Y.S.Han Finite fields Example of Groups .(Z,+).e=0 and the inverse of i is -i. .(Q-{0),).e=1 and the inverse of a/b is b/a. ·({0,l},⊕)，where⊕is exclusive-OR operation. The order of a group is the number of elements in the group. ·Additive group:({0,l,2,.,m-1},田)，where m∈Z+,and i田j=i+j mod m. -(i田)田k=i田(田k). -e=0. 一} Yo<i<m,m-i is the inverse of i. -i田j=j田i. Multiplicative group:({1,2,3,...,p-1),),where p is a prime and iij mod p. School of Electrical Engineering Intelligentization,Dongguan University of TechnologyY. S. Han Finite fields 4 Example of Groups • (Z, +). e = 0 and the inverse of i is −i. • (Q − {0}, ·). e = 1 and the inverse of a/b is b/a. • ({0, 1}, ⊕), where ⊕ is exclusive-OR operation. • The order of a group is the number of elements in the group. • Additive group: ({0, 1, 2, . . . , m − 1}, ⊞), where m ∈ Z +, and i ⊞ j ≡ i + j mod m. – (i ⊞ j) ⊞ k = i ⊞ (j ⊞ k). – e = 0. – ∀0 < i < m, m − i is the inverse of i. – i ⊞ j = j ⊞ i. • Multiplicative group: ({1, 2, 3, . . . , p − 1}, ⊡), where p is a prime and i ⊡ j ≡ i · j mod p. School of Electrical Engineering & Intelligentization, Dongguan University of Technology