ALGEBRA HANDOUT 2: IDEALS AND
... Z[i]/(n) is isomorphic to Z/(n) × Z/(n). However, there is also the matter of the multiplicative structure to consider: is it perhaps Z/(2) × Z/(2) also as a ring? The answer is no. Indeed, notice that (1+i)2 = (1+i)(1+i) = 1+2i+i2 = 2i ≡ 0 mod 2Z[i]), so that the representative r = 1 + i in the quo ...
... Z[i]/(n) is isomorphic to Z/(n) × Z/(n). However, there is also the matter of the multiplicative structure to consider: is it perhaps Z/(2) × Z/(2) also as a ring? The answer is no. Indeed, notice that (1+i)2 = (1+i)(1+i) = 1+2i+i2 = 2i ≡ 0 mod 2Z[i]), so that the representative r = 1 + i in the quo ...
Galois Field in Cryptography
... The elements of Galois Field gf (pn ) is defined as gf (pn ) = (0, 1, 2, . . . , p − 1) ∪ (p, p + 1, p + 2, . . . , p + p − 1) ∪ (p2 , p2 + 1, p2 + 2, . . . , p2 + p − 1) ∪ . . . ∪ (pn−1 , pn−1 + 1, pn−1 + 2, . . . , pn−1 + p − 1) where p ∈ P and n ∈ Z+ . The order of the field is given by pn while ...
... The elements of Galois Field gf (pn ) is defined as gf (pn ) = (0, 1, 2, . . . , p − 1) ∪ (p, p + 1, p + 2, . . . , p + p − 1) ∪ (p2 , p2 + 1, p2 + 2, . . . , p2 + p − 1) ∪ . . . ∪ (pn−1 , pn−1 + 1, pn−1 + 2, . . . , pn−1 + p − 1) where p ∈ P and n ∈ Z+ . The order of the field is given by pn while ...
