Trigonometry in Galois Fields - Polar Form

Polar Form

Let G_r and G_θ be subgroups of the multiplicative group of the nonzero elements of GI(p), of orders (p − 1)/2 and 2(p + 1), respectively. Then all nonzero elements of GI(p) can be written in the form ζ = α·β, where α ∈ G_r and β ∈ G_θ.

Considering that any element of a cyclic group can be written as an integral power of a group generator, it is possible to set r = α and εθ = β, where ε is a generator of . The powers εθ of this element play the role of ejθ over the complex field. Thus, the polar representation assumes the desired form, where r plays the role of the modulus of ζ. Therefore, it is necessary to define formally the modulus of an element in a finite field. Considering the nonzero elements of GF(p), it is a well-known fact that half of them are quadratic residues of p. The other half, those that do not possess square root, are the quadratic non-residue (in the field of real numbers, the elements are divided into positive and negative numbers, which are, respectively, those that possess and do not possess a square root).

The standard modulus operation (absolute value) in always gives a positive result.

By analogy, the modulus operation in GF(p) is such that it always results in a quadratic residue of p.

The modulus of an element, where p = 4k + 3, is

The modulus of an element of GF(p) is a quadratic residue of p.

The modulus of an element a + jb ∈ GI(p), where p = 4k + 3, is

In the continuum, such expression reduces to the usual norm of a complex number, since both, a2 + b2 and the square root operation, produce only nonnegative numbers.

• The group of modulus of GI(p), denoted by G_r, is the subgroup of order (p − 1)/2 of GI(p).
• The group of phases of GI(p), denoted by G, is the subgroup of order 2(p + 1) of GI(p).

An expression for the phase as a function of a and b can be found by normalising the element (that is, calculating ), and then solving the discrete logarithm problem of /r in the base over GF(p). Thus, the conversion rectangular to polar form is possible.

The similarity with the trigonometry over the field of real numbers is now evident: the modulus belongs to GF(p) (the modulus is a real number) and is a quadratic residue (a positive number), and the exponential component ) has modulus one and belongs to GI(p) (e also has modulus one and belongs to the complex field ).