## Introduction to the Affine Cipher and its Two Building Blocks (Additive and Multiplicative Ciphers)

A few weeks ago, I was asked by a viewer of my YouTube channel (“Cryptography for everybody”) if there was an “affine cipher component” in CrypTool 2. After taking a look at all our components, I realized that there was no component that implemented this cipher. I made a video for new CrypTool 2 developers and how they may implement the affine cipher, as an example of how to build new CrypTool 2 components. But I never added such an affine cipher component to CrypTool 2. So I sat down and created a brand new component and also made a YouTube video about the cipher (see the video at the end of this blog article). Also, in this blog article I summarize the video and how the encryption process works. You’ll also learn about the size of the key space and the unicity distance of the affine cipher. I hope you enjoy reading the article and that you understand how the cipher works afterwards.

### 1. Introduction to the Affine Cipher

The affine cipher is essentially a monoalphabetic substitution cipher, which means each letter of the plaintext is substituted with another letter to form the ciphertext. This method can trace its origins back to the time of the Caesar cipher, a cipher used in ancient Rome.

Over the course of history, as the mathematical domain grew, so did the complexity and strength of ciphers. The affine cipher is a one of the first testaments to this growth, combining the principles of two basic ciphers: the additive and the multiplicative cipher.

An interesting aspect of the affine cipher, and the ciphers we’ll be discussing, is that they operate on numbers. The ciphers essentially translate each letter of the used alphabet (e.g. the Latin alphabet) into a number (and back), providing a platform for mathematical operations. This translation is quite straightforward:

Let’s start with the basics. The additive cipher functions by adding an offset number to each letter, shifting it to the right. This is the underlying principle behind the famous Caesar cipher.

For this cipher:

• Key: The shift value, denoted as `𝒃`
• Encryption: `𝑐 = (𝑝 + 𝑏) 𝑚𝑜𝑑 26`
• Decryption: `𝑝 = (𝑐 - 𝑏) 𝑚𝑜𝑑 26`

For example, consider a shift value (b) of 5:

``````Plaintext:  HELLO WORLD = 7 4 11 11 14 22 14 17 11 3
Ciphertext: MJQQT BTWQI = 12 9 16 16 19 1 19 22 16 8``````

### 3. The Multiplicative Cipher

The multiplicative cipher involves multiplying each letter with a specific number. This process essentially “randomly selects” a letter from the alphabet, but still for each plaintext letter always the same ciphertext letter.

Key elements:

• Key: The multiplication value, denoted as `𝒂`. However, there needs to be an inverse (`𝑎⁻¹`) of this value for decryption. For a number `𝒂` to have a multiplicative inverse modulo m (here the alphabet size, which is 26), `𝒂` and m must be coprime, which means their greatest common divisor (GCD) is 1. In mathematical terms: gcd⁡(a,m)=1.
• Encryption: `𝑐 = (𝑝 ∙ 𝑎) 𝑚𝑜𝑑 26`
• Decryption: `𝑝 = (𝑐 ∙ 𝑎⁻¹) 𝑚𝑜𝑑 26`

To illustrate, let’s use `𝑎 = 5` (with its inverse `𝑎⁻¹ = 21`):

``````Plaintext: HELLO WORLD = 7 4 11 11 14 22 14 17 11 3
Ciphertext: JUDDS GSHDP = 9 20 3 3 18 6 18 7 3 15``````

Note: Computing the inverse of a number in modular arithmetic is crucial. Techniques like the extended Euclidean algorithm (see https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm) come in handy, and we’ll delve into this in a future discussion (and youtube video :-))!

### 4. The Affine Cipher

Building on the previous concepts, the affine cipher is a combination of both additive and multiplicative ciphers.

Key components:

• Key: Comprises a multiplication value `𝒂` and a shift value `𝒃`.
• Encryption: `𝑐 = (𝑝 ∙ 𝑎 + 𝑏) 𝑚𝑜𝑑 26`
• Decryption: `𝑝 = (𝑐 − 𝑏) ∙ 𝑎⁻¹ 𝑚𝑜𝑑 26`

For a hands-on example, using `𝑎 = 5` (inverse `𝑎⁻¹ = 21`) and `𝑏 = 5`:

``````Plaintext: HELLO WORLD = 7 4 11 11 14 22 14 17 11 3
Ciphertext: OZIIX LXMIU = 14 25 8 8 23 11 23 12 8 20``````

### 5. Keyspace Size and Unicity Distance

Understanding key spaces and unicity distances is essential for appreciating the security of a cipher:

• Keyspace size: For the affine cipher, we have `25` possible values for `𝒂` and `26` for `𝒃`. However, `𝒂` and `26` need to be coprime. Thus, only specific values are valid for `𝑎` as they possess an inverse (`𝑎⁻¹`). These values are `1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, and 25`. The total key space is then `12 x 26 = 312`.
• Unicity distance (𝑈): This concept helps determine the number of characters required to uniquely identify plaintext from its ciphertext. For the affine cipher, using the entropy of the key space (`𝐻(𝐾)`) and the redundancy of the English language (`𝐷`), the unicity distance is approximately `3`. The equation to compute the unicity distance is U = H(K)/D.

In essence, the affine cipher, with its mathematical underpinnings and historical significance, offers an exciting glimpse into the world of cryptography. Whether you’re a beginner or an aficionado, diving into these ciphers can be both intriguing and rewarding. Stay tuned for more cryptographic adventures also with CrypTool 2

Of course, I also made a YouTube video about the affine cipher. You may watch it here:

## Simplified AES (S-AES) Cipher Explained: Understanding Cryptographic Essentials

In the world of cryptography, security and simplicity are often at odds. But what if there was a way to bridge the gap between understanding cryptography and actually doing robust encryption? A few months ago, I found out that there is a simplified version of AES, called Simplified AES (S-AES). It is a very intriguing cipher intended as a teaching tool, analogous to Simplified DES (S-DES) for DES (which I had already implemented in CrypTool 2 many years ago). I implemented S-AES as a new CrypTool 2 component and also created a YouTube video about the cipher (see end of blog article). Also, my blog article here explains the main components of S-AES, breaks down the two rounds, and demonstrates the basic operations. I also suggest that if you really want to know how the cipher works, in addition to reading the article and watching my YouTube video, you implement the cipher yourself. As for me, I don’t understand a cipher 100% until I implement it myself :-).

We’ll start with an overview of the cipher. The following figure therefore shows the complete algorithm:

S-AES is a block cipher and it has a keysize of 16 bit and a blocksize of 16 bit. It consists of two rounds and a key expansion, which generates two additional round keys based on the provided 16-bit key. The first round consists of four building blocks, while the last round only uses three of these. In the following, we first shortly discuss the history of the S-AES cipher and after that each of the building blocks. Finally, we have a look at the key scheduling.

### 1. The S-AES Cipher

Simplified AES, or S-AES, made its debut in 2003 thanks to the work of Musa et al. [2]. Just like its more complex counterpart, the Advanced Encryption Standard (AES), S-AES is a block cipher. However, it is designed primarily for educational purposes, making it a learning tool for the classroom. While AES operates on 128-bit blocks and employs 128, 192, or 256-bit keys, S-AES works with 16-bit blocks and 16-bit keys. Furthermore, S-AES comprises only two rounds, as opposed to AES, which has 10, 12, or 14 rounds depending on the key length [1]. This design allows cryptography enthusiasts to grasp the core concepts without the overwhelming complexity of AES.

[1] Holden Joshua, Rose Hulman Institute of Technology ” A Simplified AES Algorithm “. 2010 (Figures by Holden)
[2] Musa, Mohammad A., Edward F. Schaefer, and Stephen Wedig. “A Simplified AES Algorithm and its Linear and Differential Cryptanalyses“. Cryptologia 27.2 (2003): 148 177.

The first step in an S-AES round is AddRoundKey. This operation involves XORing a 16-bit round key onto the 16-bit state. The state is shown here always as 4×4-table, each cell contains a nibble and the first column is the first byte and the second column is the second byte. XORing is a bitwise operation that combines the bits of two inputs, returning a new value based on their differences. Consider the following example:

In this case, each corresponding bit of the state and the round key is XORed together, producing the result 4E 52. Example XORing of a single state nibble:

### 3. SubstituteNibbles Operation

Next up is SubstituteNibbles, which applies a 4-bit S-box to the 16-bit state. The S-box is a lookup table that replaces each 4-bit input with a corresponding 4-bit output. For instance:

The S-AES 4-bit S-box is a key element of this operation, performing a specific substitution for each possible 4-bit input value. An example S-box lookup for a specific value looks like:

The corresponding S-box table is defined as follows:

### 4. ShiftRows Operation

ShiftRows involves exchanging the last two nibbles of the 16-bit state. It’s important to note that this operation is self-inverse, meaning it can be reversed to decrypt the data. For example:

An example computation of ShiftRows looks like:

This is the only primitive, which is self inverse. All other primitives have an inverse which is used for decryption instead of the encryption primitive.

### 5. MixColumns Operation

MixColumns applies a matrix operation on the 16-bit state within the Galois field GF(16). This operation can be quite complex, but it’s essential for ensuring strong encryption. For example:

It uses the reducible polynomial 𝑥^4+𝑥+1 for GF(16).

The matrix to mix the columns is is:

An example computation of both state bytes looks like:

The matrix multiplication is performed within GF(16). Best is to use a precomputed lookup table for the multiplication. To see how computing in finite fields works, have a look at https://en.wikipedia.org/wiki/Finite_field.

### 6. KeyExpansion

Inspired by the AES key expansion algorithm, S-AES’s ExpandKey operation computes round keys for each round. It employs a round constant array (Rcon) to generate the necessary keys. For instance, roundKey1 and roundKey2 can be computed using this scheme:

And the g function is defined as follows:

If you want some more explanations and details, please have a look at my YouTube video about S-AES, where I explain each step in more detail:

## Unveiling the Secrets of the Lorenz SZ42 Cipher Machine: A Dive into German High-Level WWII Encryption

The annals of World War II are replete with tales of ingenious inventions, some designed to bring destruction, and others to guard secrets. Among the latter is the SZ42, or “Schlüsselzusatz 42,” a German rotor encryption machine. I was always fascinated of this machine, which is one of the first “stream ciphers” you could say. Since we had no implementation in CrypTool 2, I created an implementation based on the very good descriptions of the cipher written by my friend George Lasry. Also, I made a YouTube video about the machine which you can find at the end of this article.

Of course, I also wanted to write a blog article (this one here :-)), which shows all the components of the machine in detail. Therefore, in this article we’ll unravel the mysteries of the SZ42, exploring its history, inner workings, and the encryption it offered during wartime.

### Introduction to the SZ42

The SZ42, short for “Schlüsselzusatz 42,” translates to “cipher attachment 42” in English. It was one of Germany’s cryptographic machines deployed during World War II. Notably, there were several models of this machine, including the SZ40 and SZ42, along with variants like SZ42a, SZ42b, and SZ42c. The Germans called it “Sägefisch” (“sawfish” in English), while at Bletchley Park, it went by the codename “Tunny,” akin to a tunafish.

Developed by the German company “C. Lorenz AG,” the SZ42 served the role of encrypting radio teletype (RTTY) communications. It began with an experimental link using SZ40 machines in June 1941, but it was the enhanced SZ42 machines that came into substantial use from mid-1942 onwards, primarily for high-level communications.

### 2. Baudot Code

To understand the SZ42’s significance, we must first delve into the Baudot code, the lingua franca of early teleprinter communications. Émile Baudot invented this pioneering bit-based code in the 1870s. It preceded the International Telegraph Alphabet No. 2 (CCITT-2), the most common teleprinter code before the advent of ASCII.

Baudot code words are composed of 5 bits, allowing for a total of 32 possible code words. However, this proved insufficient for encoding letters, digits, and special characters. To circumvent this limitation, Baudot introduced special symbols known as “figure shift” and “letter shift” to alter the code word representation. The version of Baudot code used in the SZ42 was the “Baudot-Murray-Code,” which optimized code assignments for commonly used letters, reducing strain on the teletypewriter mechanics.

### 3. How Does the SZ42 Encryption/Decryption Work?

The SZ42 operated by encrypting and decrypting Baudot code transmitted and received by teletype printers. It accomplished this by generating a “pseudo-random” stream called K, consisting of 5-bit code words, which was then XOR-ed with the plaintext (P) or ciphertext (C).

``````Encryption: C = P ⨁ K
Decryption: P = C ⨁ K``````

This method marks the SZ42 as one of the early stream ciphers, a critical precursor to modern encryption techniques.

### 4. The Inner Mechanism of the SZ42

The SZ42 boasted an intricate design with a total of 12 wheels, categorized into Chi, Psi, and Mu wheels. Each wheel had a unique pin count, with the Chi wheels having 41, 31, 29, 26, and 23 pins, Psi wheels with 43, 47, 51, 53, and 59 pins, and the Mu wheels with 61 and 37 pins.

Whenever a pin was in an active position, it added a 1 to the keystream, while in inactive position it added a 0. For each of the 5 Baudot bits, there were one Chi, and one Psi wheel.
The Chi wheels stepped regularly after each encryption, whereas Psi wheels stepped irregularly if Mu2 had an active pin, with Mu1 also stepping regularly after each encryption. Mu2 only stepped when Mu1 had an active pin.

### 5. Key Generation and Motor Limitations

Key generation in the SZ42 was governed by specific rules for setting the allowed numbers of active pins on each wheel. For example, Chi wheel 1 had a rule like “Allowed number of crosses in Chi1 is 20 or 21.” These rules ensured the machine operated effectively.

Motor limitations, like “CHI2_1BACK,” were introduced to make cryptanalysis more challenging. They compelled Psi wheels to move at specific positions, increasing the complexity of decryption. These limitations aimed to reduce the number of motor stops for Psi wheels, enhancing the machine’s security.

For details have a look at: James Reeds, Whitfield Diffie, and J.V. Field. 2015. Breaking Teleprinter Ciphers at Bletchley Park: An Edition of I.J. Good, D. Michie and G. Timms: General Report on Tunny with Emphasis on Statistical methods (1945). John Wiley & Sons.

### 6. Keyspace Size and Unicity Distance

The SZ42’s keyspace size without any wheel setting rules was staggering:

Considering these immense sub-keyspaces for each wheel-set, the total keyspace size reached:

This immense keyspace made brute-force attacks infeasible.

Furthermore, the SZ42 had a unicity distance U of 157, making it an incredibly secure encryption tool. Unicity distance can be camputed by dividing the Entropy of the keyspace by the redundancy of the (English) language:

Unicity distance is a concept in cryptography that refers to the minimum amount of ciphertext (encrypted text) required for an attacker to uniquely determine the corresponding plaintext (original message) while performing a brute-force attack. Since there are no “half” letters, the unicity distance value is always rounded up. With the (non-key rules-restricted) SZ42 a minimum number of 157 letters is required to obtain only one single valid solution when performing an attack. With less letters, we obtain multiple plaintexts and we can not distinguish which one is the correct one.

### 7. Conclusion

The SZ42 cipher machine stands as a testament to the ingenuity of its time, showcasing advanced encryption techniques during World War II. Its large keyspace and wheel-based encryption made it a challenging adversary for codebreakers. This machine’s historical significance and cryptographic complexity serve as a testament to the ever-evolving world of encryption and information security. Nevertheless, using Colossus, one of the first “computers”, the code breakers of Bletchley Park were able to frequently break into the encryption offered by the SZ42. Today, using modern techniques like hillclimbing and simulated annealing, we are also able to break the cipher

## The Grandpré Cipher Explained

Some days ago, I saw a very interesting hand cipher called the “Grandpré cipher”. It is not interesting because it was very secure or original. It is interesting because the “keying process”, i.e. the search for words for usage as the key(s), was kind of tedious. In this blog article, I will explain the background of the cipher, how it works, and its keyspace size and unicity distance.

### History of the Grandpré Cipher

The cipher is said to be first published by A. de Grandpré in his 1905 French book “Cryptographie pratique”. It was later named after the author. Unfortunately, I did not find any information on the author despite his (or her?) last name. So, I actually don’t know what the “A.” stands for. The description of the cipher itself can be found in the book on page 31. The chapter is named “Méthode de carre de 10×10”, which can be translated to “10×10 square method”. Grandpré defined the method for squares of 10×10, but squares with smaller sizes (9, 8, 7, 6) can also be used. We did also implement the cipher in a CrypTool 2 component. So if you want to try it by yourself, you may download CrypTool 2 to do so. Here, we show the original cover of Grandpré’s book:

Cryptographie pratique (1905) by A. de Grandpré

The American Cryptogram Association (ACA) has added the Grandpré cipher to its portfolio of “standard ciphers”. Their description of the cipher can be found here: https://www.cryptogram.org/downloads/aca.info/ciphers/Grandpre.pdf

The Grandpr´é cipher is a homophonic substitution cipher based on a keyword and several additional words. The cipher encrypts plaintext letters into two-digit ciphertext numbers. It uses a table to do so. We will discuss how this works in the next section.

### Table Creation Based on Keyword and Words

First, we need to find 10 words (or less), depending on our selected table size. The table size depends on our chosen secret keyword. Let’s say our secret keyword is “VIMOUTIERS” as used by Grandpré in his book. This keyword has 10 letters, so our table has to be a 10×10-sized table.

Now, we have to find 10 more words, each starting with a letter of our secret keyword. So a word for “V”, let’s take “VOLUPTUEUX”, a word for “I”, let’s take “INQUIETUDE”, and so on. Of course, here, for the example, we used the same words which Grandpré used in his book. To create the table, we write the keyword in the first column and the additional words in the rows. Each word starts with one of the letters of the keyword. The final table looks like the one below. But besides adding only the words, we also add digit coordinates (from 1 to “keyword length”; 10 = 0) to the rows and columns:

10×10-table based on the keyword “VIMOUTIERS” as shown by Grandpré in his book

### Encryption and Decryption

Now, we can use this table to encrypt a text. To encrypt a letter, we need to find it in the table. Then, we use the row R and column C to create our ciphertext symbol “RC”. Examples: E = “34”, A = “47”, E = “66”. Here, you can also see why the cipher is a homophonic substitution cipher. We have several options to choose from for most of the letters. But you can also see the drawback of the cipher. It is troublesome to find words for the table creation that contain all letters of our alphabet. Especially rarely used letters like Q and X are difficult to add to the table.

As an example, here is the encryption of the plaintext “HELLO WORLD”:

As you can see, we have several valid ciphertexts. Since the Grandpré cipher is a homophonic substitution cipher, we can use different homophones to create a variety of valid ciphertexts.

Clearly, the decryption of a given ciphertext is the inverse process. Here, you always take two digits and lookup the corresponding plaintext letter in the same table as used for encryption.

### Keyspace Size and Unicity Distance

Here, we calculate the size of the keyspace and the unicity distance of the Grandpré cipher. We compute the largest possible keyspace obtained by using a 10×10 table.

For the 10 x 10 case, there are 10 ∙ 10 = 100 cells in the table. Thus, we have a total number of 26^100 = 2^470 tables. But we use English words and don’t use the complete “table space”. Thus, let’s consider English has about 2,000 10-letter words. Then, we would “only” have 2000^10 valide tables, which is about 2^110 different tables.

Now, based on the above computed keyspace, lets compute the unicitiy distance U. It is the minimum number of letters needed when cryptanalyzing a ciphertext which allows us to be able to obtain only one valid solution. Below this number, we can find multiple valid English texts. To compute the distance, we have to divide the entropy of the keyspace H(K) by the redundancy D of the English language:

We need a minimum of 35 letters to be able to obtain only a single valid solution through cryptanalysis. Clearly, a given ciphertext can be solved like any other digit-based homophonic substitution cipher :-).

## The British “Typex” Cipher Machine Explained

The Typex is a cipher machine used by the British during World War II. It is, similar to the German Enigma cipher machine, an elector-mechanical rotor encryption machine. In contrast to the Enigma, the Typex was not broken during WWII. The Germans believed that Enigma is unbreakable and since Typex is very similar, they did not even attempt to break the machine.

I recently wrote a new CrypTool 2 component that implements the Typex cipher machine. If you are interested in testing the component (and the machine) yourself, you should download the latest nightly build of CrypTool 2.

## History and Usage of the Typex

The Typex machine was used for
a) Encryption of the own communication
b) Deciphering German Enigma messages

It was developed by Wing Commander Oswyn G.W. Lywood, Flight Lieutenant Coulson, Mr. E. W. Smith, and Sergeant Albert Lemon.

This image here shows Oswyn George William Gifford Lywood; a photo by Walter Stoneman; bromide print, February 1945; NPG x186070 © National Portrait Gallery, London.

There were several different versions of Typex including: Typex Mark I up to Mark VIII, Mark 22 and Mark 23.

A very nice and more detailed overview of the history of Typex can be found here: https://typex.virtualcolossus.co.uk/typex.html

## The Typex Components

In the following, we have a look at the machine’s components. The Typex machine consists of:

When pressing a key on the keyboard, the plaintext letter is printed by the plaintext printer. Also, current flows through the plugboard, the two stators, the three rotors and is then reflected by the reflector. Then it flows back through the three rotors and the two stators as well as the plugboard. Finally, the ciphertext letter is printed by the ciphertext printer.

Clearly, every time a key is pressed, between one and all three rotors move (Stators of course don’t move). In contrast to Enigma, a Typex rotor moves much more often. This is because the rotors have between 4 and 7 notches, while Enigma rotors had at most two notches.

## The Typex Plugboard

The Typex plugboard is the first (and last) component (despite the printers), which current is lead through after a key is pressed on the keyboard. It allows to “plug” letters, creating an initial monoalphabetic substitution.

The plugboard is not reciprocal (like the Enigma‘s plugboard. With Enigma, if we have letter X to letter Y, then we would also have letter Y to letter X). It, thus, offers a larger keyspace than Enigma’s plugboard.

## The Typex Rotors/Stators

Typex consists of two stators and three rotors. A rotor has more „notches“ than Enigma rotors (in CrypTool 2’s Typex implementation between 4 and 7). A rotor’s electrical contacts are doubled to improve reliability. Unfortunately, the original rotors are not published and still kept secret, thus, the simulators use no official rotor definitions.

## The Typex Reflector

The Typex reflector “reflects” the current coming from the rotors back through the rotors. In later Typex versions the reflector was replaced by an additional plugboard which allowed to change the reflector’s wiring easily.

## Keyspace Size and Unicity Distance

Since no original rotor definitions are known, the computation of keyspace size and unicity distance is based on the “CyberChef” Typex simulator written and published by GCHQ (see https://gchq.github.io/CyberChef/).

With this implementation, we have to choose 5 rotors (3 actual rotors and 2 stators) from a set of 8 rotors. Since a rotor can be put into the machine in forward or reversed position, they basically doubled the amount of usable rotors to 16. Here, we assume that we can use each rotor as many times as we like in parallel. Thus, the “rotor keyspace size” is:

We have to set the rotor start positions. We have five rotors (3 actual rotors and 2 stators). Each rotor can be in one of 26 different positions (A-Z). Thus, we have a total “start position keyspace” of:

The plugboard is basically a simple monoalphabetic substitution cipher. That means, for the first letter we have 26 different letters to choose from, for the second letter, we have 25 different remaining letters to choose from,…
Thus, the “plugboard keyspace” is:

To compute the overall keyspace size, we have to multiply all “sub-keyspace” sizes:

To compute the unicity distance U, we have to divide the entropy of the keyspace by the redundancy of the (English) language:

This means, we need a minimum of 42 letters to be able to obtain a single valid solution when we perform cryptanalysis of a Typex message.

I also created a YouTube video about the Typex cipher machine. Here, I discuss the machine as well as its keyspace size and unicity distance. Also, I show how to use the Typex component in CrypTool 2:

Finally, if you want to “play” with a really nice simulator (and also want to learn much more about the Typex), you should have a look at the “Virtual Typex”: https://typex.virtualcolossus.co.uk/Typex/

The usage of the simulator is also shown in the above linked YouTube video :-).

## The GRANIT / 160 Cipher – A GDR Stasi Hand Cipher for Spies

The GRANIT / 160 cipher is a hand cipher which was used for communication between the GDR’s “Ministerium für Staatssicherheit” (Engl. Ministry for State Security) or MfS or Stasi and their agents in West Germany. The Stasi’s “Geheime Dienstvorschrift” (Engl. secret service regulation) GVS 1064/59 and GVS 1065/59 describe its usage. Copies of the original (German) service regulations can be found on Jörg Drobicks’s homepage.

The GRANIT cipher is a variant of the “Doppelwürfel” (= double columnar transposition cipher). The Stasi communicated with their agents in West Germany via numbers stations: The agent had to turn on a radio and wait for his call sign. Then, the agent notes down the spoken numbers and after that he decrypts the received message using the GRANIT cipher.

The West German “Zentralstelle für das Chiffrierwesen” (Engl. Central Office
for Ciphering) was able to capture and decipher messages, because they could guess some of the used keys.

## The Günter Guillaume Case – An East German Spy Near Willy Brandt

Günter Guillaume and his wife Christel Guillaime were East German spies deployed in West Germany. Günter Guillaume became officer in the economic, financial and social policy department of German Chancellor Willy Brandt. The Guillaume couple used the GRANIT cipher for communication with the Stasi. But the West German “Zentralstelle für das Chiffrierwesen” was able to decipher some of their messages.

Matching birthday date wishes and well wishes for the birth of a son were found in a deciphered message. The Guillaumes were finally caught and arrested in 1974. Ironically, mentioned birthday dates of the Guillaumes were their “agent fake birth dates”…

Despite having good evidence police raided his home but Guillaume instantly confessed being a Stasi spy when approached by police. More about the story can be read (in German) on Klaus Schmeh’s blog.

## How the GRANIT / 160 Cipher Works

The GRANIT / 160 cipher is a hand cipher and consists of five steps:

1. Create a straddling checkerboard based on a keyword
2. Encrypt the plaintext using the straddling checkerboard
3. Create two rectangles for a double columnar transposition based on two key phrases
4. Encrypt the numbers (result of step 2) using the first rectangle
5. Encrypt the numbers (result of step 4) using the second rectangle

Clearly, the Stasi defined how to create the keys, how to create message indicators so that the receiver of a message is able to decrypt, etc. The details of these procedures and the details of the actual cipher can be seen in the video below :-).

If you want to know, how the details of the GRANIT cipher work, please watch my YouTube video on “Cryptography for everybody” about the GRANIT cipher:

## The Bazeries Cipher Explained – A Classical Cipher Based on Substitution and Transposition

The Bazeries cipher was invented by and named after Étienne Bazeries, a French cryptographer. Bazeries was active between 1890 and the First World War.

Bazeries is probably most famously known for the “Bazeries Cylinder”, a cipher device similiar to the Jefferson Disk or the M-94 cipher. Bazeries was a good code breaker: He solved messages encrypted with the official French military transposition system (lead to improvements of the ciphers). He further exposed weaknesses in French cipher systems. He assisted in solving German military ciphers during World War I, after he retired from the army. And in the 1890s he broke the famous nomenclator system called the “Great Cipher”, created by the Rossignols in the 17th century.

### How Does the Cipher Work?

The cipher is a combination of substitution and transposition. For encryption, Bazeries only used a single number key, e.g. 123. In the following, we encrypt an example plaintext (“HELLOWORLD”).

We create two Polybius squares. In the first square, we put the Latin alphabet (I=J; filled from left to right and top to bottom row-wise). In the other square, we write a text representation of the number key, e.g. ONEHUNDREDTWENTYTHREE, followed by the remaining unused letters of the Latin alphabet. We fill the second square from top to bottom and left to right column-wise:

To encrypt a plaintext, we first substitute it using the two created Polybius squares. We look for the plaintext letter in the left square and use the corresponding letter of the right square for the ciphertext (For eample A->O, B->D, etc.). When we encrypt HELLOWORLD, we obtain BQEELYLWEI.

Then, we transpose the intermediate ciphertext using the digits of the number key. We split the text into blocks of sizes defined by the digits of the number key. Then, we reverse each of these blocks to create the final ciphertext:

So our final ciphertext here is BEQYLELWEI. Of course, the decryption is the inverse process of the above shown steps :-).

### Keyspace Size and Unicity Distance

Here, we compute the keyspace size as well as the unicity distance (https://en.wikipedia.org/wiki/Unicity_distance). In the original version, as written above, Bazeries created the second Polybius square using the same key as he used for transposition. So for e.g. a maximum number key length of four digits, we compute:

• For a 4-digit key, we have 10^4=10,000 different keys
• For a 3-digit key, we have 10^3=1,000 different keys
• For a 2-digit key, we have 10^2=100 different keys
• For a 1-digit key, we have 10^1=10 different keys

Then, we have to add all these number. Thus, we have a toal keyspace size of 11,110

If we consider that the encrypter uses an independent (other) key for the Polybius square creation, we would have to compute 26! ≈ 2^88.4 for the number of possible different Polybius squares. In this case, we have to compute for the “complex” Bazeries cipher 11,110 ∙ 2^88.4 which is about 2^101.8.

To compute the unicity distance (of the complex case), we have to divide the entropy of the keyspace with the redundancy of the language:

So we would need a ciphertext with a minimum length of 32 letters to obtain only one valid (and the correct) solution via cryptanalysis.

## The Four-Square Cipher Explained

This is the third cipher of Félix-Marie Delastelle, a French hobby cryptographer, which I implemented in CrypTool 2. Delastelle published the four-square cipher in his book “Traité Élémentaire de Cryptographie“. He wrote the book in 1901 but it was published after Delastelle’s death in 1902.

The four-square cipher is a bigraphic monoalphabetic substitution cipher. Bigraphic means, that it always encrypts two plaintext letters at the same time. The cipher uses four Polybius squares, two of which are created using keywords.

### Key Generation – Preparing the Four Polybius Squares

First, you have to prepare the four polybius squares. Let’s assume our keywords are “secret” and “keyword”:

Here, we created the second and third polybius squares using the previously chosen keywords. To create one of these polybius squares, the corresponding keyword is first written into the square from left to right and top to bottom. Here, if a letter occurs more than once, we omit it. In our example, we do not write the second “E” of “secret” again in the square. After writing the keyword, we fill the rest of the square with the remaining alphabet letters, which we did not already use for the keyword. After we created the second and third square, we fill the first and fourth square just with the alphabet. Since all the squares have only 25 positions, our alphabet consists of only 25 letters. Delastelle used “I” = “J” and did not include a “J” in his alphabet.

### How Encryption works

Now, we can encrypt a plaintext using our four polybius squares. Here, for example we want to encrypt “HELLOWORLD”. To do so, we search the letter “H” in the first square and the letter “E” in the fourth square:

We create a connected “rectangle” where “H” is the upper corner and “E” is the lower corner. Now, we find our ciphertext letters in the two other corners of the so-created rectangle. Here, we encrypt the “H” by “G” and the “E” by “Y”. We continue encrypting our plaintext using this method to obtain the complete ciphertext:

To decrypt a given ciphertext, we just reverse the process. We look up the ciphertext letters (pair-wise) in the second and third square and find our plaintext letters in the first and fourth square.

### Keyspace Size and Unicity Distance Computation

We compute the keyspace size by k = 25! * 25! = 2^167.36. This is, because we have to fill two polybius squares and a single polybius square has 25! possibilities to be filled with 25 letters.

We compute the unicity distance U by

Here, H(k) is the entropy of the keyspace and D is the redundancy of the language (here English). From the English Wikipedia: “In cryptography, unicity distance is the length of an original ciphertext needed to break the cipher by reducing the number of possible spurious keys to zero in a brute force attack”. In our case, we need more than 53 letters to be able to obtain only one valid plaintext.

I also created a YouTube video about the four-square cipher (and how you can use it in CrypTool 2). Watch it here:

## The Bifid and Trifid Cipher Explained

I recently made two videos about two interesting classical ciphers invented by Félix Marie Delastelle. Delastelle wrote a book on cryptography in 1901. Unfortunately, he died before his book was published in 1902. In his book, he describes several ciphers he invented. This blog post is about two of them: the bifid cipher and the trifid cipher.

### The Bifid Cipher

The bifid cipher is a cipher which combines a Polybius square with transposition, and uses fractionation:

1. First, we use a keyword to create a 25-letter Polybius square
For example: “SECRET KEYWORD”:

2. Then, we encrypt the plaintext using the square, by writing the coordinates of the square below the plaintext. Example:

3. After that, we write the digits (transposed/fractionated) in a single row:

4. Finally, we decrypt the digits using the square to obtain the ciphertext:

The decryption is the reverse process. It is also possible to not encrypt the plaintext in one go. Instead, you can encrypt the ciphertext in blocks of n (n for example being 5).

The keyspace size and unicity distance (minimal number of letters needed in a ciphertext that allows having only a single valide solution) can be computed as follows:

### The Trifid Cipher

The trifid cipher was invented by Félix Marie Delastelle as an extension of the above shown bifid cipher.

1. First, we use a keyword to create three 9-letter Polybius squares. For example: “SECRET KEYWORD”

2. Then, we encrypt the plaintext using the squares, by writing the number of the used square number and the coordinates below the plaintext. Example:

3. After that, we write the digits (transposed/fractionated) in a single row

4. Finally, we decrypt the digits using the three squares:

The decryption is the reverse process. It is also possible to not encrypt the plaintext in one go. Instead, you can encrypt the ciphertext in blocks of n (n for example being 5).

The keyspace size and unicity distance (minimal number of letters needed in a ciphertext that allows having only a single valide solution) can be computed as follows:

I made a YouTube video about the Bifid cipher. Here, you can also see how to use the bifid cipher component of CrypTool 2:

I also made a YouTube video about the Trifid cipher. Here, you can also see how to use the trifid cipher component of CrypTool 2:

## The Book Cipher Explained

A book cipher is a cipher where the plaintext letters (or words) are encrypted using a book (or other text document) as a kind of lookup table. Sender and receiver of encrypted messages can agree to use any book or other publication available to both of them. A book cipher has a considerable advantage for a spy in enemy territory since it does not raise suspicion (like e.g. a code book). The main strength of a book cipher is the key, because only being in possession of the original “book” allows the decryption.

# Famous Examples of Book Ciphers

1. The most famous book ciphers are probably the “Beale ciphers”
• The Beale ciphers are three encrypted documents
• Only one of the documents has been successfully deciphered (using the United States Declaration of Independence as key)
• The two other messages are still unsolved… (it is unclear, how these were encrypted)
2. The “Arnold Cipher” was a book cipher used by John André and Benedict Arnold in 1780 during the American Revolutionary War
• The book used as a key to the cipher was either “Commentaries on the Laws of by William Blackstone or Nathan Bailey’s Dictionary
• The cipher consisted of a series of three numbers separated by periods:
page number . Line number . word number
3. The “Cicada 3301 online puzzle” series also contained book ciphers

## The Book Cipher

First, the sender and the receiver have to agree on the (exact) same “book”. They also have to agree on an “encoding scheme”:
1. Encode single letters
2. Encode complete words

Also, they need to know “what” is encoded:
1. Page
2. Line
3. Word

1. Single Letter Scheme:
In the following, we show an example of a book cipher with the “single letter scheme”. It uses this sample text as key:

To encrypt a plaintext, take a random word from the “book” above which starts with the plaintext letter you want to encrypt. Then, write the position of the word into the ciphertext. Go on, until you have encrypted the complete plaintext. In the following are two examples, how to encrypt plaintexts:

Example 1 (Write the offset of the word into the ciphertext):
H E L L O W O R L D –> 6 37 100 42 56 72 12 53 42 52

Example 2 (Write the line number and position of the word in the particular line into the ciphertext):
H E L L O W O R L D –> 09.04 04.08 12.03 05.08 05.09 08.06 06.03 07.07 12.03 06.09

Hint: With a real book, we could also prepend the page number.

2. Complete word scheme:
The “complete word scheme” is described in my YouTube video about the book cipher (see below). Also, I explain how to use a book cipher in CrypTool 2.