Thus, we define addition and multiplication modulo n , denoted +_n and *_{n ,} as follows:

[a] _n + _n [b] _n = [a + b] _n (addition modulo n)

[a] _n * _n [b] _n = [a *b] _n (multiplicative modulo n)

Using this definition of addition modulo n , we define the additive group modulo n as ( Z _{n ,} + _n ). The size of the additive group modulo n is | Z _n | = _n . Modular addition over the group ( Z _{6 ,} + ₆ ) is defined as follows:

Closure: If a ∈ Z_n and b ∈ Z_n then from the definition of addition modulo n a +_n b = [a + b]_n ∈ Z _n .

Identity: 0 is the identity element of Z_n

Inverse: Inverse of [a] _n is [-a] _n ≡ [n-a] _n

Using this definition of multiplication modulo n , we define the multiplicative group modulo n as ( Z^*_{n ,} *_n ) where Z^*_n={[ a ] _n ε Z _n | gcd( a , n )=1} . For e.g. when n =15,

Z^*₁₅= {1, 2, 4, 7, 8, 11, 13, 14}. Modular multiplication over the group ( Z^*_{15 ,} * ₁₅ ) is defined as follows:

Identity: [1] _n