**Modular arithmetic** is the central mathematical concept and fundamental in cryptography. Almost any cipher from the **Caesar Cipher** (Julius Caesar) to the **RSA Cipher** use modular arithmetic. Initially, starting with **Caesar cipher** (also known as a **shift cipher**) is one of the simplest forms of encryption. Caesar cipher is a **substitution cipher** that replaces each plaintext letter by another one.

**Note:** The replacement rule is very simple, where it takes each letter that follows the **key (k)** positions in the alphabet. The table below shows the mapping letters with numbers.

The table above represents the letters in plaintext. In order to send a message using these letters in a ciphertext form, the sender and receiver should generate a secret key (k). For example, if ** k = 8**, the table would be updated to the following:

Plaintext |
ATTACK | 0, 19, 19, 0, 2, 10. |

Ciphertext |
IBBIKS | 8, 1, 1, 8, 10, 18. |

So, the table below is a comparison between the plaintext table and the cipher text table:

Another simple way to look at the **Caesar cipher** (shift cipher) would be:

Caesar cipher is an elegant mathematical description of the cipher, since *k, x, y ε {0, 1, 2, 3, …, 23, 24, 25}.*

- Encryption:
*y = e*_{k}(x) ≡ x + k mod 26. - Decryption:
*x = d*_{k}(x) ≡ y – k mod 26.

The question we keep asking is – – “Is the shift cipher secure?”; The answer is **No**, because there are several attacks occurred on this cipher, including:

- The limitation of the key (Exhaustive key search) – key space is only 26.
- The letter frequency analysis is similar to attack against substitution cipher.

Eventually, Caesar’s cipher is not that nowadays, but **asymmetric cryptography** (RSA and elliptic curve crypto) uses modular arithmetic extensively.