Trigonometry in Galois fields

In mathematics, trigonometry analogies are supported by the theory of quadratic extensions of finite fields, also known as Galois fields. The main motivation to deal with a finite field trigonometry is the power of the discrete transforms, which play an important role in engineering and mathematics. Significant examples are the well-known discrete trigonometric transforms (DTT), namely the discrete cosine transform and discrete sine transform, which have found many applications in the fields of digital signal and image processing. In the real DTTs, inevitably, rounding is necessary, because the elements of its transformation matrices are derived from the calculation of sines and cosines. This is the main motivation to define the cosine transform over prime finite fields. In this case, all the calculation is done using integer arithmetic.

In order to construct a finite field transform that holds some resemblance with a DTT or with a discrete transform that uses trigonometric functions as its kernel, like the discrete Hartley transform, it is firstly necessary to establish the equivalent of the cosine and sine functions over a finite structure.

Trigonometry over a Galois field

The set GI(q) of gaussian integers over GF(q) plays an important role in the trigonometry over finite fields (hereafter the symbol := denotes equal by definition).

GI(q) := q = p^r,

r being a positive integer, p being an odd prime for which...
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