The Hartley Transform in a Finite Field

The k-trigonometric functions over the Galois Field GF(q) are introduced and their main properties derived. This leads to the definition of the cask(.) function over GF(q), which in turn leads to a finite field Hartley Transform. The main properties of this new discrete transform are presented and areas for possible applications are mentioned.


Introduction
Discrete transforms play a very important role in engineering. A significant example is the well known Discrete Fourier Transform (DFT), which has found many applications in several areas, specially in Electrical Engineering. A DFT for finite fields was introduced by Pollard in 1971 [1] and applied as a tool to perform discrete convolutions using integer arithmetic. Since then several new applications of the Finite Field Fourier Transform (FFFT) have been found, not only in the fields of digital signal and image processing [2][3][4][5], but also in different contexts such as error control coding and cryptography [6][7][8].
A second relevant example concerns the Discrete Hartley Transform (DHT) [9], the discrete version of the integral transform introduced by R. V. L. Hartley in [10]. Although seen initially as a tool with applications only on the numerical side and having connections to the physical world only via the Fourier transform, the DHT has proven over the years to be a very useful instrument with many interesting applications [11][12][13].
In this paper the DHT over a finite field is introduced. In order to obtain a transform that holds some resemblance with the DHT, it is necessary firstly to establish the equivalent of the sinusoidal functions cos and sin over a finite structure. Thus, in the next section, the k-trigonometric functions cos k and sin k are defined from which the cas k (cosine and sine) function is obtained and used, in section 3, to introduce a symmetrical discrete transform pair, the finite field Hartley transform, or FFHT for short. A number of properties of the FFHT is presented, including the cyclic convolution property and Parseval's relation. In section 4, the condition for valid spectra, similar to the conjugacy constraints for the Finite Field Fourier Transform, is given. Section 5 contains a few concluding remarks and some possible areas of applications for the ideas introduced in the paper. The FFHT presented here is different from an earlier proposed Hartley Transform in finite fields [14] and appears to be the more natural one.

k-Trigonometric Functions
The set G(q) of gaussian integers over GF(q) defined below plays an important role in the ideas introduced in this paper (hereafter the symbol := denotes equal by definition).
Definition 1: 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), is the set of gaussian integers over GF(q).
Trigonometric functions over the elements of a Galois field can be defined as follows.
For simplicity suppose α to be fixed. We write cos k (∠α i ) as cos k ( i ) and sin k (∠α i ) as sin k ( i ). The ktrigonometric functions satisfy properties P1-P8 below. Proofs are straightforward and are omitted here.
P5. Double Arc: . Symmetry : P7. cos k ( i ) Summation: P8. sin k ( i ) Summation: A simple example is given to illustrate the behavior of such functions.
Example 1 -Let α = 3, a primitive element of GF (7). The cos k (i) and sin k (i) functions take the following values in GF (7): Proof: By definition 2, and, from P9, the result follows. where Gand Vdenotes, respectively, the sequences {G N-k } and {V N-k }.

Valid Spectra
The following lemma states a relation that must be satisfied by the components of the spectrum V for it to be a valid finite field Hartley spectrum, that is, a spectrum of a signal v with GF(q)-valued components. The relation for valid spectra shown above implies that only two components V k are necessary to completely specify the vector V, namely V 0 and V 1 . This can be verified simply by calculating the cyclotomic classes induced by lemma 1 which, in this case, are C 0 = (0) and C 1 = (1, 8, 9, 6, 4, 10, 3, 2, 5, 7).

Conclusions
In this paper, trigonometry for finite fields was introduced. In particular, the k-trigonometric functions of the angle of the complex exponential α i were defined and their basic properties derived. From the cos k (∠α i ) and sin k (∠α i ) functions, the cas k (∠α i ) function was defined and used to introduce a new Hartley Transform, the Finite Field Hartley Transform (FFHT). The FFHT seems to have interesting applications in a number of areas. Specifically, its use in Digital Signal Processing, along the lines of the so-called number theoretic transforms (e.g. Mersenne transforms) should be investigated. In the field of error control codes, the FFHT might be used to produce a transform domain description of the field, therefore providing, possibly, an alternative to the approach introduced in [6]. Digital Multiplexing is another area that might benefit from the new Hartley Transform introduced in this paper. In particular, new schemes of efficient-bandwidth code-division-multiple-access for band-limited channels based on the FFHT are currently under development.