Trigonometry in Galois Fields - Trigonometry Over A Galois Field

Trigonometry Over A Galois Field

The set GI(q) of Gaussian integers over the finite field GF(q) plays an important role in the trigonometry over finite fields. If q = pr is a prime power such that −1 is a quadratic non-residue in GF(q), then GI(q) is defined as

GI(q) = {a + jb; a, b ∈ GF(q)},

where j is a symbolic square root of −1 (that is j is defined by j2 = −1). Thus GI(q) is a field isomorphic to GF(q2).

Trigonometric functions over the elements of a Galois field can be defined as follows:

Let be an element of multiplicative order N in GI(q), q = pr, p an odd prime such that p 3 (mod 4). The GI(q)-valued k-trigonometric functions of in GI(q) (by analogy, the trigonometric functions of k times the "angle" of the "complex exponential" i) are defined as

for i, k = 0, 1,...,N − 1. We write cos_k(i) and sin_k (i) as cos_k(i) and sin_k(i), respectively. The trigonometric functions above introduced satisfy properties P1-P12 below, in GI(p).

• P1. Unit circle:

• P2. Even/Odd:

• P3. Euler formula:

• P4. Addition of arcs:

• P5. Double arc:

• P6. Symmetry:

• P7. Complementary symmetry: with

• P8. Periodicity:

• P9. Complement: with

• P10. summation:

• P11. summation:

• P12. Orthogonality: