# Starting to understand Encryption

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Encryption is the principal application of cryptography; it makes data incomprehensible in order to ensure its confidentiality. Encryption uses an algorithm called a cipher and a secret value called the key; if you don’t know the secret key, you can’t decrypt, nor can you learn any bit of information on the encrypted message—and neither can any attacker.

## The basics of encryption

When we’re encrypting a message, plaintext refers to the unencrypted message and ciphertext to the encrypted message. A cipher is therefore composed of two functions: encryption turns a plaintext into a ciphertext, and decryption turns a ciphertext back into a plaintext. But we’ll often say “cipher” when we actually mean “encryption.” For example, the figure below shows a cipher, E, represented as a box taking as input a plaintext, P, and a key, K, and producing a ciphertext, C, as output. I’ll write this relation as C = E(K, P). Similarly, when the cipher is in decryption mode, I’ll write D(K, C).

## History of Encryption

One of the earliest forms of encryption is symbol replacement, which was first found in the tomb of Khnumhotep II, who lived in 1900 B.C. Egypt. Symbol replacement encryption is “non-standard,” which means that the symbols require a cipher or key to understand. This type of early encryption was used throughout Ancient Greece and Rome for military purposes.]

One of the most famous military encryption developments was the Caesar Cipher.

### The Caesar Cipher

The Caesar cipher is so named because the Roman historian Suetonius reported that Julius Caesar used it. It encrypts a message by shifting each of the letters down three positions in the alphabet, wrapping back around to A if the shift reaches Z. For example, ZOO encrypts to CRR, FDHVDU decrypts to CAESAR, and so on, as shown in the next figure.

There’s nothing special about the value 3; it’s just easier to compute in one’s head than 11 or 23.

The Caesar cipher is super easy to break: to decrypt a given ciphertext, simply shift the letters three positions back to retrieve the plaintext. That said, the Caesar cipher may have been strong enough during the time of Crassus and Cicero. Because no secret key is involved (it’s always 3), users of Caesar’s cipher only had to assume that attackers were illiterate or too uneducated to figure it out—an assumption that’s much less realistic today. (In fact, in 2006, the Italian police arrested a mafia boss after decrypting messages written on small scraps of paper that were encrypted using a variant of the Caesar cipher: ABC was encrypted to 456 instead of DEF, for example.)

Could the Caesar cipher be made more secure? You can, for example, imagine a version that uses a secret shift value instead of always using 3, but that wouldn’t help much because an attacker could easily try all 25 possible shift values until the decrypted message makes sense.

### The Vigenère Cipher

It took about 1500 years to see a meaningful improvement of the Caesar cipher in the form of the Vigenère cipher, created in the 16th century by an Italian named Giovan Battista Bellaso. The name “Vigenère” comes from the Frenchman Blaise de Vigenère, who invented a different cipher in the 16th century, but due to historical misattribution, Vigenère’s name stuck. Nevertheless, the Vigenère cipher became popular and was later used during the American Civil War by Confederate forces and during WWI by the Swiss Army, among others.

The Vigenère cipher is similar to the Caesar cipher, except that letters aren’t shifted by three places but rather by values defined by a key, a collection of letters that represent numbers based on their position in the alphabet. For example, if the key is DUH, letters in the plaintext are shifted using the values 3, 20, 7 because D is three letters after A, U is 20 letters after A, and H is seven letters after A. The 3, 20, 7 pattern repeats until you’ve encrypted the entire plaintext. For example, the word CRYPTO would encrypt to FLFSNV using DUH as the key: C is shifted three positions to F, R is shifted 20 positions to L, and so on. The next figure illustrates this principle when encrypting the sentence THEY DRINK THE TEA.

The Vigenère cipher is clearly more secure than the Caesar cipher, yet it’s still fairly easy to break. The first step to breaking it is to figure out the key’s length. For example, take the example in Figure 1-3, wherein THEY DRINK THE TEA encrypts to WBLBXYLHRWBLWYH with the key DUH. (Spaces are usually removed to hide word boundaries.)

Notice that in the ciphertext WBLBXYLHRWBLWYH, the group of three letters WBL appears twice in the ciphertext at nine-letter intervals. This suggests that the same three-letter word was encrypted using the same shift values, producing WBL each time. A cryptanalyst can then deduce that the key’s length is either nine or a value divisible by nine (that is, three). Furthermore, they may guess that this repeated three-letter word is THE and therefore determine DUH as a possible encryption key.

The second step to breaking the Vigenère cipher is to determine the actual key using a method called frequency analysis, which exploits the uneven distribution of letters in languages. For example, in English, E is the most common letter, so if you find that X is the most common letter in a ciphertext, then the most likely plaintext value at this position is E.

Despite its relative weakness, the Vigenère cipher may have been good enough to securely encrypt messages when it was used. First, because the attack just outlined needs messages of at least a few sentences, it wouldn’t work if the cipher was used to encrypt only short messages. Second, most messages needed to be secret only for short periods of time, so it didn’t matter if ciphertexts were eventually decrypted by the enemy. (The 19thcentury cryptographer Auguste Kerckhoffs estimated that most encrypted wartime messages required confidentiality for only three to four hours.)

## How Ciphers Work

Based on simplistic ciphers like the Caesar and Vigenère ciphers, we can try to abstract out the workings of a cipher, first by identifying its two main components: a permutation and a mode of operation. A permutation is a function that transforms an item (in cryptography, a letter or a group of bits) such that each item has a unique inverse (for example, the Caesar cipher’s three-letter shift). A mode of operation is an algorithm that uses a permutation to process messages of arbitrary size. The mode of the Caesar cipher is trivial: it just repeats the same permutation for each letter, but as you’ve seen, the Vigenère cipher has a more complex mode, where letters at different positions undergo different permutations.

### The Permutation

Most classical ciphers work by replacing each letter with another letter— in other words, by performing a substitution. In the Caesar and Vigenère ciphers, the substitution is a shift in the alphabet, though the alphabet or set of symbols can vary: instead of the English alphabet, it could be the Arabic alphabet; instead of letters, it could be words, numbers, or ideograms, for example. The representation or encoding of information is a separate matter that is mostly irrelevant to security. (We’re just considering Latin letters because that’s what classical ciphers use.)

A cipher’s substitution can’t be just any substitution. It should be a permutation, which is a rearrangement of the letters A to Z, such that each letter has a unique inverse. For example, a substitution that transforms the letters A, B, C, and D, respectively to C, A, D, and B is a permutation, because each letter maps onto another single letter. But a substitution that transforms A, B, C, D to D, A, A, C is not a permutation, because both B and C map onto A. With a permutation, each letter has exactly one inverse.

Still, not every permutation is secure. In order to be secure, a cipher’s permutation should satisfy three criteria:

• The permutation should be determined by the key, so as to keep the permutation secret as long as the key is secret. In the Vigenère cipher, if you don’t know the key, you don’t know which of the 26 permutations was used; hence, you can’t easily decrypt.
• Different keys should result in different permutations. Otherwise, it becomes easier to decrypt without the key: if different keys result in identical permutations, that means there are fewer distinct keys than distinct permutations, and therefore fewer possibilities to try when decrypting without the key. In the Vigenère cipher, each letter from the key determines a substitution; there are 26 distinct letters and as many distinct permutations.
• The permutation should look random, loosely speaking. There should be no pattern in the ciphertext after performing a permutation because patterns make a permutation predictable for an attacker, and therefore less secure. For example, the Vigenère cipher’s substitution is pretty predictable: if you determine that A encrypts to F, you could conclude that the shift value is 5 and you would also know that B encrypts to G, that C encrypts to H, and so on. However, with a randomly chosen permutation, knowing that A encrypts to F would only tell you that B does not encrypt to F.

We’ll call a permutation that satisfies these criteria a secure permutation. But as you’ll see next, a secure permutation is necessary but not sufficient on its own for building a secure cipher. A cipher will also need a mode of operation to support messages of any length.

### The Mode of Operation

Say we have a secure permutation that transforms A to X, B to M, and N to L, for example. The word BANANA, therefore, encrypts to MXLXLX, where each occurrence of A is replaced by an X. Using the same permutation for all the letters in the plaintext thus reveals any duplicate letters in the plaintext. By analyzing these duplicates, you might not learn the entire message, but you’ll learn something about the message. In the BANANA example, you don’t need the key to guess that the plaintext has the same letter at the three X positions and another same letter at the two L positions. So if you know, for example, that the message is a fruit’s name, you could determine that it’s BANANA rather than CHERRY, LYCHEE, or another six-letter fruit.

The mode of operation (or just mode) of a cipher mitigates the exposure of duplicate letters in plaintext by using different permutations for duplicate letters. The mode of the Vigenère cipher partially addresses this: if the key is N letters long, then N different permutations will be used for every N consecutive letters. However, this can still result in patterns in the ciphertext because every Nth letter of the message uses the same permutation. That’s why frequency analysis works to break the Vigenère cipher, as you saw earlier.

Frequency analysis can be defeated if the Vigenère cipher only encrypts plaintexts that are of the same length as the key. But even then, there’s another problem: reusing the same key several times exposes similarities between plaintexts. For example, with the key KYN, the words TIE and PIE encrypt to DGR and ZGR, respectively. Both end with the same two letters (GR), revealing that both plaintexts share their last two letters as well. Finding these patterns shouldn’t be possible with a secure cipher.

To build a secure cipher, you must combine a secure permutation with a secure mode. Ideally, this combination prevents attackers from learning anything about a message other than its length.

This article was based on the book Serious Cryptography – A Practical Introduction to Modern Encryption by Jean-Philippe Aumasson. To learn more about encryption and cryptography in general we recommend this great book.

This is the index from the book:

• Chapter 1: Encryption
• Chapter 2: Randomness
• Chapter 3: Cryptographic Security
• Chapter 4: Block Ciphers
• Chapter 5: Stream Ciphers
• Chapter 6: Hash Functions
• Chapter 7: Keyed Hashing
• Chapter 8: Authenticated Encryption
• Chapter 9: Hard Problems
• Chapter 10: RSA
• Chapter 11: Diffie–Hellman
• Chapter 12: Elliptic Curves
• Chapter 13: TLS
• Chapter 14: Quantum and Post-Quantum

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