Euler's theorem is a fundamental statement in number theory that establishes a relationship between the totient function and modular exponentiation. The theorem states that for any integer \( a \) and any integer \( n \) that is coprime to \( a \), the following congruence holds:
\( a^{\phi(n)} \equiv 1 \ (\text{mod} \ n) \)
where \( \phi(n) \) is Euler's totient function, which counts the number of integers up to \( n \) that are coprime to \( n \). This theorem has important applications in the field of cryptography, particularly in the RSA encryption algorithm.