Trigonometry in Galois fields

In mathematics, the theory of quadratic extensions of finite fields supports analogies with trigonometry.

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

: G("q") := {"a" + "jb", "a", "b" ∈ GF("q")} "q" = "p"^"r",

"r" being a positive integer, "p" being an odd prime for which "j"² = −1 is a quadratic non-residue in GF("q"); it is a field isomorphic to GF("q"²).

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

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

:

:

for "i, k" = 0, 1,...,"N" − 1. We write cos_k($anglezeta$^"i") and sin_k ($anglezeta$^"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: $sin\_k^2(i)+cos\_k^2(i)equiv\; 1$.

P2. Even/Odd: i) $cos\_k(i)equiv\; cos\_k(-i)$. ii) $sin\_k(i)equiv\; -sin\_k(-i).$

P3. Euler formula: $zeta^\{ki\}\; equiv\; cos\_k(i)+jsin\_k(i).$

P4. Addition of arcs:

i) $cos\_k(i+t)equiv\; cos\_k(i)cos\_k(t)-sin\_k(i)sin\_k(t),$

ii) $sin\_k(i+t)equiv\; sin\_k(i)cos\_k(t)+sin\_k(t)cos\_k(i).$

P5. Double arc:

i)

ii)

P6. Symmetry:

i) $cos\_k(i)equiv\; cos\_i(k),$

ii)$sin\_k(i)equiv\; sin\_i(k).$

P7. Complementary symmetry: with $(k+t)=(i+r)=\; N$,

i) $cos\_k(i)\; equiv\; cos\_r(t),$

ii) $sin\_k(i)\; equiv\; sin\_r(t).$

P8. Periodicity:

i)$cos\_k(i+N)\; equiv\; cos\_k(i),$

ii)$sin\_k(i+N)\; equiv\; sin\_k(i).$

P9. Complement: with $(i+t)=\; N,$

i)$cos\_k(i)\; equiv\; cos\_k(t),$

ii) $sin\_k(i)equiv\; -sin\_k(t).$

P10. $cos\_k(i)$ summation: $sum\_\{k=0\}^\{N-1\}\; cos\_k(i)equiv.$

P11. $sin\_k(i)$ summation: $sum\_\{k=0\}^\{N-1\}\; sin\_k(i)equiv\; 0.$

P12. Orthogonality:$sum\_\{k=0\}^\{N-1\}\; [cos\_k(i)sin\_k(t)]\; equiv\; 0.$

Examples

i) With ζ = 3, a primitive element of GF(7), then cos_"k"("i") and sin_"k"("i") functions take the following values in GI(7):

:::"Table I - Finite field cosine and sine functions over GI(7)."

Figure 1 illustrates the 12 roots of unity in GF(11^{2). Clearly, G₁ is isomorphic to C12, the group of proper rotations of a regular dodecagon. $zeta$=8+j6 is a group generator corresponding to a counter-clockwise rotation of 2π/12 = 30o. Symbols of the same colour indicate elements of same order, which occur in conjugate pairs.}

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 ε^θ = β, where ε is a generator of $G\_\; heta$. The powers ε^θ of this element play the role of "e"^"j"θ over the complex field. Thus, the polar representation assumes the desired form, $zeta=r\; cdotepsilon^\; heta$, 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 factFact|date=August 2008 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 $mathbb\{R\}$ 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 $ain\; GF(p)$, where "p" = 4"k" + 3, is

:

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

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

:$mid\; a+jbmid\; =left\; |\; sqrt\{mid\; a^2+b^2\; mid\}\; ight\; |.$

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_$heta$, is the subgroup of order 2("p" + 1) of GI("p").

An expression for the phase $heta$ as a function of a and b can be found by normalising the element $zeta$ (that is, calculating $zeta\; /r\; =\; epsilon^\; heta$), and then solving the discrete logarithm problem of $zeta$/"r" in the base $epsilon$ 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 $epsilon$^$heta$) has modulus one and belongs to GI("p") (e^{$j\; heta$} also has modulus one and belongs to the complex field $mathbb\{C\}$).

The "Z" plane in a Galois field

The complex "Z" plane (Argand diagram) in GF("p") can be constructed from the supra-unimodular set of GI("p"):

* The supra-unimodular set of GI("p"), denoted G_s, is the set of elements ζ = ("a" + "jb") ∈ GI("p"), such that ("a"² + "b"²) $equiv$−1 (mod "p").
* The structure s,*>, is a cyclic group of order 2("p" + 1), called the supra-unimodular group of GI("p").

The elements ζ = "a" + "jb" of the supra-unimodular group "G"_"s" satisfy ("a"² + "b"²)²$equiv$1 (mod "p") and all have modulus 1. "G"_"s" is precisely the group of phases $G\_\; heta$.

* If "p" is a Mersenne prime ("p" = 2^"n" − 1, "n" > 2), the elements ζ = "a" + "jb" such that "a"² + "b"² $equiv$−1 (mod "p") are the generators of G_s.

Examples

* Let "p" = 31, a Mersenne prime, and ζ = 6 + "j"16. Then $r=|sqrt\{(|6^2+16^2|)\}|equiv|sqrt$

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 $zeta$ 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"=("p"²-1)/"N" and "k"=0,1,2,.....,"N" − 1. Given a prime "p"$equiv$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.

:: "Table V - Some elements and their orders in GI(7)."

References

* [1] R. M. Campello de Souza, H. M. de Oliveira and A. N. Kauffman, "Trigonometry in Finite Fields and a New Hartley Transform," "Proceedings of the 1998 International Symposium on Information Theory", p. 293, Cambridge, MA, Aug. 1998.
* [2] M. M. Campello de Souza, H. M. de Oliveira, R. M. Campello de Souza and M. M. Vasconcelos, "The Discrete Cosine Transform over Prime Finite Fields," "Lecture Notes in Computer Science", LNCS 3124, pp. 482–487, Springer Verlag, 2004.
* [3] R. M. Campello de Souza, H. M. de Oliveira and D. Silva, "The Z Transform over Finite Fields," "International Telecommunications Symposium", Natal, Brazil, 2002.
* [4] http://www2.ee.ufpe.br/codec/FFtools.htm