Fermat’s Little Theorem is used in cryptography in several ways. One of the most common applications is in the generation of so-called “public-key” cryptography systems, which are used to securely transmit messages over the internet and other networks.

In a public-key cryptography system, each user has a pair of keys: a public key, which is widely known and can be used by anyone to encrypt a message intended for that user, and a private key, which is known only to the user and is used to decrypt messages that have been encrypted with the corresponding public key.

One way to generate these keys is to use prime numbers and Fermat’s Little Theorem. For example, suppose we want to generate a public-key cryptography system for a user with the initials “ABC”. We might choose two large prime numbers, ppp and qqq, and then compute the product n=pqn = pqn=pq.

The user’s public key would then be the pair (n,a)(n, a)(n,a), where aa is any integer not divisible by ppp or qqq. The user’s private key would be the pair (n,b)(n, b)(n,b), where bbb is the modular multiplicative inverse of a modulo nnn. This means that when we multiply aaa and bbb together, the result is congruent to 111 modulo nnn.

To encrypt a message with the user’s public key (n,a)(n, a)(n,a), we first convert the message into a number mmm (using some agreed-upon scheme), and then compute the encrypted message ccc as c=ma mod nc = m^a \bmod nc=mamodn. To decrypt the message, the recipient simply computes m=cb mod nm = c^b \bmod nm=cbmodn, which (by Fermat’s Little Theorem) is equivalent to m=(ma)b mod n=m(ab) mod n=m1 mod n=m mod nm = (m^a)^b \bmod n = m^(ab) \bmod n = m^1 \bmod n = m \bmod nm=(ma)bmodn=m(ab)modn=m1modn=mmodn.

In this way, Fermat’s Little Theorem allows us to perform modular exponentiation efficiently, which is a crucial operation in public-key cryptography. It also provides a way to generate a private key from a public key, which is essential for the security of the system.