone you want and get help from others, so long as it is help in the spirit of learning how to do things yourself not getting answers you don't understand. You should understand everything you turn in for the assignment well enough to be able to produce it completely on your own.

Remember to follow the course pledge you read and signed at the beginning of the semester. For this assignment, you may consult any outside resources, including books, papers, web sites and people, you wish except for materials from previous cs1120 courses or direct solutions to the given problems. You may consult anyone you want, but that person cannot type anything in for you and all work must remain your own and outside sources should never give you specific answers to problem set questions.

If you use resources other than the class materials, lectures and course staff, you should document this in your turn-in.

As usual, you are strongly encouraged to take advantage of the cs1120 slack group and the scheduled office hours.


Cryptography means secret writing. The goal of much cryptography is for one person to send a message to another person over a channel that an adversary may be eavesdropping on without the eavesdropper understanding the message. We do this by having functions that encrypt and decrypt messages. The encrypt function takes a plaintext message and produces a ciphertext message. Encrypt scrambles and alters the letters of the plaintext message so that an eavesdropper will not be able to understand the message. The decrypt function takes a ciphertext message and produces the corresponding plaintext message. Encryption works as intended if the only person who can perform the decrypt function is the intended recipient of the message.

Since making up good encrypt and decrypt functions and keeping them secret is hard, most cryptosystems are designed to be secure even if the encryption and decryption algorithms are revealed. The security relies on a key which is kept secret and known only to the sender and receiver. The key alters the encryption and decryption algorithm in some way that would be hard for someone who doesn't know the key to figure out. If the sender and receiver use the same secret key we call it a symmetric cipher. If the sender and receiver can use different keys we call it an asymmetric cipher.

Ciphertext = Encrypt(KE, Plaintext)
Plaintext = Decrypt(KD, Ciphertext)

If KE == KD it is symmetric encryption.
If KD cannot (feasibly) be derived from KD it is asymmetric ("public key") encryption.

In this project, you will explore a symmetric cipher based on the Lorenz Cipher that was used by the German Army High Command to send some of the most important and secret messages during World War II. The Lorenz Cipher was broken by British Cryptographers at Bletchley Park.

Arguably the first electronic programmable computer, Colossus, was designed and built by Tommy Flowers, a Post Office engineer working at Bletchley Park during the war. (There is a lot of arguing about what should be considered the first computer. Who Invented the Computer? The Legal Battle That Changed Computing History supports John Atanasoff's case; this Scientific American post summarizes some of the other candidates.)

Ten Collosi were built in 1943 and 1944, and used to break some of the most important German messages during World War II. Messages broken by Colossus were crucial to the D-Day invasion since the allies were able to learn that their campaign to deceive Hitler about where the attack would come was succeeding and knew where German troops were positioned.

Bletchley Park (Summer 2004)

Optional: For more background on encryption, the Lorenz Cipher and how it was broken, see Udacity cs387: Applied Cryptography (Lesson 1). It is not necessary to do this to complete this project, but I hope you'll find it interesting and worthwhile! It will also be very helpful if you attempt the bonus question at the end.

It is regretted that it is not possible to give an adequate idea of the fascination of a Colossus at work: its sheer bulk and apparent complexity; the fantastic speed of thin paper tape round the glittering pulleys; the childish pleasure of not-not, span, print main heading and other gadgets; the wizardry of purely mechanical decoding letter by letter (one novice thought she was being hoaxed); the uncanny action of the typewriter in printing the correct scores without and beyond human aid; the stepping of display; periods of eager expectation culminating in the sudden appearance of the longed-for score; the strange rhythms characterizing every type of run; the stately break-in, the erratic short run, the regularity of wheel-breaking, the stolid rectangle interrupted by the wild leaps of the carriage-return, the frantic chatter of a motor run, the ludicrous frenzy of hosts of bogus scores.

D. Mitchie, J. Good, G. Timms. General Report on Tunny, 1945. (Released to the Public Record Office in 2000).

Project 4 Repository

Set up your project4 repository in the same way as you did for project 3.

The project4 repository contains these files:

Unlike the first three projects, the only code we are providing for this one is tedious code for converting characters to Baudot codes that you could have written yourself, but probably in a more tedious way than is doing by the provided code. You will be writing all of the interesting code yourself for this project.

Measuring Cost

For each of the following subquestions, you will be given two functions, g and f that each take a single parameter n (which must be a non-negative integer) and asked to determine which of O(f(n)), Ω(f(n)), and Θ(f(n)) contain g. It is (of course!) possible that more than one of the sets contain g.

In addition to identifying the properties that hold, you should justify your answer by:

For example, if g is n2 and f is n3 you would argue:

  1. n2 is in O(n3) since if we pick c = 1 and n0 = 1 then n2cn3 for all n > 1.

  2. n2 is not in Ω(n3) since for any c value we know n2 is not greater or equal to cn3 for all n > 1/c. This is the case since we can simplify the inequality to 1 ≥ cn which is not true if we choose n > 1/c.

  3. n2 is not in Θ(n3) since n2 is not in Ω(n3).

Problem 1. For each g and f pair below, argue convincingly whether or not g is (1) in O(f), (2) in Ω(f), and (3) in Θ(f) as explained above. For all questions, assume n is a non-negative integer.

  1. g(n) = n + 3; f(n) = n
  2. g(n) = n2 + n; f(n) = n2
  3. g(n) = 2n; f(n) = 3n
  4. g(n) = 2n; f(n) = nn
  5. g(n) = the federal debt n years from today; f(n) = the US population n years from today (this one requires a more informal argument)
(You may write your answers to Problem 1 as a comment in, or by hand or using another document editor and print out a separate page.)

The Exclusive Or (XOR)

The exclusive-or (xor, sometimes pronounced "zor") function is every cryptographers favorite function. The xor function takes two Boolean inputs, and returns True if exactly one of the inputs is True, and returns False otherwise. Often in English, when people say "or", what they really mean is "xor". For example, when someone asks "Would you like the chicken or fish?", they do not consider "both" to be a valid answer.

In cryptography, it is usually easier to deal with the binary digits 1 and 0 instead of True and False. We use 1 to represent true and 0 to represent false, and call each 0 or 1 a bit. (The term bit was introduced by Claude Shannon.)

Problem 2. Define the xor function that takes two bits as parameters and returns to 1 if exactly one of the parameters is 1 and returns to 0 otherwise. (Note that this xor function may be slightly different from others you may see, since the inputs are bits (represented as numbers) not Booleans.

Your xor function should produce these interactions:

>>> xor(0,0)
>>> xor(0,1) 
>>> xor(1,0) 
>>> xor(1,1)

The xor function has several properties that make it extremely useful in cryptography:

The second property means that if a message is encrypted by converting the message into a sequence of bits, and xor-ing each bit in the message with a perfectly random, secret bit known only to the sender and receiver, then we can send a message with perfect secrecy! This is known as a one-time pad (and is, essentially, the only perfect cipher possible, as was proven by Claude Shannon).

Fortunately for the Allies, the Nazis did not have a way of generating and distributing perfectly random bit sequences. Instead, they generated non-random sequences of bits using rotors. Because the sequences of bits generated was determined by the structure of the machine used to generate them, the Allies were able to figure out a way to break the code.

We will look at how the Lorenz cipher used xor to encrypt messages soon, but first, consider how to turn messages into sequences of bits.

The Baudot Code

The Nazis used the Baudot code to represent the letters in their messages as sequences of bits. The Baudot code translates letters and other common characters into a 5-bit sequences. With five bits, we have 25 = 32 possible values. This is enough to give each letter in the alphabet a different code and have a few codes left over for spaces and other symbols.

Modern character encodings. Until a few years ago, most computers typically used either 7-bit ASCII encodings to have 27 = 128 possible characters. This is enough to cover lowercase and uppercase letters and some additional punctuation characters. As computing became more international (not to mention the important need for many emoji like 😜 ("FACE WITH STUCK-OUT TONGUE AND WINKING EYE", added in 2010), computing systems adapted to support more complex character sets with unicode encodings. Unicode can encode over a million different characters (only 120,000 are currently assigned), using between one and 6 bytes. (One of the biggest changes between Python 2 and Python 3 was to support Unicode throughout Python 3. This means you can have variable names like ΘąƸІڜঐ if you really want (although this is most definitely not recommended!).

Table 1 shows the letter mappings for the Baudot code. For example, the letter H is represented by 10100 and I is 00110.

We can put letters together just by concatenating their encodings. For example, the string 'HI' is represented in Baudot as 1010000110.

There are some values in the Baudot code that are awkward to print: carriage return, line feed, letter shift, figure shift and error. For our purposes we will use printable characters unused in the Baudot code to represent those values so we can print encrypted messages as normal strings. Table 1 shows the replacement values in parenthesis.

A 00011      H 10100      O 11000      V 11110      space 00100
B 11001      I 00110      P 10110      W 10011      carriage return (,) 01000
C 01110      J 01011      Q 10111      X 11101      line feed (-) 00010
D 01001      K 01111      R 01010      Y 10101      letter shift (.) 11111
E 00001      L 10010      S 00101      Z 10001      figure shift (!) 11011
F 01101      M 11100      T 10000           error (*) 00000
G 11010      N 01100      U 00111     
Table 1. Baudot Code mappings.

We can use lists of bits to represent Baudot codes. H is represented as the list [1, 0, 1, 0, 0]. A string is represented as a list of these lists: 'HI' is [[1, 0, 1, 0, 0], [0, 0, 1, 1, 0]].

We have provided two functions in

char_to_baudot(char): Character → Baudot
Takes a character as input, and outputs the corresponding Baudot code represented as a list of five bits.
baudot_to_char(bcode): Baudot → Character
Takes a baudot code, represented as a list of five bits, as input and outputs the corresponding character.

Problem 3. Define a function, string_to_baudot, that takes a string of characters and transforms it into a list of Baudot codes.

Problem 4. Define the inverse function, baudot_to_string, that takes a list Baudot codes (which is a list of lists of 5 bits), and returns the corresponding string of characters.

Your string_to_baudot and baudot_to_string functions should be inverses. Hence, you can test your code by evaluating compositions like, baudot_to_string(string_to_baudot("HELLO")) (which should return "HELLO").

Problem 5. Describe the running time of your baudot_to_string procedure. You may do this using precise English, or using Θ notation. Make sure to explain carefully what every variable you use means.

Lorenz Cipher Machine

The Lorenz Cipher

The Lorenz cipher was an encryption algorithm developed by the Germans during World War II. It was used primarily for communications between high commanders in Axis headquarters and conquerer capitals. The original Lorenz machine consisted of 12 wheels, each one having 23 to 61 unique positions. Each position of a wheel represented either a 0 or a 1.

The first 5 wheels were called the K wheels. Each bit of the Baudot representation of a letter was xor-ed with the value showing on the respective wheel.

The same process was repeated with the next 5 wheels, named the S wheels. The resulting value represented the encrypted letter.

After each message letter the K wheels turn one rotation. All five of the wheels advance one position (this coordination of movement across the wheels is one of the biggest weaknesses in the cipher design).

The movement of the S wheels was determined by the positions of the final two wheels, called the M wheels.

Lorenz Cipher Wheels

Like most ciphers, the Lorenz machine also required a key. The key determined the starting position of each of the 12 wheels. To decipher the message you simply need to start the wheels with the same position as was used by the sender to encrypt the message, and enter the ciphertext. Because of the invertability of xor, when the receiver's machine produces the same sequence of encryption bits the receiver obtains the original message by xoring the ciphertext with the generated bits.

There were 16,033,955,073,056,318,658 possible starting positions. This made the Nazis very confident that without knowing the key (starting positions of the wheels), no one would be able to break messages encrypted using the Lorenz machine. There confidence was futher bolstered by knowing that the Allies had not acquired a single Lorenz machine, since the machines were all kept in highly secure locations controled by the Axis (this is different from Enigma machines which were widely deployed in the field, with every army unit and submarine having one).

Their confidence was misplaced, however, and Allied cryptographers at Bletchley Park were able to deduce the structure of the Lorenz machine from intercepted re-transmissions (with errors), identify statistical weaknesses in the encryption, and build a computer to rapidly decrypt intercepted messages, providing great benefits to the Allies during the last years of the war.

Since the full Lorenz cipher would be too hard for this project, we will implement a simplified version. You should be suitably amazed that the allied cryptographers in 1943 were able to build a computer to solve a problem that is still hard for us to solve today! (Of course, they did have more that a week to solve it, and more serious motivation than we can provide in this course.)

Our Lorenz machine will use 11 wheels, each with only 5 positions. The first five wheels will be the K wheels and the second five the S wheels. Each of these will only have a single starting position for all 5 — that is, unlike the real Lorenz machine, for this project we will assume all five K wheels must start in the same position and all five S wheels must start in the same position.

The final wheel will act as the M wheel. After each letter all the K wheels and the M wheel should always rotate. Before the M wheel rotates, if it shows a 1 the S wheels should also rotate, but if the M wheel shows a 0 the S wheels do not rotate.

Describing the Simplified Lorenz Machine

We have provided 3 lists that represent the wheels in The first is called K_wheels and is a list of lists, each of the inner lists containing the 5 settings:

K_wheels = [[1,1,0,1,0], [0,1,0,0,1], [1,0,0,1,0], [1,1,1,0,1], [1,0,0,0,1]]

There is a similar list called S_wheels to represent the S wheels of our simulated machine.

S_wheels = [[0,0,0,1,1], [0,1,1,0,0], [0,0,1,0,1], [1,1,0,0,0], [1,0,1,1,0]]

The final list represents the M wheels and is just a single list (it is just one wheel, so is a single list representing that wheel, not a list of wheels):

M_wheel = [0,0,1,0,1]

To rotate our wheels, we will take the number at the front of the list and move it to the back. Thus, the first number in the list represents the current position of the wheel.

Problem 6. Define a function rotate_wheel that takes as input a wheel (represented as a list). It should return a new list that represents the wheel rotated once (without modifying the input wheel). For example, rotate_wheel([1, 2, 3, 4, 5]) should return [2, 3, 4, 5, 1]. Although all the wheels in our simulated Lorenz cipher machine have five bits, your rotate_wheel function should work for any non-empty list.

We also want a procedure that can rotate a wheel zero or more times:

Problem 7. Define a function rotate_wheel_by that takes two inputs: a wheel as the first input and a number as the second. It should return a new wheel that represents the result of rotating the wheel the input number of times. For example, rotate_wheel_by([1, 0, 0, 1, 0], 2) should return [0, 1, 0, 1, 0] and rotate_wheel_by(wheel, 0) (where wheel is any list) should return wheel.

Next, we define similar functions that work on a list of wheels at a time instead of a single wheel.

Problem 8. Define a function rotate_wheel_list that takes a list of wheels (like K_wheels) as its input and returns a new list of wheels where each of the wheels in the parameter list of wheels has been rotated once. For example, rotate_wheel_list(K_wheels) should return [[1, 0, 1, 0, 1], [1, 0, 0, 1, 0], [0, 0, 1, 0, 1], [1, 1, 0, 1, 1], [0, 0, 0, 1, 1]].

Avail yourself of these means to communicate to us at seasonable intervals
a copy of your journal, notes & observations of every kind,
putting into cipher whatever might do injury if betrayed.

Thomas Jefferson's instructions to Captain Lewis for the Expedition to the Pacific.

Problem 9. Define a function rotate_wheel_list_by that takes a list of wheels and a number as parameters, and returns a new list where each of the wheels in the parameter list of wheels has been rotated the number parameter times. For example, rotate_wheel_list_by(K_wheels, 5) should return the same list as K_wheels.

Problem 10. Describe the running time of your rotate_wheel_list_by procedure (preferably using Θ notation, but any clear description is acceptable). Be sure to explain what all variables you use mean.

Simulating the Lorenz Machine

Now that we can rotate our wheels, we simulate the Lorenz machine using our K and S wheels. Since both sets of wheels are doing the same thing, we should be able to write one procedure that will work with either the K wheels or the S wheels.

Problem 11. Define a function, wheel_encrypt that takes a Baudot-encoded letter (a list of 5 bits) and a list of 5 wheels as parameters. The function should return a list that is the result of xor-ing each bit of the Baudot list with the first value of its respective wheel.

Your wheel_encrypt function should have running time in Θ(n) where n is the length of the list passed as the first parameter to wheel_encrypt. Even though our Lorenz crypotography will always involve lists of length five, your answer must work for any length inputs as long as both lists have the same length.

For example, wheel_encrypt([0, 0, 0, 1, 1], K_wheels) should produce [1, 0, 1, 0, 0].

We now have all the procedures we need to implement our simplified Lorenz machine. A quick review of how the machine should work:

This process is repeated for each letter in the message.

Problem 12. Define a function, do_lorenz that takes a list of Baudot values, the K wheels, S wheels and M wheel. The function should encrypt the first Baudot code with the K and S wheels, then continue encrypting the rest of the Baudot codes with the wheels rotated. The function should return the encrypted values in the form of a list of Baudot values.

For example:

>>> do_lorenz(string_to_baudot("COOKIE"), K_wheels, S_wheels, M_wheel)
[[1, 1, 0, 1, 0], [0, 0, 0, 0, 1], [1, 1, 0, 0, 1], [1, 0, 0, 0, 1], \ 
 [0, 0, 1, 1, 1], [1, 1, 0, 1, 1]]

Problem 13. Define a function lorenz_encrypt that takes four parameters: a string and three integers. The integers represent the starting positions of the three wheels (zero indexed), respectively. The function should call do_lorenz with the string converted to Baudot and the wheels rotated to the correct starting positions. The function should return the ciphertext in the form of a string.

You should now be able to encrypt strings using the simplified Lorenz cipher. To test it, call your lorenz_encrypt function with a string and offsets of your choice to produce ciphertext. Since our encryption and decryption functions are the same, if you apply lorenz_encrypt again using the ciphertext and the same offsets you should get your original message back:

>>> lorenz_encrypt("CAKE", 1, 2, 3)
>>> lorenz_encrypt("BNR!", 1, 2, 3)

Cracking the Code

The first Lorenz-encrypted messages were intercepted by the British in early 1940. The intercepts were sent to Bletchley Park, the highly secret British base set up specifically to break enemy codes. The code-breakers at Bletchley Park had little success with the Lorenz Cipher until the Germans made a major mistake in late 1941. A German operator had nearly finished sending a long message using a Lorenz machine when the receiver radioed back to tell him that the message had not been received correctly. The operator then reset his machine back to the same starting position and began sending the message again. But the operator, probably frustrated at having to resend the message, abbreviated some of the words he had typed out completely the first time. This led to two nearly identical messages encrypted using the same starting positions.

The messages were sent to John Tiltman at Bletchley Park. Tiltman was able to discern both messages and determine the generated key. The messages were then passed on to Bill Tutte who, after two months of work, figured out the complete structure of the Lorenz machine only from knowing the key it generated. The British were then able to break the Lorenz codes, but much of the work needed to be done by hand, which took a number of weeks to complete. By the time the messages were decrypted they were mostly useless.

The problem was given to Tommy Flowers, an electronics engineer from the Royal Post Office. Flowers designed and built a device called Colossus that worked primarily with electronic valves. The Colossus was the first electronic programmable computer. It was able to decrypt the Lorenz messages in a matter of hours, a huge improvement from the previous methods. The British built ten more Colossi and were able to decrypt an enormous amount of messages sent between Hitler and his high commanders. The British kept Colossus secret until the 1970s. After the war, eight of the Colossi were quickly destroyed and the remaining two were destroyed in 1960 and all drawings were burnt. The details of the breaking of the Lorenz Cipher were kept secret until 2000, but are now available at

Colossus (Original, 1943)

Colossus (Rebuilt, 2004)

Our simplified cipher will be much easier to break. There are only 5 starting positions for the K wheels, 5 starting positions for the S wheels, and 5 starting positions for the M wheel. Hence, the keyspace for our simplified Lorenz machine is 125. This is a small enough keyspace that it can be broken using "brute force": go through all the possible keys looking for a likely message. Because of the tiny keyspace, no clever cryptanalysis or even automated message recognition is required (unlike what was necessary for the real Lorenz cipher, with an astronomically huge keyspace).

Problem 14. Define a function, brute_force_lorenz, that takes as input a ciphertext string. Your procedure should call lorenz_encrypt on CIPHERTEXT (defined in for all 125 possible starting configurations, printing out the result of decrypting the ciphertext using that key.

If your procedure works correctly, one of the messages generated will look like sensible English and you will have found the key and message! (Your procedure does not need to return any value, although a more useful procedure would return the most likely plaintext message.)

Problem 15. (Optional bonus question) The actual Lorenz cipher operated similarly to the one in this problem set except instead of only having 5 positions in each wheel, each wheel had many more positions (up to 61). Suppose n is the number of possible positions for each Lorenz cipher wheel, and the procedure used to cryptanalyze the cipher is the same as your code (except extended as necessary to handle the extra wheel positions).

  1. Describe how the amount of work needed to break a Lorenz-encrypted message grows as a function of the number of wheel positions, n, using Θ notation. Assume the message length and number of wheels are fixed.
  2. Suppose instead that it was possible to add S and K wheels to the Lorenz cipher, and w represents the number of S and K wheels (for example, for the cipher machine in the problem set, w = 5). Describe how the amount of work needed to break a Lorenz-encrypted message grows as a function of w, using Θ notation. Assume the message length and number of wheel positions are fixed.

  3. If the Nazis had learned of Bletchley Park's success in breaking the Lorenz cipher during the war, what changes that could be done without building and redistributing completely new cipher machines, would be most likely to prevent further successful cryptanalysis?

Triple-Gold-Star Challenge! Decrypt the ciphertext defined as CHALLENGE_CIPHERTEXT in (this is not included in the project4 repository, use the link to download). You should also download, which includes the wheel definitions for the challenge and the code used to generate it.

The challenge ciphertext message is standard English. You should be happy that it is long (although should debug your code on shorter message): the more ciphertext you have, the easier it is to find statistical patterns. This message was encrypted using a simulated Lorenz cipher, without any of the simplifications used for the rest of this project. Hence, the keyspace is far to large for a brute-force attack, even with modern computing resources. The parts of the Udacity cs387: Applied Cryptography (Lesson 1) about the Lorenz cipher and how it was cryptanalyzed should be helpful for solving this.

Anyone who solves the "Triple-Gold-Star Challenge" will be offered a paid summer position in Dave's research group. (A good attempt is also probably worthy of a summer position, but not guaranteed.)

Credits: This problem set was originally created for CS200 Spring 2002 by Jon Erdman and David Evans, and tested by Stephen Liang. It was revised for CS200 Spring 2003 and CS150 Fall 2005 by David Evans, for CS 150 Spring 2009 by Wes Weimer, and for cs1120 Fall 2009 by David Evans. It was revised for Python in 2012 by Jonathan Burket and Lenny Li, and updated for Spring 2016 by David Evans.