A NEW CLASS OF SPREADING SEQUENCES A.Z.Tirke1, T.E.Hall*, C.F.0sborne' Scientific Technology, 8 Cecil St, E.Brighton 3 187, Australia. *Department of Mathematics, Monash University, Clayton 3 168. Australia. 'Department of Physics, Monash University, Clayton 3 168. Australia. phone/fax: +61-3-95922206, emaitAn&w. [email protected]

ABSTRACT are produced analogously. This shift between columns is relative and the last shiftin the sequence is the cyclic wraparound. An example of this is shown in Fig.1.

This paper is concerned with the construction and analysis of a new class of spreading sequences. A known pseudonoise sequence and a selection of its cyclic shills is folded into a new type of array and unfolded along its diagonal. The resulting sequence is equivalent to a decimation of a shuftled ensemble of cyclic shifts of the original pseudonoise sequence. The new sequences form a family comparable in size and figure of merit to that of Gold Codes, but available in many different (composite) lengths. They possess enhanced immuniq against cryptographic attack., owing to their large hear and nonlinear span. These sequences are well suited to CDMA, especially for microcell applications. INTRODUCTION The concept of folding of m-sequences into arrays has been studied by MacWUiams and Sloane [SI. The reverse process of unfolding along the diagonals has been examined by Like et al [SI, [6]. In this paper, we present a new method of folding sequences into arrays, which generates high figwre of merit arrays from columns which have good autocorrelation properties. These arrays can be unfolded along their diagonals, resulting in sets of sequences with similar auto and cross-correlation properties to Gold Codes, but with many choices of sequence length. The compound construction process results in a large linear and non-linear span, enhancing immunity to cryptographic attack. The arrays described here are based on binary or ternary alphabets, but the principles are not restricted to these alphabets. The authors have applied the same theory to fold and unfold 3 and higher dimensional arrays [9]. The method of folding presented here was originally developed by the authors to generate pxp two dimensional and pxqxpq three dimensional arrays suitable for embedding as watermarks on digital images. It was modified to construct non-square arrays, which can be unfolded into new sequences.

ARRAY CONSTRUCTIQN The following example qustrates the construction method. The array is genebted by placing a suitable sequence in the left column, (the "seed" column). The adjoining right cdumn to the seed is derived from an appropriate cyclic shift of the seed Subsequent columns

0-7803-4281-X/97/$10.00 01998 IEEE

m=4 Fig.1. pxp dsp Array Construction

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The construction method is based on the concept of distinct sums property (dsp): i.e. all accumulated cyclic shifis between columns separated by distance k are different. For a seed sequence of prime length p s it is possible to deduce such sequences of shifts of length p analytically, using finite field themy [lo]. dsp guarantees that an array can only have 0 or 1 columns agreeing with a displaced version of itself. Low crosscorrelationbetween two arrays is ia consequence of 0,1,or 2 columns from p columns matching, which is also obtainable analytically. The above arrays are almost balanced. Typical auto and crosscorrelations are shown in Fig.2. below.

compared with the pxp arrays. Examples of some modificationsare presented in the next section.

ARRAY MODIFICATION The authors have investigated several methods of m-ng the pxp arrays, with relatively small degradation in correlation. These include the obvious cases of puncturing and extension of the number of columns from the pxp construction. The method of convolving the pxp arrays with suitable non-square arrays, as described in [l] and [Ialso yields new, nonsquare arrays with predictable correlation. We call these compound arrays. Finally, the authors have developed a method of construction of qxp arrays with dsp, where q the seed column length is greater than p , but not necessarily prime. The following examples illustrate the method of construction and correlation results for each of these modifications. Unfolded sequences are also shown. px@-1) “Punctured Arrays”. The p q arrays with dsp can be punctured by removing the column with 0 relative sh&, which preserves the wraparound condition. An example of such 7x6 arrays, their auto and crosscorrelation is shown in Fig.3. below. These arrays have single diagonals and hence lead to sequences of length 42. Further puncturing is possible, but has not been investigated, since it causes s i w c a n t increases to the correlation sidelobes.

(a) Autocorrelation of a pxp array

(b) Crosscorrelationof arrays m:=l with m=2

Fig.2. Auto and Cross Correlation of 5x5 Distinct Sums Arrays Any pseudonoise sequences [2], [3] ane suitable column seeds. Their two-valued autocorrelation is useful in our construction, if the out of phase value is small, as the arrays then have three-valued auto and cross-correlation, with all out of phase values being small.

ARRAY UNFOLDING A two-dimensional array of size U x v with gcd(u,v) = 1 can be unfolded into sequences in four ways, by following diagonal entries as described in [8] : SI={aJ, SZ={a,.$ and S3={a,,-L},s4={a-,,) All these sequences are decimations of each other, as shown in the Appendix. When arraiys of equal size, with known correlation are unfolded using the same method, the correlation of unfolded sequences is identical to that of the parent arrays. The p q arrays above are unsuitable for diagonal unfolding, since they have p clisjoint diagonals kcd(p,p)-p]. However, there are a number of modifications which can transform the above to arrays of size U x v ,with gcd(u,v) = 1, without insistence on U or v king prime. Such modifications result in some predictable degradation to the correlation results when

m=2 (a) Two ‘‘punctured’’ 7x6 arrays

(b) Autocorrelation

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(b) Autocorrelation

(e) Crosscorrelation

0+++0+-0-+000+-+++-+++-~-0-----

++-++---+o-+-

Fig.3. Punctured Arrays

(c) Unfolded Ternary Sequence px@+l) “Extended DSP Arrays”. A pxp array with

dsp can be extended by appending a column of zeros to form apx(p+l) array, An example of a 7x8 array and its autocorrelation is shown in Fig.4. below. This leads to sequences of length 56.

Fig.5 5x9 Extended Array

Compound Arrays. Arrays with dsp can be convolved with other binary or ternary arrays 111, [I with desirable properties, to generate new arrays of Merent sizes or dimensions, with predictable auto and crosscorrelation properties. For example, Fig.6. shows the Legendre sequence of Fig. l., multiplied using the Kroneker Product by a perfect aperiodic 2x2 array. This results in the 10x2 array, with a desirable autocorrelation shown below. A similar process can be applied to pxq arrays with dsp to produce new sequences along their diagonals.

(a) Extended 7x8 Array

(b) Autocorrelation 01111101010111100011001100100011Q1101101000100 0001001110 (e) Unfolded Binary Sequence

Fig.4. Extendedp+I)

Array

Fig.6. Compound Array

The above process of extension can be repeated, with various column r e p t patterns. The resulting sequence of shifts does not possess dsp and the autocorrelation depends on the number of columns which can match simultaneously. In general, this is a difFicult problem to solve analytically. However, px(2p-1) arrays have small sidelobes, nearly symmetric about 0 (the next best thing to dsp). An example of such an array and its unfolded Fig.5 . below. ternary sequenc

q x p DSP Constructions, Sequences of shifts of length p a satisfyrng dsp can be found analytically 191 or by computer search. For p prime, a family of arrays result. A balanced pseudonoise seed sequence of length p has autocorrelation values p or -1. The autocorrelation ,8 of an array constructed from this seed assumes values of

e,=*h(-mP-4+1)

a sequence of shifts for p=3 is shown in Fig.7. below.

un

(a)

(1)

An example of a seed of an m-sequence of length 7, with

5x9px(Zp-l) Array

(a)7x3 Array

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CONCLUSIONS

This paper presents a new class of sequences by appending cyclic shifts of a pseudonoise sequence to form columns of an array and unfolding the array along one of its diagonals. Where the series of cyclic shifts possesses the distinct sums property, the array and the unfolded sequences possess a high figure of merit for auto and crosmmlation, which is only slightly inferior to that for pseudonoise sequences and similar to that of Gold Codes of comparable length. The new sequences are available for composite lengths, for which pseudonoise sequences do not exist or are not known to exist.

(b) Autocorrelation 001110110111000001111

(c) Unfolded Seqiienee ETg.7 q xp DSP Amay

REFERENCES NEW SEQUENCE PROPERTIES

1. M.F.MAntweiler, L.Bomer, H-D.Liike. “Perfect Ternary Arrays“. IEEE Transactions on Information Th€Wy, V01.36, NO.3, May 1990, p.696-705. 2. Everett. “Periodic Digital Sequences With Pseudonoise Properties”. GEC Journal, 1966, Vol. 33, NO.3, p. 115-126. 3. G.Gong, P.Gaal, S.W.Golomb. “A Suspected New Infinite Class of (2n-1,2n-’-1,2n-2-1) Cyclic Difference Sets”. EEE IT Workshop, Svalbard, July 6-10, 1997, p.13-14. 4. D.H.Green Structural Properties of Pseudorandom Arrays and Volumes and Their Related Sequences, IEE Proceedings, vol 132, R.E, No.3, May 1985, p.133-145. 5. H-D.Liike. BMre Folgen und Arrays mit optimalen ungeraden Autokorrelatiomfwktionen. Frequenz 48, (1994) p.213-220. 6. H.D.Liike. Sequences and Arrays With Perfect Periodic Correlation. IEEE Trans on Aerospace and Electronic Systems, vol24, No 3., May 1988, p.287294. 7. H-D.Liike, L.Bijmer, M.Antweiler. Perfect Binary Arrays. Signal Processing 17 (1989) Elsevier Science Publishing, p.69-80. 8. F.J.MacWilliams, N.J.A.Sloane. Pseudo-Random Sequences and Arrays. Proceedings of the IEEE, vol 64, N0.12, Dec.1976, p.1715-1729. 9. A.Z.Tirke1, T.E.Hall, C.F.@borne. “SteganographyApplications of Coding Theory”. IEEE Information Theory Workshop, Svdbard, Norway, July 6-10, 1997. p.57-59. 10.T.E.H&l, C.F.Osborne, AZTirkel. “Some Binary Arrays and their Auto and Cross-Correlation”. In submission.

Figure of Merit In terms of their auto and crosscorrelation, the new class of sequences described here approximate the performance of Gold Codes. For large p, the px@+l) array results in a sequence of length I Ep’, whilst the auto and crosscorrelation sidelobes are E kp = 4’. The Figure of Merit is thus 4. Sequences derived fkom the px@-l) arrays an:slightly inferior. For large p and q, the general qxp arralys (pp)offer a Figure of Merit of orderp. Number of sequences. The number of Werent sequences of length I generated by a single seed column sequence is approximately p = d. A choice of another pseudonoise seed column sequence results in another family of identical autoconelation sequences of length I. However, the crosscorrelation between the Merent families of sequences has no speciad properties. It should be noted that this method yields sequences of many composite lengths. Balance. The px(p+l) sequences are perfectly balanced, whilst the sequences derived fkom the qxp arrays have an imbalance of p . Susceptibility to cryptographic attack The unfolded sequences can be considered as scrambled or “shuffled“ decimations of collections of repeats of the seed pseudonoise sequence. If the seed column and the sequence of shifts are both subject to linear recursion relations, the resultant diagonal sequence is as well. However, the window size of the sequence i s much larger than that of the seed and so consequently are the nonlinear and linear spans. For example, the sequence of length 21 of Fig.3(c) has non-linear and linear recursions of minimum order 6 and 9 respectively. Where the sequence of shifts is not subject to recursions, these orders are likely to be larger. This offers security against cryptographic attack. These features are currently being analysed [lo].

APPENDIX

MacWilliams and Sloane [8] and Green [4] folded an msequence of composite length into a matrix diagonal and showed that the matrix has the same periodic autocorrelation as the m-sequence. The converse is true only if the unfolding diagonal makes a single pass through the entire matrix. Even in this case, there are four ways of unfolding a matrix, since each matrix entry

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has four diagonal neighbom. We examine the relationship between these. So, consider a two dimensional matrix in FigS. with m rows and n columns, where m>n and gcd(m,n)= 1. To ensure cyclic boundary conditions, this matrix can be considered to be on the outer surface of a torus. Paths in opposite directions along the same diagonal lead to sequences which are reverses of each other (decimations by -1). The latter is equivalent to a generalized diagonal containing all entries of the matrix [8]. We show that the perpendicular diagonals are also decimations of each other and we derive the decimation d. We consider the two peapendicular diagonals, D1 and DZ to both start with u11. Dl consists of ( ~ 1 1 ,uzZ, u339 Q ~ and } DZ of

.....,

fall, aZn,

U3(wl)? .-*v &Z}.

The successor of a11 in Dz is ut,,. Denote by p. the position of uznin Dl. Thus d, the required decimation is p-I. Shift uzn to the top left position in the matrix by

moving its row up by one and its column to the first column p i t i o n . Then ~ 3 f ~ will 4 ) now be in the previous position of UZ,, and so ~ 3 + 1 ) occu~sd positions later than uznin Dl;and likewise for the further entries in D2.TkrusI92 is obtained from Dl by a decimation by a! We now calculate d from m and n.

- '2,

(2)

I

a(d+l~d+x)

Where ~ t a + l ) ( a + t" )m

U*

,with

d + 1 = r [mod( m 11 and d + for 19- and 1 ~ ~ 9 . We want r=2 and s==n, or: d+I=km +2 and

: 1 = s[md(

n)]

d+l=In for integers k,l. So now, we solve

En = km + 2 or In = I[mod(m)] Since m and n are coprime, n has a multiplicative inverse, say n-lt*dI, with n-' among 1,2,3,... ,m-l. Then k ( 2 n-l)tmadI,with 1 among I,2,3,...,m-I. Finally, d+l+ (as integers). Therefore: d=(2 n-l)tmod(nlln-l(as integers). As an example, consider the array with m=7 and n 4 . Then &15.

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