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Modding (from "modifying") is the act of modifying hardware, software program, or the rest to perform a function not originally meant by the designer, or to realize bespoke specification or appearance. Given the Euler's totient perform φ(m), any set of φ(m) integers which might be relatively prime to m and mutually incongruent beneath modulus m is named a reduced residue system modulo m.

They form an abelian group underneath multiplication; its order is φ(m), where φ is Euler's totient perform.

If c ≡ d (mod φ(m)), where φ is Euler's totient function, then ac ≡ ad (mod m)-offered that a is coprime with m. If ok a ≡ ok b (mod m) and okay is coprime with m, then a ≡ b (mod m). If ax ≡ b (mod m) and a is coprime to m, then the answer to this linear congruence is given by x ≡ a−1b (mod m). Congruence modulo m is a congruence relation, which means that it's an equivalence relation that's compatible with addition, subtraction, and multiplication.

The congruence relation is an equivalence relation. Some operations, like discovering a discrete logarithm or a quadratic congruence seem like as arduous as integer factorization and thus are a place to begin for cryptographic algorithms and encryption.