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Discrete Cosine Transform Over Finite Prime Fields

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Discrete Cosine Transform Over Finite Prime Fields THE DISCRETE COSINE TRANSFORM OVER PRIME FINITE FIELDS M.M. Campello de Souza, H.M. de Oliveira, R.M Campello de Souza, M. M. Vasconcelos Federal University of Pernambuco - UFPE, Digital Signal Processing Group C.P. 7800, 50711-970, Recife - PE, Brazil E-mail: {hmo, marciam, ricardo}@ufpe.br, [email protected] Abstract - This paper examines finite field the FFFT have been found, not only in the fields trigonometry as a tool to construct of digital signal and image processing [3-5], but trigonometric digital transforms. In particular, also in different contexts such as error control by using properties of the k-cosine function coding and cryptography [6-7]. over GF(p), the Finite Field Discrete Cosine The Finite Field Hartley Transform, which is a Transform (FFDCT) is introduced. The digital version of the Discrete Hartley Transform, FFDCT pair in GF(p) is defined, having has been recently introduced [8]. Its applications blocklengths that are divisors of (p+1)/2. A include the design of digital multiplex systems, special case is the Mersenne FFDCT, defined multiple access systems and multilevel spread when p is a Mersenne prime. In this instance spectrum digital sequences [9-12]. blocklengths that are powers of two are Among the discrete transforms, the DCT is possible and radix-2 fast algorithms can be specially attractive for image processing, because used to compute the transform. it minimises the blocking artefact that results when the boundaries between subimages become Key-words: Finite Field Transforms, Mersenne visible. It is known that the information packing primes, DCT. ability of the DCT is superior to that of the DFT and other transforms. Furthermore, the DCT is not 1. Introduction data dependent like the Karhunen-Loève transform, the best of all linear transforms for The manifold applications of discrete transforms energy compaction [13]. The DCT is now a tool in defined over finite or infinite fields are very well the international JPEG standard for image known. The Discrete Fourier transform (DFT) processing, but there is no equivalent transform has, since long, been playing a decisive role in over finite fields. A natural question that arises is Engineering. Another powerful discrete transform whether it is possible to find a DCT representation is the Discrete Cosine Transform (DCT), which over such fields. The first task towards such a new became a standard in the field of image transform is to establish the equivalent of the compression [1]. Such transforms, despite being cosine function over a finite structure. discrete in the variable domain, have coefficients Based on the trigonometry over finite fields with amplitudes belonging to an infinite field. presented in [8], this paper introduces a new They can therefore be understood as some kind of digital transform, the finite field discrete cosine "analog transforms" (something as Pulse transform (FFDCT). The FFDCT is defined for Amplitude Modulation systems). In contrast, signals over GF(p), p≡3 (mod 4) and its spectrum transforms defined over finite fields, are discrete has components over GF(p). in both variable and transform domain. Their coefficients are from a finite alphabet, so they can 2. Background and Preliminaries be understood as "Digital Transforms". They can possibly be attractive in the same extent as digital 2.1 Fundamentals on finite field complex numbers systems are, compared to analog systems. A Fourier analysis can also be applied for analyzing Definition 1. The set of Gaussian integers over GF(p) is the set GI(p) = {a + jb, a, b ∈ GF(p)}, signals over a finite field. 2 A very rich transform related to the DFT is where p is a prime such that j = -1 is a quadratic the Finite Field Fourier Transform (FFFT), nonresidue over GF(p). Only primes of the type p introduced by Pollard in 1971 [2] and applied as a ≡ 3 (mod 4) meet such a requirement [14]. tool to perform discrete convolution using integer arithmetic. Since then several new applications of 1 The extension field GF(p2) is isomorphic to the replicating x[n] and computing its DFT, from “complex” structure GI(p) [15]. From the above which the DCT coefficients can be obtained. definition, GI(p) elements can be represented in Therefore, to construct a length N DCT, a kernel the form a + jb and are referred to as complex of order 2N is required. finite field numbers. 3. The Discrete Cosine Transform in a Finite Definition 2. (unimodular set): The elements ζ = Field (a+jb) ∈ GI(p), such that a2+b2≡1 (mod p) are referred to as unimodular elements. Let f = (fi) be a length N vector over GF(p). To define its DCT using k-cosines, in a similar way to Unimodular elements are used in proposition 3 to the classical DCT, the following lemma is required. construct a GF(p)-valued DCT. Lemma 1 (k-cos lemma): If ζ ∈ GI(p) has 2.2 Finite Field Trigonometry multiplicative order 2N, then This session describes briefly trigonometric N −1, if i = 0 N −1 functions over a finite field, which hold many A = ∑cosk (i) = −1, if i is even (≠ 0) k =1 properties similar to those of the standard real- 0, if i is odd valued trigonometric functions [16]. In what Proof: By definition follows, the symbol := means equal by definition. N −1 N −1 A cos (i) 1 ( ki −ki ) = ∑∑k = 2 ζ +ζ , Definition 3. Let ζ be a nonzero element of GI(p), k =1 k =1 where p ≡ 3 (mod 4). The k-trigonometric i functions cosine and sine of ∠(ζ ) (arc of the so that, clearly, A = N -1 if i = 0. Otherwise, element ζi) over GI(p), are ζi (ζi(N −1) −1) ζ−i (ζ−i(N −1) −1) i −1 ik -ik A = 1 [ + ]. cosk (∠ζ ) := (2 mod p) (ζ + ζ ) 2 ζi −1 ζ−i −1 and i −1 ik -ik sink (∠ζ ) := (2 mod p) (ζ - ζ )/ j Since ζ has order 2N, then ζ N = −1. Multiplying the second term by ( −ζ i ), yields i, k = 0, 1,..., N-1, where ζ has order N. (−1)i − ζi ) 1 − (−1)i ζi For the sake of simplicity, these are denoted by A = 1 [ + ]. 2 ζi −1 ζ−i −1 cosk(i) and sink(i). These definitions make sense only if GI(p) is a field. This is the reason for 1− ζi +1− ζi requiring p ≡ 3 (mod 4). Therefore, for i even, A = 1 [ ] = −1 2 ζi −1 Over the field of real numbers, the DCT is and, for i odd, defined by the pair −1− ζi +1+ ζi A = 1 [ ] = 0. 2 ζi −1 N −1 (2n +1)kπ C[k] := ∑ x[n]cos , n=0 2N From this k-cos lemma, it is possible to define a new digital transform, the finite field discrete cosine transform (FFDCT). N −1 (2n + 1)kπ x[n] = ∑β[k]C[k]cos 2N k=0 Definition 4: If ζ ∈ GI(p) has multiplicative order 2N, then the finite field discrete cosine transform of where β[k] is the weighting function the sequence f = ( f i ), i = 0,1,... N -1, fi ∈ GF( p) , is the sequence C = (Ck ), k = 0,1,... N -1, Ck ∈ GI( p) , 1 , if k = 0 β[k] = 2 . of elements 1, if k = 1 N −1 2i+1 . The steps leading to the expression for C[k], the Ck := ∑2 fi cosk ( 2 ) DCT of the length N time sequence x[n], involve i=0 2 The inverse FFDCT is given by theorem 1 i) If r+i+1 = 0, then r = -i –1, which implies fr = 0. below. Therefore, in this case, gi = 0. Teorema 1 (The inversion formula): The inverse ii) If r-i = 0, then r = i. In this case finite field discrete cosine transform of the sequence 2 1 1 1 gi = N fi [ 2 + 2 (0) + 2 (N −1)] = fi . C = (Ck ), k = 0,1,.. N -1, Ck ∈ GI( p) , is the sequence f = ( fi ), i = 0,1,.. N -1, f i ∈ GF( p) , of elements iii) If both, r+i+1 and r-i are different from zero, considering the parity for these terms it is possible N −1 to write 1 2i+1 , fi = N ∑βk Ck cosk ( 2 ) k=0 N −1 2 1 1 1 , gi = N ∑ f r [ 2 + 2 (0) + 2 (−1)] = 0 r=0 where the weighting function βk is given by so that −1 gi = fi , i = 0,1,... N -1. (2 mod p), if k = 0 . βk = 1, if k ≠ 0 To conclude this session, the elements of an FFDCT of length 8 over GF(31) are presented in Proof: To establish the inversion formula, it is example 1. sufficient to show that gi = fi , i = 0,1,... N -1, where Example 1: For p = 31, the element ζ = (7+j13) ∈ GL(31) has order (p+1)/2 = 16. The FFDCT of N −1 1 2i+1 . length (p+1)/4 = 8 of the sequence f = (1, 2, 3, 4, 5, gi := N ∑βk Ck cosk ( 2 ) k=0 6, 7, 8) is the sequence C = (10, 20, 0, 17, 0, 12, 0, 5). The transform matrix {2cos ( 2i+1)}, i, k = 0,1,..,7, k 2 From definition 4, one may write is N −1 N −1 2 2 2 2 2 2 2 2 g = 1 β [ 2 f cos ( 2r+1)]cos ( 2i+1) , i N ∑∑k r k 2 k 2 27 10 20 22 9 11 21 4 k=0 r=0 14 5 26 17 17 26 5 14 which is the same as 10 9 4 11 20 27 22 21 .
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