1 Quantization Index Modulation using the E8 lattice Qian Zhang and Nigel Boston Dept of Electrical and Computer Engineering University of Wisconsin Madison 1415 Engineering Drive, Madison, WI 53706 Email: [email protected], [email protected] Abstract— Quantization Index Modulation (QIM) is a watermarking is not robust at all; even trivial changes method for digital watermarking. The host signal is quan- can cause errors. Patchwork was proposed in 1995 by tized to the nearest point in the codebook indexed by the Bender et al [3]. In this algorithm the cryptographic key message. The performance depends greatly on the lattice selects n pixel pairs in the host image pseudorandomly. quantization codebook construction. This paper applies E8 Assume the luminance values (a ; b ) of the n pairs of Lattice Quantization to the QIM algorithm and compares P i i S = (a ¡ b ) = 0 (a ; b ) it with the Chen-Wornell quantization scheme. It is proved the pixel satisfies i i . The i i are modified by adding 1 to every a and substracting that the E8 lattice works better than the Chen-Wornell i quatization scheme in both theory and simulation. 1 from every bi. The decodingP process uses the same key, and computes Sˆ = (ˆai ¡ ˆbi). If a watermark is embedded in the image, Sˆ should be around 2n; I. INTRODUCTION otherwise, it is close to 0. The Patchwork watermarking As the Internet, MP3, DVD etc have greater and has good invisibility, but the data capacity is very low. greater impact on our lives, more and more digital It is robust to intensity correction (gamma, tone, :::), information security issues are brought up such as (1) while vulnerable to affine transforms (rotation, scaling, copyright protection, (2) unobtrusive communications :::). Spread Spectrum Technique is derived from CDMA systems for military and intelligence agencies, (3) communication systems. It spreads the watermark over content authentication and tamper-detection, and (4) a large number of frequency ranges and is thus robust. fingerprinting etc. [1]. Their solution lies in digital The energy inserted into the cover message in any given watermarking, which is hereby becoming a hot research frequency range is unnoticeable. topic. In the second category, the host signals are partitioned Digital watermarking is the embedding of the into subsets, and mapped to another set according to information in a host signal such as image, video or the watermarks. Quantization Index Modulation is in audio in an imperceptible way, with the recovery of this category. Chen-Wornell proposed the Quantization the information at the receiver side. Encryption tries to Index Modulation(QIM) method in [4] in 2001. In the conceal the meaning, but watermarking even tries to QIM embedding process, the mapping is based on a conceal the very existence of the message. quantization codebook. A good watermarking scheme must be invisible and In this paper we import a technique called Lattice robust. This means that the existence of information must Quantization into the QIM method. The paper is orga- be concealed, and its resistance to attack must be strong. nized as follows. Section II introduces some definitions of the lattice, and demonstrates that the E8 lattice has a Many digital watermarking schemes were proposed. much better packing property than D2 lattice. Section III [2] classifies them into two categories. In the first gives an overview of watermarking systems and propose category the watermark is added to the host signal after distortion denoted as D and correctability denoted as some processes such as encoding, modulation, etc.. Least C as measures for watermarking scheme. An important significant bit (LSB) watermarking, patchwork technique inequality C · D=2 is given. The quantization index and spread spectrum technique are representatives of this modulation method [4] is also introduced. category. The LSB process chooses a subset of the cover The QIM method depends much on the lattice image, and then exchanges the LSB of each element in quantization. This motivates us to apply QIM to some the subset to be watermarks m (0 or 1). However, LSB other lattices with better sphere packing properties. 2 Section IV applies the lattice quantization techniques The minimal squared distance between distinct lattice to the QIM method. In terms of C=D, we see that E8 vectors, or more simply the minimal norm of lattice Λ, QIM performance is better than Chen-Wornell QIM in is theory. The encoding and decoding method for the E8 lattice is also given in this section. Simulation results min fN(x ¡ y): x; y 2 Λ; x 6= yg are in section V and the conclusion is confirmed in = min fN(x): x 2 Λ; x 6= 0g section VI. [6] summarizes the records for optimal packings, coverings and quantizers for lattices in each dimension as shown in Table I (Bold stands for optimal among II. LATTICES AND SPHERE PACKING lattices). In this section we introduce lattices. Formally, a lattice is a discrete subgroup of Euclidean TABLE I space, assumed to contain the origin. That is, a lattice RECORDS FOR PACKINGS, COVERINGS AND QUANTIZERS is closed under addition and inverses, and every point has a neighborhood in which it is the only lattice point. DIMENSION 1 2 3 4 5 6 7 8 n Densest ZA2 A3 D4 D5 E6 E7 E8 Usually, a lattice in R is denoted as Λ and defined as Packing the subgroup ¤ ¤ ¤ ¤ ¤ ¤ Thinnest ZA2 A3 A4 A5 A6 A7 A8 Covering fa1v1 + ::: + anvng ¤ ¤ ¤ ¤ Best Quan- ZA2 A3 D4 D5 E6 E7 E8 where a1; : : : ; an are arbitrary integers. And tizer v1; v2; :::; vn are linearly independent vectors in n-dimensional real Euclidean space Rn. The vectors v1; v2; :::; vn are then called a basis for the lattice. Let The density ∆ of a lattice packing is defined as: the coordinates of the basis vectors be ∆ = proportion of space that is occupied v1 = v11e1 + v12e2 + ::: + v1nen; by the spheres volume of unit sphere v2 = v21e1 + v22e2 + ::: + v2nen; = volume of fundamental region volume of unit sphere ::: = detM v = v e + v e + ::: + v e ; n n1 1 n2 2 nn n The densest possible lattice packings are known in n where e1; e2;:::; en are the standard basis of R . dimensions n · 8 and are unique in each dimension as The matrix shown in the first row of the entries in Table I. 2 3 v v : : : v 11 12 1n Consider any discrete collection of points P = 6 v v : : : v 7 M = 6 21 22 2n 7 fP ; P ;:::g in Rn. The least upper bound for the 4 :::::::::::: 5 1 2 distance from any point of Rn to the closest point P is v v : : : v i n1 n2 nn called the covering radius of P, usually denoted by R. is called the generator matrix for the lattice, and the Thus, lattice vectors consist of all the vectors R = sup inf dist(x; P) x2Rn P2P "M If the upper bound does not exist we set R = 1. where " = ("1;:::;"n) is an arbitrary vector with integer components "i. The points of the Voronoi cells whose distance from P is a local maximum are called holes in P. The points The norm of the lattice vector x = (x1; x2; : : : ; xn) 2 whose distance from P is an absolute maximum are Λ is defined by called deep holes in P. Their distance from P is the X covering radius R. 2 N(x) = x ¢ x = (x; x) = xi 3 Suppose an arrangement of spheres of radius R covers and denotes the recovery of the watermark from the Rn. If the centers form a lattice Λ then the thickness is marked image. gf(x; m) = m means that in the absence defined by of any attack the correct mark is recovered. Θ = average number of spheres that contain a point of the space noise Watermarked volume of one sphere Host signal x Signal y m' = Message m Encoding Channel r Decoding detM f(x,m) g(r) V Rn = n detM Key where ¼n=2 V = n (n=2)! As shown in Table I the thinnest lattice covering, i.e. for covering with minimal thickness are known in dimension 1 through 5, and in each case the optimal Fig. 1. A general Watermarking System ¤ lattice is An ( the dual of A). The covering associated with A¤ is in fact the thinnest covering known in all n In order to give a first measure of how good the dimensions n < 23. scheme is, we might define the following related quantities. The third row of Table I indicates the best lattices for quantization. If given input x, the output is rounded off Definition 3.1: The distortion of the scheme is to the closest lattice point. If the input x is uniformly the infimum of the real numbers D for which distributed, the lattices in Table I perform best in each d(x; f(x; m)) · D for all x 2 X,m 2 M holds true. dimension, i.e. they have the minimum average mean squared error per dimension. Definition 3.2: The correctability of the scheme is the supremum of the real numbers C for which Overall from the comparison above E demonstrate 8 d(x0; f(x; m)) · C implies g(x0) = m holds true. much better packing properties than the D2 lattice. Note that so long as #M > 1, C · D. This follows since if D < C, then combining the above inequalities III. WATERMARKING SYSTEM AND QUANTIZATION we get g(x) = m for all x 2 X; m 2 M. For the most INDEX MODULATION common examples of metric space X, e.g.