Knowledge Base

Let’s start with an illustration before introducing the key terms. We begin with the plaintext that we want to encrypt. The plaintext is the readable data; it can be anything from a simple “hello”, a cat photo, credit card information, or medical health records. From a cryptography perspective, these are all “plaintext” messages waiting to be encrypted. The plaintext is passed through the encryption function along with a proper key; the encryption function returns a ciphertext. The encryption function is part of the cipher; a cipher is an algorithm to convert a plaintext into a ciphertext and vice versa.

To recover the plaintext, we must pass the ciphertext along with the proper key via the decryption function, which would give us the original plaintext. This is shown in the illustration below.

We have just introduced several new terms, and we need to learn them to understand any text about cryptography. The terms are listed below:

• Plaintext is the original, readable message or data before it’s encrypted. It can be a document, an image, a multimedia file, or any other binary data.
• Ciphertext is the scrambled, unreadable version of the message after encryption. Ideally, we cannot get any information about the original plaintext except its approximate size.
• Cipher is an algorithm or method to convert plaintext into ciphertext and back again. A cipher is usually developed by a mathematician.
• Key is a string of bits the cipher uses to encrypt or decrypt data. In general, the used cipher is public knowledge; however, the key must remain secret unless it is the public key in asymmetric encryption. We will visit asymmetric encryption in a later task.
• Encryption is the process of converting plaintext into ciphertext using a cipher and a key. Unlike the key, the choice of the cipher is disclosed.
• Decryption is the reverse process of encryption, converting ciphertext back into plaintext using a cipher and a key. Although the cipher would be public knowledge, recovering the plaintext without knowledge of the key should be impossible (infeasible).

Historical Ciphers

Cryptography’s history is long and dates back to ancient Egypt in 1900 BCE. However, one of the simplest historical ciphers is the Caesar Cipher from the first century BCE. The idea is simple: shift each letter by a certain number to encrypt the message.

Consider the following example:

• Plaintext: TRYHACKME
• Key: 3 (Assume it is a right shift of 3.)
• Cipher: Caesar Cipher

We can easily figure out that T becomes W, R becomes U, Y becomes B, and so on. As you noticed, once we reach Z, we start all over, as shown in the figure below. Consequently, we get the ciphertext of WUBKDFNPH.

To decrypt, we need the following information:

• Ciphertext: WUBKDFNPH
• Key: 3
• Cipher: Caesar Cipher

For encryption, we shift to the right by three; for decryption, we shift to the left by three and recover the original plaintext, as illustrated in the image above. However, if someone gives you a ciphertext and tells you that it was encrypted using Caesar Cipher, recovering the original text would be a trivial task as there are only 25 possible keys. The English alphabet is 26 letters, and shifting by 26 will keep the letter unchanged; hence, 25 valid keys for encryption with Caesar Cipher. The figure below shows how decryption will succeed by attempting all the possible keys; in this case, we recovered the original message with K__e__y = 5. Consequently, by today’s standards, where the cipher is publicly known, Caesar Cipher is considered insecure.

You would come across many more historical ciphers in movies and cryptography books. Examples include:

• The Vigenère cipher from the 16th century
• The Enigma machine from World War II
• The one-time pad from the Cold War

You can experiment with Caesar cipher shifts using the interactive tool at the Cryptii website. It lets you tweak the rotation and instantly see how the text transforms.

Basic Math

The building blocks of modern cryptography lie in mathematics. To demonstrate some basic algorithms, we will cover two mathematical operations that are used in various algorithms:

• XOR Operation
• Modulo Operation

XOR Operation

XOR, short for “exclusive OR”, is a logical operation in binary arithmetic that plays a crucial role in various computing and cryptographic applications. In binary, XOR compares two bits and returns 1 if the bits are different and 0 if they are the same, as shown in the truth table below. This operation is often represented by the symbol ⊕ or ^.

If this is the first time you work with a truth table, it is a table that shows all possible outcomes. The XOR truth table above states all four cases: 0 ⊕ 0 = 0, 0 ⊕ 1 = 1, 1 ⊕ 0 = 1, and 1 ⊕ 1 = 0.

Let’s consider an example where we want to apply XOR to the binary numbers 1010 and 1100. In this case, we perform the operation bit by bit: 1 ⊕ 1 = 0, 0 ⊕ 1 = 1, 1 ⊕ 0 = 1, and 0 ⊕ 0 = 0, resulting in 0110.

You may be wondering how XOR can play any role in cryptography. XOR has several interesting properties that make it useful in cryptography and error detection. One key property is that applying XOR to a value with itself results in 0, and applying XOR to any value with 0 leaves it unchanged. This means A ⊕ A = 0, and A ⊕ 0 = A for any binary value A. Additionally, XOR is commutative, i.e., A ⊕ B = B ⊕ A. And it is associative, i.e., (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C).

Let’s see how we can make use of the above in cryptography. We will demonstrate how XOR can be used as a basic symmetric encryption algorithm. Consider the binary values P and K, where P is the plaintext, and K is the secret key. The ciphertext is C = P ⊕ K.

Now, if we know C and K, we can recover P. We start with C ⊕ K = (P ⊕ K) ⊕ K. But we know that (P ⊕ K) ⊕ K = P ⊕ (K ⊕ K) because XOR is associative. Furthermore, we know that K ⊕ K = 0; consequently, (P ⊕ K) ⊕ K = P ⊕ (K ⊕ K) = P ⊕ 0 = P. In other words, XOR served as a simple symmetric encryption algorithm. In practice, it is more complicated as we need a secret key as long as the plaintext.

Modulo Operation

Another mathematical operation we often encounter in cryptography is the modulo operator, commonly written as % or as m__o__d. The modulo operator, X%Y, is the remainder when X is divided by Y. In our daily life calculations, we focus more on the result of division than on the remainder. The remainder plays a significant role in cryptography.

You need to work with large numbers when solving some cryptography exercises. If your calculator fails, we suggest using a programming language such as Python. Python has a built-in int type that can handle integers of arbitrary size and would automatically switch to larger types as needed. Many other programming languages have dedicated libraries for big integers. If you prefer to do your math online, consider WolframAlpha.

Let’s consider a few examples.

• 25%5 = 0 because 25 divided by 5 is 5, with a remainder of 0, i.e., 25 = 5 × 5 + 0
• 23%6 = 5 because 23 divided by 6 is 3, with a remainder of 5, i.e., 23 = 3 × 6 + 5
• 23%7 = 2 because 23 divided by 7 is 3 with a remainder of 2, i.e., 23 = 3 × 7 + 2

An important thing to remember about modulo is that it’s not reversible. If we are given the equation x%5 = 4, infinite values of x would satisfy this equation.

The modulo operation always returns a non-negative result less than the divisor. This means that for any integer a and positive integer n, the result of a%n will always be in the range 0 to n − 1.