New Matrices with Good Auto and Cross-Correlation

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PAPER

Special Section on Sequence Design and its Application in Communications

New Matrices with Good Auto and Cross-Correlation Andrew TIRKEL†a) and Tom HALL† , Members

SUMMARY Large sets of matrices with good auto and crosscorrelation are rare. We present two such constructions, a method of extending family size by column multiplication and a method of extending physical size by interlacing. These matrices can be applied to digital watermarking of images. key words: matrix, array, interlacing, correlation, watermark

1.

Introduction

Sequences with good auto and cross-correlation have many applications in modern communications. Some sequences can be folded into matrices (arrays). Such arrays have been used in radar, sonar and frequency hopping [2], time hopping patterns for UWB radio [3] and two-dimensional optical orthogonal coding (OOC) [4]. In these applications, the matrix representation is useful, but not essential, and often their two-dimensional nature is not exploited. Arrays with good autocorrelation have been used in Coded Aperture Optics, Structured Light and related areas. Families of arrays with good auto and cross-correlation are rare. They can be obtained by folding of sequence sets, such as Gold or Kasami sequences of composite length. However, the dimensions of these arrays are restricted and impractical, and the family sizes sub-optimal. The electronic watermark (introduced by our group at Monash [5]) is an application, which can make use of large families of arrays with good correlation. The watermark is an array or a sum of arrays which is added to an image in the spatial (pixel) or transform domain, taking advantage of psychophysical masking criteria (not considered here). It is recovered by performing a 2-dimensional cyclic correlation of the watermarked image with the watermark template(s). The detection of a correlation peak indicates the presence of a watermark, whilst its cyclic shift (and the structure of the array) can carry information. Such watermarks have been used to determine proof of ownership of the image, detect tampering and to provide an image audit trail (fingerprinting, traitor tracing etc). Watermarks have become an important part of steganography, where information is hidden in an image, video, audio or other medium. The multi-user nature of watermarking is unique, in that different recipients Manuscript received December 10, 2005. Manuscript revised March 11, 2006. Final manuscript received May 12, 2006. † The authors are with the School of Mathematical Sciences, Monash University, PO Box 28M, Victoria 3800, Australia. a) E-mail: [email protected] DOI: 10.1093/ietfec/e89–a.9.2315

may superimpose (add) individual watermarks so that an audit trail exists in the case of a leaked image. This requires good auto and cross-correlation. Families of arrays with good auto and cross-correlation have other benefits for watermarking: (a) Multiple arrays can be added to increase information capacity (b) Addition of arrays enhances watermark security In this paper, we present a construction of such families of arrays. We base our construction on special frequency hopping patterns [6], [7], which we have generalized [8]– [10]. Such patterns are described by a sequence of dots (impulses), with at most one dot per column. The patterns are designed, so that at most two dots can overly, for any twodimensional cyclic shift of the patterns (except of the full match for a pattern with an unshifted version of itself). In frequency hopping, the column number determines the time slot, and the vertical position of a dot within a column determines the frequency for that time slot. Such patterns are too sparse for watermarking [11]. We overcome this, by replacing each column with a dot by a cyclic shift of a single pseudonoise sequence, of the same column length. The cyclic shift is determined by the position of the dot within the column, and the ordered list of cyclic shifts is called a shift sequence. A sequence is pseudonoise if its autocorrelation takes on two values, a high value, V, at zero cyclic shift, and a single low value, (−1 or 0) for all other shifts. The matrix auto and cross-correlations are summations of the correlations of the overlaying columns, and are thus determined by the numbers of columns with matching shifts. This is the same as the number of dots which can overly each other in the original frequency hopping patterns. Therefore, the correlation of such matrices is constrained: (a) All columns match (autocorrelation peak) (b) 0, 1, or 2 columns match (at most 2 dots can overly) Although the construction and the correlations referred to are doubly periodic, it is clear that the above constraints apply if the horizontal shifts are treated as aperiodic. This is because the periodic correlation is a sum of two aperiodic correlations. We modify the prototype matrices by modulating the columns by a suitably chosen multiplication sequence. This results in much larger matrix families with almost no degradation in correlation values. Then, using a method called interlacing, we derive new arrays from these prototypes. The

c 2006 The Institute of Electronics, Information and Communication Engineers Copyright


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paper is organized as follows: Sect. 3 outlines the construction of and the properties of the prototype arrays. The modulated matrices are presented and analyzed in Sect. 4. Section 5 describes the process of interlacing, which is used to construct new arrays from the prototype arrays. The main application of our arrays is to digital watermarking. 2.

Preliminaries

In this section, we introduce array notation and the basic principles of watermarking. A deeper presentation of watermarks can be found in the monograph [1]. 2.1 Definition of Periodic Correlation of Sequences For (two) finite sequences S = (s1 , s2 , . . . , sn ) and T = (t1 , t2 , . . . , tn ) of complex numbers, the cross-correlation numbers form the sequence C = (c0 , c1 , . . . , cn ), where each ck is the dot product of S with the cyclic shift k of T by k places (to the left), namely with
T ∗= (tk+1 , tk+2 , . . . , tn , t1 , t2 , . . . , tk ). Thus ck = S · T k = ni=1 si ti+k which covers both cases of real or complex entries in the sequences S and T (* denotes complex conjugation). This definition also covers the case when S = T , giving the autocorrelation numbers of S . 2.2 Definition of Periodic Auto and Cross-Correlation Numbers of Matrices (Arrays) A matrix A can be cyclically shifted (or rotated) in 2 directions, horizontally by k places to the left and vertically by l places upwards. Performing the shifts in either order (horizontally and then vertically, or vertically and then horizontally) gives the same resultant shifted matrix A(k,l) , defined formally as follows: For an m × n matrix A = (ai, j ) with (i, j) entry being ai, j for i = 1, 2, . . . , n, j = 1, 2, . . . , m (m rows and n columns) the shifted matrix A(k,l) has (i, j) entry ai+k, j+l , so A(k,l) = (ai+k, j+l ), where the subscript i + k is reduced modulo n and j + l is reduced modulo m. The inner (or dot) product of two m
× n
matrices A = (ai, j ) B = (bi, j ) is defined as A · B = mj=1 ni=1 ai, j b∗i, j =
n
m ∗ i=1 j=1 ai, j bi, j , covering the two cases of matrices with real or complex entries. The cross-correlation number ck,l of A and B is the dot product of A with the (k, l) shift of B, ck,l = A · B(k,l) . All the cross-correlation numbers of ck,l , k = 0, 1, . . . , n l = 0, 1, . . . , m, form the matrix C = (ck,l ). In the case A = B, the cross-correlation numbers ck,l become autocorrelation numbers ak,l = A · A(k,l) , and C becomes the autocorrelation matrix A = (ak,l ). 3.

Prototype Arrays

The arrays in this section are based on the construction [6] and a modification of the frequency hopping patterns of [7]. Arrays similar to [6] have been used to construct sequences

with good auto and cross-correlation [12]. 3.1 Polynomial Shift Sequences mod p Consider a polynomial ϕ(x) of order n, with coefficients from Z p : ϕ(x) = an xn + an−1 xn−1 + . . . + a1 x1 + a0 where ϕ(x) is also expressed modulo p. When cyclic shifts ϕ(x) are applied to a given fixed column to obtain columns c x , x = 0, 1, . . . , p−1, this describes a p×p matrix Aϕ . There are p2 − 1 other matrices cyclically equivalent to Aϕ . These are obtained by all two-dimensional non-zero shifts of matrix Aϕ . Horizontal shifts are equivalent to the transformation x = x + k, whilst vertical shifts are achieved by the transformation ϕ (x) = ϕ(x) + l where k and l are chosen from Z p . Shift polynomials which differ only in a0 and/or an−1 describe matrices which are two dimensional cyclic shifts of each other, and are considered equivalent. The number of matrices and hence non-equivalent shift sequences is thus pn−1 − 1. The all zeros polynomial is excluded. Consider another matrix, built from the same column sequence, but using the shift sequence: ψ(x) = bm xm + bm−1 xm−1 + . . . + b1 x1 + b0 .

(1)

The number of matching columns between the matrices is obtained by solving ψ(x) − ϕ(x) = 0 for all shifts of one of them. The difference polynomial is of degree at most z = max(m, n), so this is an upper bound on the number of solutions, and thus on the number of matching columns. Hence, for columns with autocorrelation numbers: V and −1, the cross-correlation of these matrices can take the values: 0V − (p − 0), 1V − (p − 1), 2V − (p − 2), . . . , zV − (p − z). For the autocorrelation of matrix Aϕ , the number of matching columns is obtained from: ϕ(x + k) + l − ϕ(x) = 0. Substituting for ϕ, we have an (x + k)n + an−1 (x + k)n−1 + . . . +a1 (x + k)1 + a0 + l − an xn − an−1 xn−1 − . . . −a1 x1 − a0 = 0

(2)

This polynomial is of degree n − 1, so n − 1 is an upper bound on the numbers of matching columns, apart from the full match. Hence, the autocorrelation can take the values: 0V − (p − 0), 1V − (p − 1), 2V − (p − 2), . . . , (n − 1)V − (p − (n − 1)). The quadratic is the lowest degree polynomial that yields meaningful matrices [13]. Example: p = 7 One such quadratic shift sequence is: ϕ(x) = 3x2 + 3x: 0, 6, 4, 1, 4, 6, 0 (mod 7). A matrix built from this shift sequence using a binary msequence column is shown in Fig. 1 below. The autocorrelation values are: p2 , +1, −p.

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Mapping produced by T .

Table 2

Fig. 1

Permutation for h = 2.

A matrix constructed using a quadratic shift sequence. Table 3

Fig. 2

Another quadratic shift sequence matrix.

Another matrix can be constructed from a different shift sequence: ψ(x) = 6x2 + 6x: 0, 5, 1, 2, 1, 5, 0 (Fig. 2). Its cross-correlation with the first matrix is: −p, 1, p + 2. 3.2 Moreno-Maric Shift Sequences mod (pn + 1) Moreno and Maric [7] present an ingenious construction of frequency hopping patterns, which are considered as cyclic shifts of an impulse column. The patterns are designed so that the number of off-peak auto and cross hits between two patterns is constrained to 0, 1, or 2. Their shift sequences are constructed as follows. Whenever x2 + x + α is irreducible, and α is a primitive element in GF(pn ), then the polynomial −α n x+1 gives a cycle of full length (p + 1). Take the recursion −α T : xi+1 = xi +1 . Start with x0 = 0. This leads to the following sequence: x1 = −α −α x2 = 1−α α2 − α x3 = 1 − 2α 2α2 − α x4 = 2 α − 3α + 1 −α3 + 3α2 − α x5 = 3α2 − 4α + 1 −3α3 + 4α2 − α x6 = −α3 + 6α2 − 5α + 1 ...   r Consider each x as a vector where r is the numerator s and s is the denominator. The recursion relation becomes:  i   0 −α 0 −α 0 xi+1 = xi and in general xi = . 1 1 1 1 1 Berlekamp and Moreno [14] show that this recursion cycles through all the elements of the finite field GF(pn ) once, and then produces an extra element called ∞, before repeating

Inverse mapping.

the cycle. Adding to their treatment involving GF(pn ) for n > 1, we cover the case n = 1 of the base field Z p . Any element xi+k of GF(pn ) ∪ {∞} is obtained by applying T k to any other element xi , where i, k ∈ Z pn +1 . Consider Z 7 and α = 3. The recursion T : xi+1 = xi4+1 , is shown in Table 1, starting from 0. Now consider two natural permutations of Z 7 ∪ {∞}: x = hx

and

x = hx−1 , h = 1, 2, 3, . . . 6.

(3)

Example: x = 2x Apply the inverse mapping x∗ of the mapping x in Table 1, to the second row of Table 2, back to the integers {0, 1, . . . , 7}, as shown by the third row of Table 3. In general, the sequence of integers si can be used to (cyclically) shift pn + 1 columns of a matrix, each of length pn + 1, and we call it a Moreno-Maric shift sequence. Theorem 1: 1. The cyclic differences mod (pn + 1) between entries in the above shift sequences taken any distance k apart contain at most 2 occurrences of each difference. 2. Thus, the off-peak autocorrelation values have a bound calculated from knowing that, at most, two columns match. 3. For pseudonoise columns, with autocorrelation values of pn + 1 and −1, the matrix off-peak autocorrelation is at most pn + 3, compared with the peak of (pn + 1)2 . Proof: Apply T recursively k times to xi . The result xi+k +B is of the form: xi+k = CAxxii+D , (with A, B, C, D in Z p ), a rational function. The permutations of the form x = hx and x = hx−1 do not alter this form. In both cases, denote the A(h)xi +B(h) . Consider differences between result as: xi+k = C(h)x i +D(h) entries k apart: ∆k = xi+k − xi =

A(h)xi +B(h)−C(h)x2i −D(h)xi . C(h)xi +D(h)

(4)

For each occurring value of ∆k , Eq. (4) is quadratic in xi , and so there are at most two values of xi , and thus of i (since the “function” x is one-to one), that satisfy (4). Note: An entirely similar theorem covers cross-correlation, with (b) and (c) remaining true with “autocorrelation” replaced by “cross-correlation.” Permutations (and therefore

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Distinct Moreno-Maric shift sequences of length 8.

with such a sequence as a column, have off-peak autocorrelation and cross-correlation below 2 × 42 + 40 = 124, or 7% of the autocorrelation peak of 42 × 42 = 1764. 3.4 Derived Constructions The quadratic and Moreno-Maric constructions appear to be the only constructions of large families of shift sequences, which lead to large families of matrices with the number of matching columns being bounded by 2. These matrices can be modified by column lengthening or shortening and column insertion or deletion. The resulting upper bounds on the numbers of matching columns increase in a predictable manner [18]. In the next section we present a modification which enlarges the family size for each construction, with minimum effect on the correlation values. We call these modulated matrices.

Fig. 3 An 8 × 8 matrix constructed using a Moreno-Maric shift sequence and a Milewski column sequence.

shift sequences) are called distinct if they lead to nonequivalent matrices. Table 4 shows the 6 distinct shift sequences out of a total of 11, produced by the permutations from Eq. (3), for this example. A suitable column sequence for building a matrix from the shift sequences of Table 4 is the Milewski sequence [15], 1, 1, i, −1, 1, −1, i, 1. It is a perfect sequence, with autocorrelation numbers 8, 0. A matrix using shift sequence a, of Table 4 is shown in Fig. 3. There are 6 such 8 × 8 matrices, all with off-peak autocorrelation and cross-correlation values of 0, 8, 16, and an autocorrelation peak of 64. 3.3 Column Sequences Column sequences over 2 or 4 roots of unity (and 0) are useful for watermarking, since they can be embedded without quantization errors. The former can be added (+1, −1, 0) to greyscale image values or their transform. The latter can be added to any two of the three channels of colour images: say (0, +1, −1) to R and (0, +i, −i) to G. For polynomial shift matrices, Binary Legendre sequences exist for p = 4k − 1, (or binary m/GMW sequence for p = 2n −1), and the ternary Legendre sequence for p = 4k + 1. For the Moreno-Maric shift sequence matrices, the GCL, Milewski [15], or Schotten and Lee [16] sequences make suitable columns over 4 roots of unity (and 0). Also, it is possible to use other sequences with low off-peak autocorrelation. In [17] we constructed binary sequences of length q(q − 1) (q prime), by diagonal unfolding of exponential shift arrays. There are many cases where (pn + 1) = q(q − 1). E.g. a sequence of length 42 is obtained from the shift sequence si = gi mod 7 applied to a binary m-sequence column (g is a primitive root of 7). Such sequences have off-peak autocorrelation values of −q + 1 and +2. Therefore, 42 × 42 Moreno-Maric arrays,

4.

Modulated Matrices

The matrices of section 3 can be modified by multiplying each column by +1 or −1 (using a binary sequence r(i)). For the resulting matrices, the columns which match in shift contribute +V to the matrix correlation, when their polarities match, and −V when they do not. Similarly, for columns mismatched in shift, the correlations are −1 or +1 depending on column polarities. For non-zero 2D matrix displacement, the number of columns with matching shifts is constrained to 2. Hence, these columns contribute −2V, or −V, or 0, or +V, or +2V to the matrix correlation. All other columns contribute +1 or −1 to the matrix correlation. For the multiplying sequence r(i), let n 

r(i) = ∆

i=1

Then the matrix auto and cross-correlation takes on the values: −2V − 2 − ∆, −V − 1 − ∆, −∆, V + 1 − ∆, 2V + 2 − ∆. In the special case of one matrix multiplied by two sequences r1 (i) and r2 (i), then the zero shift cross-correlation is: A·B=

n m   j=1 i=1

ai, j b∗i, j = V

m 

r1 ( j)r2 ( j)

j=1

i.e. the product of the column autocorrelation peak and the cross-correlation of the multiplying sequences. The multiplying sequences are thus chosen to be orthogonal: for example, the set of cyclic shifts of a pseudonoise sequence, or the set of balanced Walsh-Hadamard sequences. The above analysis can be extended to the case where the column or multiplication sequences are perfect sequences which include entries of 0. 4.1 Polynomial Shift Sequence Construction An example of the method is shown below in Fig. 4 for the quadratic shift sequence matrix of Fig. 1 of Sect. 3.1. The

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Fig. 4

Matrix of Fig. 1 modulated by m-sequence.

Fig. 6

Matrix of Fig. 3 modulated by Walsh-Hadamard sequence.

column contributes +7 and all other columns −1 for a total correlation of +1. 7 occurrences. Fig. 5

Linear shift matrix modulated by an m-sequence.

complemented columns are shown in grey. The multiplication sequence is an m-sequence of length 7 or any one of its cyclic shifts: −1, −1, 1, −1, 1, 1, 1. This is the reverse of the column sequence, since there are few suitable sequences of such short length available. A more interesting case of polynomial based shifts sequences is the linear ϕ(x) = a1 x + a0 . case: In the absence of a modulating sequence, such matrices are not useful since they exhibit an autocorrelation peak along the leading diagonal. This is because ϕ(x + k) + l = a1 (x + k) + a0 + l = a1 x + a0 has solutions independent of x: a1 k + l = 0 i.e. l = −a1 k with k
0 when all columns match i.e. when matrices are equal. However, for k
0, the multiplication sequences do not match, so the modulated matrices are not equal. As an illustration, we present the example of Fig. 5. The autocorrelation of such a matrix is: a) 49 for k = l = 0 (peak) (all columns match in shift and multiplication sequence) 1 occurrence b) −7 for k = 0, l
0 (no columns match in shift, but all match in multiplication sequence) 6 occurrences c) −7 for l = −a1 k (all columns match in shift but 3 match in multiplication sequence and 4 do not) 6 occurrences d) +1 for all other cases (no columns match in shift i.e. column correlation = −1, 3 match in multiplication sequence and 4 do not) 36 occurrences Other matrices can be constructed using different multiplier values and different cyclic shifts of the m-sequence. The number of columns matching in shift can be obtained by solving a1 (x + k) + a0 + l = b1 x + b0 . This is a linear equation in Z 7 and thus a column matches for every value of k, l. a) Multiplication sequences match. A single shift-matched

b) Multiplication sequences do not match. The column matching in shift matches in multiplication. It contributes +7 to the matrix correlation. The remaining columns contribute four +1’s (multiplication disagreements) and two −1’s (multiplication agreements) for a total of +9. 18 occurrences. c) Multiplication sequences do not match. The column matching in shift does not match in multiplication. A similar argument to a) yields a correlation value of −7. 24 occurrences. This theory is true not just for 7, but for any prime p. The correlation values for such matrices are restricted to: p2 (autocorrelation peak) and −p, +1, p + 2. The number of matrices in the family is p(p − 1) + 1. There are p cyclic shifts of the multiplication sequence and (p − 1) non-constant linear shift sequences. One constant shift sequence matrix is included. For that case, all cyclic shifts of the multiplication sequence yield equivalent matrices. The theory also applies to matrices constructed from higher degree polynomials representing the shift sequence. In such cases, more columns can match in shift and therefore the number of correlation values rises accordingly. For example, modulated quadratic shift sequence matrices have correlation values: −2p − 1, −p, +1, p + 2, 2p + 3. For higher degree polynomials, these correlation values degrade linearly, but the family size grows exponentially. It is also possible to use perfect ternary sequences of prime length as column sequences or multiplication sequences or both. Such arrays have similar correlation properties. They have reduced efficiency, but increased cryptographic immunity. However, they are relatively scarce. 4.2 Moreno-Maric Shift Sequence Construction A similar method can be applied to matrices constructed using the Moreno-Maric shift sequence. Consider the example of Fig. 3. In this case, Walsh-Hadamard sequences of length 8 are suitable multiplication sequences. An example is shown in Fig. 6. There are 6 parent matrices built using the Moreno-

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Comparison with Kasami construction.

wi = si/2

i ∈ Z 2q , i even wi = t(i+1)/2

i ∈ Z 2q , i odd

We call this an interlaced array, Ai j because of its similarity to the interlaced line format of television images. Theorem 2: A q×2q interlaced array Ai j constructed using shift sequence wi , applied to a single pseudonoise column has at most 2m matching columns, when compared with any non-zero cyclic 2d shift of itself. Maric shift sequence. The total number of modulated matrices is 48. The off-peak auto and cross-correlations are constrained to 0, ±8, ±16, compared with an autocorrelation peak of 64. In general, there are 22m − 2m+1 such 2m × 2m matrices for 2m − 1 prime, with an autocorrelation peak of 22m and off-peak auto and cross-correlation values of 0, ±2m , ±2m+1 . 5.

Comparison with Classical Constructions

Sequences of composite length, which can be factored into two co-prime factors, can be folded into arrays along any single diagonal. Such arrays possess the same 2D cyclic correlations as the original sequences. This applies to two classical sequence sets: the small and large Kasami codes. A comparison between these and our modulated linear and quadratic shift arrays is presented in Table 5. When p ≈ 2n + 1, the Small Kasami set has a similar correlation bound to the linear shift. However, the family size of the linear shift far exceeds that of the small Kasami set. Similarly, the Large Kasami and the quadratic shift family have similar correlation bounds (about double the previous case). However, the Large Kasami family is much larger. Therefore, the linear shift matrices are preferred for lowest correlation, whilst the large Kasami set are preferred for largest family size. This is true if other factors are not important. The Kasami matrices are available in very few sizes which grow exponentially, have strange aspect ratios and many of them are unbalanced. By contrast, all our constructions are available for many sizes (namely primes or powers of primes, which are much denser than powers of 2), they are square and balanced. In the next section, we present a new modification, interlacing, which extends these arrays horizontally, vertically, or both, without changing the normalized auto and crosscorrelation. 6.

Interlacing

Consider a shift sequence si : i ∈ Z q with the property that any cyclic difference (mod q), (si+k − si ) for any fixed k ∈ Z q , appears at most m times. Consider another, nonequivalent shift sequence ti : i ∈ Z q , with the same difference property. Now, examine the cyclic differences (mod Zq ): (ti+k − si ). Further, assume the frequency of occurrence of any of these differences is bounded by m as well. Both families of constructions in Sect. 2 are of this type. Now consider an array of size q × 2q constructed using a shift sequence wi of length 2q:

Proof: Consider a shift (k, l) of Ai j , (defined in Sect. 1). Case I—k is even. Columns with shift si+k fall upon those with shift si , whilst columns with shift ti+k fall upon those with shift ti . There are at most m (autocorrelation) matches within each of the two sub-arrays, so an upper bound on the number of matching columns is 2 m. Case II—k is odd. Columns with shift si+k fall upon those with shift ti , whilst columns with shift ti+k fall upon those with shift si . There are at most m (cross-correlation) matches within each of the two sub-arrays and so an upper bound on the number of matching columns is 2m. Suppose F is a family of shift sequences such that any distinct pair s, t of sequences in F have the properties assumed at the beginning of this section. Take distinct s, t, u, v from F, and construct Ai j from s, t as above, and construct Bi j similarly from u, v. Consider the number of columns in a cyclic 2d shift of array Ai j which match overlying columns in array Bi j . Theorem 3: An upper bound on the number of matching columns between arrays Ai j and Bi j is 2m. Generalization: An interlaced q × nq array can be constructed using n shift sequences with the above properties. An upper bound on the number of matching columns is mn. Another q×nq array can be constructed from non-equivalent shift sequences with the same difference properties. An upper bound on the number of matching columns between the two arrays is also mn. Result: The off-peak autocorrelation of an interlaced array of size q × nq constructed in the manner described above, from pseudonoise columns with autocorrelation values of q and −1, is constrained to be between −nq and n(m−1)q+mn. The autocorrelation peak is nq2 . For large q, the normalized off-peak autocorrelation and cross-correlation values range between −1/q and (m − 1)/q + m/q2 . Corollary 1: Let F have M distinct shift sequences. The number of arrays with the auto and cross-correlation properties above is M/n (the number of arrays without reuse of a shift sequence). Corollary 2: The interlacing can also be performed vertically, without affecting the result. Both horizontal and vertical interlaces can be performed simultaneously. Note: The interlacing in this paper uses a single column sequence and different shift sequences. Therefore, all array

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2321 Table 6

New array properties.

correlations are sums of sequence autocorrelations. It is possible to use different column sequences, but then, the array correlations are sums of cross-correlations of columns, and therefore difficult to contain for all shifts. 7.

Conclusions

This paper presents matrix constructions with good auto and cross-correlation, based on cyclic shifts of a pseudonoise column. Two prototype families are considered: polynomial shift sequences mod p and Moreno-Maric shift sequences mod pn + 1. The size of the families of these prototype matrices is enlarged by modulating them through the use of a column multiplication sequence. We also present a simple construction, interlacing, which extends the matrix size. The properties of these new matrices are summarized in Table 6. These matrices can be applied to digital watermarking of images. References [1] S. Katzenbeisser and F.A.P. Petticolas, ed., Information Hiding Techniques for Steganography and Digital Watermarking, Computer Security Series, Artech House, 2000. [2] M.R. Schroeder, Number Theory in Science and Communication, 3rd ed., Springer-Verlag, 1997. [3] R.A. Scholtz, P.V. Kumar, and C.J. Corrada-Bravo, “Signal design for ultra-wideband radio, sequences and their applications,” SETA ’01, May, 2001. http://www.ee.vt.edu/˜ha/research/uwb/signalings/ Scholtz SignalDesign4UWB.pdf [4] E.S. Shivaleela, K.N. Sivarajan, and A. Selvarajan, “Design of a new family of two-dimensional codes for fiber-optic CDMA networks,” J. Lightwave Technol., vol.16, no.4, pp.501–508, 1998. [5] A.Z. Tirkel, G.A. Rankin, R.M.van Schyndel, W.J. Ho, N.R.A. Mee, and C.F. Osborne, “Electronic water mark,” DICTA 1993, pp.666– 673, Macquarie University, Sydney, 1993. [6] P.V. Kumar, “On the existence of square dot-matrix patterns having a special three-valued periodic-correlation function,” IEEE Trans. Inf. Theory, vol.34, no.2, pp.271–277, 1988. [7] O. Moreno and S.V. Maric, “A new family of frequency-hop codes,” IEEE Trans. Commun., vol.48, no.8, pp.1241–1244, Aug. 2000. [8] A.Z. Tirkel, C.F. Osborne, and T.E. Hall, “Steganography— Applications of coding theory,” IEEE-IT Workshop, pp.57–59, Svalbard, Norway, 1997. [9] A.Z. Tirkel and T.E. Hall, “Matrix construction using cyclic shifts of a column,” ISIT’05, pp.2050–2054, 2005. [10] T.E. Hall, C.F. Osborne, and A.Z. Tirkel, “Families of matrices with good auto and cross-correlation,” Ars Combinatoria, vol.61, pp.187– 196, 2001. [11] S.W. Golomb, Private communication, ISSSTA’94. [12] K.G. Paterson, “Binary sequence sets with favourable correlation properties from difference sets and MDS codes,” IEEE Trans. Inf.

Theory, vol.44, no.1, pp.172–180, 1998. [13] A.Z. Tirkel and T.E. Hall, “A unique watermark for every image,” IEEE Multimedia, vol.8, no.4, pp.30–37, Oct.-Dec. 2001. [14] F.J. MacWilliams and N.J.A. Sloane, The Theory of ErrorCorrecting Codes, North-Holland, 1977. [15] A. Milewski, “Periodic sequences with optimal properties for channel estimation and fast start-up equalization,” IBM J. Res. Dev., vol.27, no.5, pp.426–431, 1983. [16] H.D. Schotten and H.D. Luke, “New perfect and ω-cyclic perfect sequences,” ISITA’97, pp.82–85, 1997. [17] A.Z. Tirkel and T.E. Hall, “Pseudo-noise patterns for watermarking,” Proc. Int. Symposium on Multimedia Information Processing, pp.324–327, Sydney, 2000. [18] E.I. Krengel, A.Z. Tirkel, and T.E. Hall, “New binary and ternary sequences with low correlation,” in Sequences and Their Applications, LNCS 3486, pp.220–235, Springer-Verlag, 2005. [19] A.Z. Tirkel, C.F. Osborne, and T.E. Hall, “Image and watermark registration,” Signal Process. J., vol.66, pp.373–383, 1997.

Andrew Tirkel received a B.Sc. degree in 1970 and a Ph.D. degree in Physics in 1975, from Monash University. From 1976 to 1981 he worked for the US Aerospace Industry in microwaves and IR. From 1982 to 1985 he was a senior lecturer in Communications/Physics at RMIT. Since 1986 he has been an independent consultant in the electronics industry. In 1999 he joined Mathematics at Monash University.

Tom Hall received a B.Sc. degree from the University of Queensland in 1965 and a Ph.D. degree in Mathematics from Monash University in 1969. He was a Nuffield Foundation Fellow at Stirling, Scotland, in 1970 and since 1971 he has been an academic (currently a reader) at Monash University.