Caesar cipher

A Caesar cipher is a super simple type of cipher where we shift the letters of the alphabet by a certain number of places. For example, if we shift by 3, then A becomes D, B becomes E, C becomes F, and so on.

Mathematical function

E(x) = (x + k) \mod 26

E(x) is the encryption function, which takes a letter (represented as a number from 0 to 25) and produces the ciphertext letter.
x is the plaintext letter, represented as a number from 0 to 25 (where A=0, B=1, …, Z=25).
k is the key, which is the number of positions we shift the letters by. It can be any whole number from 0 to 25.

Decryption function

To decrypt a message encrypted with a Caesar cipher, we can use this:

D(y) = (y - k) \mod 26

As you can see, all we do is subtract the key from the encrypted character, to find the plaintext letter.

Why is it so easy to crack?

The Caesar cipher isn’t used in any real encryption schemas, because it’s so insecure. The main reasons for that are:

There are only 26 possible keys (shifts), so an attacker can easily try all of them to see which one produces a readable message.
It doesn’t change the frequency of letters, so if you analyze the frequency of letters in the ciphertext, you can easily guess the key by looking at the most common letters (e.g., if ‘G’ is the most common letter in the ciphertext, it’s likely that the key is 2, as ‘E’ is the most common letter in English text, so shifting back by 2 would give you ‘E’ as the most common).

Examples

Encrypt the text “HELLO” with a pad (key) of 3

We can first write out a table of the letters and their corresponding encrypted form:

Plain:  ABCDEFGHIJKLMNOPQRSTUVWXYZ
Cipher: DEFGHIJKLMNOPQRSTUVWXYZABC

Then we can encrypt each letter of “HELLO”:

H (7) becomes K (10)
E (4) becomes H (7)
L (11) becomes O (14)
L (11) becomes O (14)
O (14) becomes R (17)

So our final encrypted message is “KHOOR”.

You may already be able to see an issue - we can immediately see there is a double letter in the 3rd and 4th positions of the word, which may help us crack the cipher!

Decrypt the text “KHOOR” with a pad (key) of 3

Obviously, we know what our answer will be, but let’s go through the decryption process to check it works!

With a pad of 3, we can get from the encrypted letter to the plaintext letter by subtracting 3 from the encrypted letter’s position in the alphabet:

Cipher: ABCDEFGHIJKLMNOPQRSTUVWXYZ
Plain:  DEFGHIJKLMNOPQRSTUVWXYZABC

Then we can decrypt each letter of “KHOOR”:

K (10) becomes H (7)
H (7) becomes E (4)
O (14) becomes L (11)
O (14) becomes L (11)
R (17) becomes O (14)

So our final decrypted message is “HELLO”, which is what we started with!

flashcards

Question	Answer
What is a Caesar cipher?	A super simple type of cipher where we shift the letters of the alphabet by a certain number of places.
What is the mathematical formula for Caesar cipher encryption?	E(x) = (x + k) \mod 26
What does x represent in the encryption formula E(x) = (x + k) \mod 26?	x is the plaintext letter, represented as a number from 0 to 25 (where A=0, B=1, …, Z=25).
What does k represent in the Caesar cipher?	k is the key, which is the number of positions we shift the letters by, and can be any whole number from 0 to 25.
What is the mathematical formula for Caesar cipher decryption?	D(y) = (y - k) \mod 26
How do you decrypt a message encrypted with a Caesar cipher?	Subtract the key from the encrypted character to find the plaintext letter.
Why is the Caesar cipher considered insecure?	Because there are only 26 possible keys, so an attacker can easily try all of them, and it doesn’t change the frequency of letters.
How can frequency analysis be used to crack a Caesar cipher?	By analyzing the frequency of letters in the ciphertext, you can guess the key by looking at the most common letters (e.g., if ‘G’ is most common, the key is likely 2 since ‘E’ is most common in English).
Encrypt “HELLO” with a Caesar cipher using key 3.	KHOOR
Show the encryption steps for “HELLO” with key 3.	H (7) becomes K (10), E (4) becomes H (7), L (11) becomes O (14), L (11) becomes O (14), O (14) becomes R (17), resulting in “KHOOR”.
Decrypt “KHOOR” with a Caesar cipher using key 3.	HELLO
Show the decryption steps for “KHOOR” with key 3.	K (10) becomes H (7), H (7) becomes E (4), O (14) becomes L (11), O (14) becomes L (11), R (17) becomes O (14), resulting in “HELLO”.