moduulilaskennalla

Moduulilaskennalla, or modular arithmetic, is a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value called the modulus. In this system, two integers are considered equivalent if they have the same remainder when divided by the modulus. The notation a ≡ b (mod n) expresses that a and b differ by a multiple of n. Modular arithmetic was first rigorously studied by the Iranian mathematician Omar Khayyam in the 12th century and later by European mathematicians in the 17th and 19th centuries. It is now a fundamental tool in number theory, cryptography, computer science, and coding theory.

A key operation in moduulilaskennalla is the modulo operation, which extracts the remainder. For example, 17 modulo 5 equals 2, so 17 ≡ 2 (mod 5). Addition, subtraction, and multiplication obey the property that the result of the operation modulo n equals the modulo of the operands combined. Exponentiation also reduces nicely: a^b ≡ (a mod n)^b (mod n). Inverse elements exist only for numbers that are coprime with the modulus; in that case, multiplicative inverses can be found using the extended Euclidean algorithm.

Applications include cryptographic protocols such as RSA and Diffie-Hellman, hashing algorithms, pseudo-random number generation, and error detection codes like checksums. Moduulilaskennalla also underlies the arithmetic of finite fields and group theory. The Chinese remainder theorem, Fermat's little theorem, and Euler’s theorem are important results that rely on modular arithmetic to solve congruences and evaluate large powers efficiently.