Trigonometry in Galois Fields - Trajectories Over The Galois Z Plane in GF(p)

Trajectories Over The Galois Z Plane in GF(p)

When calculating the order of a given element, the intermediate results generate a trajectory on the Galois Z plane, called the order trajectory. In particular, If has order N, the trajectory goes through N distinct points on the Z plane, moving in a pattern that depends on N. Specifically, the order trajectory touches on every circle of the Galois Z plane (there are ||G_r|| of them), in order of increasing modulus, always returning to the unit circle. If it starts on a given radius, say R, it will visit, counter-clockwise, every radius of the form R+k.r, where r=(p2−1)/N and k = 0, 1, 2, ....., N − 1. Given a prime p 3 (mod 4), there are a (finite) number of (p − 1)/2 distinct circles over the Galois Z plane GI(p), and the number of distinct finite field ellipses is (p − 1).(p − 3)/4.

• Table V lists some elements ζ ∈ GI(7) and their orders N. Figures 3–5 show the order trajectories generated by ζ.

• Figure 3. Order trajectory for ζ = j2, an element of order N = 12 of GI(7), on the Galois Z-plane over GF(7).

• Figure 4. Order trajectory for ζ = 3 + j3, an element of order N = 24 of GI(7), on the Galois Z-plane over GF(7).

• |Figure 5. Order trajectory for ζ = 6 + j4, an element of order N = 48 of GI(7), on the Galois Z Plane over GF(7).