【正文】 ontents.169。 ={M1, M2,...,Mr} and using CRT, a large integer Z canbe represented by a set of smaller integers {R1, R2,...,Ri}. It is extremely di?cult to get back the original integer Zwithout knowing 181。. The rank of A is defined as the number of nonzero diagonal elements of D.The SVDbased watermarking scheme proposed by Chang et al. [14] uses a blockbased SVD technique. The scheme isquite promising, however, the main weakness of this scheme is its reliance on the use of rank in selecting an image block444 . Patra et al. / Digital Signal Processing 20 (2020) 442–453Fig. 1. Watermarking Scheme 1. The change in rank results in a shifted Panda in the extracted watermark: (a) original 16 16 watermark, (b) extracted16 16 watermark, (c) difference between original and extracted watermarks.for embedding watermark bits. This reliance causes the scheme to be less resistant to attacks, which alters the rank of theimage blocks. The host image is a greyscale image and the watermark is a binary image. First, the host image is dividedinto blocks of n n pixels. A single watermark bit is embedded in a block. The blocks are chosen randomly using a pseudorandom number generator (PRNG) based on their rank. The blocks with higher ranks are selected first before those of lowerranks.In order to embed a watermark bit in a block the following procedure is adopted. We first perform a SVD transformationon the selected block. Let c1and c2denote the second and the third elements (u21and u31) of the first column of U matrix,respectively. The embedding rules are as follows:To embed a watermark bit ‘1,’ the value of c1? c2should be positive and its magnitude is greater than a strengthfactor, s. If this condition is not satisfied, c1and c2are modified as cprime1and cprime2, respectively, and are given bycprime1= a + s/2,cprime2= a ? s/2, (2)where a = (|c1|+|c2|)/2.To embed a watermark bit ‘0,’ the value of c1? c2should be negative and its magnitude is greater than the strengthfactor, s. If this condition is not satisfied, c1and c2are respectively modified as cprime1and cprime2given bycprime1= a ? s/2,cprime2= a + s/2. (3)To reconstruct the watermarked block, an inverse SVD transformation is performed on the modified U matrix withthe original D and V matrices. This watermarked block then replaces the original selected block in the host image. Thisembedding process is repeated until all the watermark bits have been embedded.In this scheme, the selection of image blocks based on higher ranks would lower the distortion level of the watermarkedimage. However, the rank is not a reliable feature as it is not stable. The rank of the selected block could change after modifications are made to the elements c1and c2. Therefore, without any tampering to the watermarked image, the extractedwatermark could be corrupted due to a change in rank of the block. Fig. 1 shows the corruption of the extracted watermarkdue to changes in rank. In general, when the strength factor increases, the likelihood of changing of the rank of a blockincreases. This leads to a higher level of corruption and distortion in the watermark and watermarked image, respectively.The initial steps of the extraction procedure are similar to that of the embedding process up to and including theselection of the coe?cients c1and c2.Thevalueofc1? c2determines the value of the extracted watermark bit. A positivedifference indicates that the watermark bit is a ‘1.’ Whereas, a negative difference would imply that a watermark bit ‘0’ isextracted.. Watermarking Scheme 2The embedding procedure of the scheme proposed by Patra et al. [15] is an improved version of Scheme 1. The hostimage is divided into a number of superblocks that is equal to the number of watermark bits. From each superblock,a subblock is selected which is transformed via SVD to embed the watermark. Instead of embedding exclusively in theU matrix, the V matrix is also utilized. A counter is introduced to allow random selection of embedding the watermark bitin either U or V matrix to improve the security and robustness.In addition, the coe?cients (c1and c2) are selected randomly from the first column of the U or V matrices using a PRNG.This flexibility gives an edge over the scheme proposed by Chang et al. [14], as malicious tampering cannot be targeted atthe two modified coe?cients due to their random selection. However, to minimize the distortion to the watermarked image,the position of c2must be consecutive to c1. Experiments show that the watermarked image quality reduces as the distancebetween the modified coe?cients is increased. The modifications of c1and c2are similar to the procedure in Scheme 1. Thereconstruction of the new block, replacement of the block into the image, and extraction procedure are also similar to thatof Scheme 1.This scheme removes the reliance on rank and also improves security by using random selection of coe?cients forembedding the watermark bits. The weakness of this scheme is that the robustness of the scheme is inversely proportional. Patra et al. / Digital Signal Processing 20 (2020) 442–453 445to the watermarked image quality. A higher strength factor would increase the watermarked image to be resistant to attacks,but it also means a drop in image quality. On the other hand, too low a strength factor would reduce its robustnessagainst attacks. Moreover, the concept of using a superblock in this case causes a sharp drop in the capacity of embeddedwatermark bits. It may be insu?cient for a smallsized watermark to provide evidence of ownership of a document or animage.3. Chinese remainder theorem. The CRTThe CRT can be pactly stated as follows. Let 181。5+8 M21