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, p and
q, and then compute the
product n = pq.
The user’s public key would then be the pair
(n,a), where aa is any integer not divisible by
p or
q. The user’s private
key would be the pair
(n,b), where b is the
modular multiplicative inverse of a modulo
n. This means that when
we multiply a and
b together, the result
is congruent to 1 modulo
n.
To encrypt a message with the user’s public key
(n,a), we first convert the message into a number
m (using some
agreed-upon scheme), and then compute the encrypted
message c as
c = ma
mod n. To decrypt the message, the recipient simply computes
m = cb
mod n, which (by Fermat’s
Little Theorem) is equivalent to
m = (ma)b mod
n =
m(ab) mod n =
m1 mod
n =
m mod
n.
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.