Local Negative Base Transform and Image Scrambling

Scrambling transform is an important tool for image encryption and hiding. A new class of scrambling algorithms is obtained by exploiting negative integer as the base of number representation to express the natural numbers. Unlike Arnold transform, the proposed scrambling transform is one-dimensional and nonlinear, and an image can be shuffled by using the proposed transform to rearrange the rows and columns of the image separately or to permute the pixels of the image after scanned into a sequence of pixels; it can be also applied to shuffle certain part region of an image. Firstly, the transformation algorithm for converting nonnegative integers in base B to the corresponding integers in base −B is given in this paper, which is the computational core of scrambling transform and the basis of studying scrambling transform.Then, the three kinds of transforms are introduced, that is, negative base transform (abbreviated as NBT), modular negative base transform (MNBT), and local negative base transform (LNBT) with three parameters, where NBT is an injection and MNBT a surjection and LNBT a bijection. The minimum transform periods of LNBT are calculated for some different values of the three parameters, and the algorithm for calculating the inverse transform of LNBT is given. The image scrambled by LBNT can be recovered by the transform period or the inverse transform. Numerical experiments show that LNBT is an efficient scrambling transform and a strong operation of confusing gray values of pixels in the application of image encryption. Therefore, the proposed transform is a novel tool for information hiding and encryption of two-dimensional image and one-dimensional audio.

Most of image scrambling transforms are twodimensional, such as Arnold transform, baker map, Peano-Hilbert curve, John Conway game, knight's tour, magic square transform, and Fibonacci-Q transform [10].For two-dimensional scrambling transforms, the popular scrambling transform is Arnold transform, but it requires the sizes of images are the form of  ×  [10]; Fibonacci-Q Mathematical Problems in Engineering transform is the special case of Arnold transforms [10].Unlike Arnold transform, knight's tour [8] and Peano-Hilbert curve [19], John Conway game [15], and magic square transform [9] are not easy to calculate and are rarely used in the application of image scrambling.In general, if using John Conway game to shuffle images we first need to generate a random matrix of 0 and 1, and thus there are difficulties to restore scrambled images.In fact, Baker map is usually exploited to generate chaotic sequence on the interval [0, 1) × [0, 1) [4], and if using baker map to shuffle images we must modify the formula of baker map.
One-dimensional scrambling transform can also be used in two-dimensional image encryption and information hiding.Battisti used Fibonacci transform to construct Fibonacci-Haar matrix and then studied digital watermarking and encryption algorithms in Fibonacci-Haar domain [7], while Li and Tsai exploited Fibonacci transform to shuffle watermark information [13,14].Gunjal combined Fibonacci transform with Lucas transform to construct Fibonacci-Lucas transform which is applied to shuffle watermark image [16].Zou investigated a class of generalized Gray code and applied it to scramble color image.Two-dimensional scrambling transform or higher-dimensional scrambling transform [10] generally require that the shape of the region of scrambled image must be rectangular (i.e., the size of  × ) or even square such as Arnold transform.One-dimensional scrambling transform can not only permute a sequence of data, but also permute the pixels in an image region of any shape.In fact, as long as the pixels in the image region are scanned into a sequence in accordance with certain rule, we can exploit one-dimensional scrambling transform to permute these pixels.
In this paper, we use negative number as the base of numeral system to represent the natural numbers and then define negative base transform (abbreviated as NBT) with respect to radix −; construct modular negative base transform (abbreviated as MNBT) with respect to radix −; and modulo   followed by constructing local negative base transform (abbreviated as LNBT) whose radix is − and modulo   and shift , where  is an integer greater than 1 and ,  are natural numbers.In particular, LNBT is a bijective map from the subset {,  + 1, . . .,  +   − 1} of natural numbers to the subset itself.Since LNBT is one-dimensional and nonlinear, we can use it to shuffle one-dimensional audio signal and also to shuffle two-dimensional image.The rest of the paper is organized as follows.Section 2 studies the conversion rules and conversion algorithms from base  integers to the corresponding base − integers, which is the key of calculating NBT, MNBT, and LNBT.Section 3 gives the definitions of NBT, MNBT, and LNBT and then studies their properties and inverse transforms and last calculates the minimum transformation period of LNBT for some parameter values.Section 4 verifies the scrambling effect of LNBT by simulation experiments, which includes three kinds of scrambling algorithms: scrambling separately along xand y-direction, scrambling a sequence of pixels after image scanned to a sequence line-by-line or scanned in zigzag order, and then tests the effect against cropping attack in digital watermarking application and the effect of image encryption by shuffling the binary bits of image pixels.Section 5 analyzes the scrambling effect and compares it with other scrambling algorithms; finally, the conclusion is given.

Using Negative Base to
Express Nonnegative Integers The method of seeking remainder can achieve the conversion from decimal integers to base − integers [29].Next, we shall investigate the algorithm for converting base  integers to base − integers.Although we can apply the method in [29] to convert successfully base  integers to base − integers after the base  integers being converted to decimal integers, we still need to investigate the method of straightforward conversion of base  integers to base − integers.There are two main reasons.First, Theorem 7 needs to use the conclusion of Theorem 1, that is, we need to get the relation of the digit capacities between any base  integer and the corresponding base − integer.Second, sometimes we need to convert base  integers directly to base − integers.The proposed algorithm can achieve simply the conversion from base  integers to base − integers; especially the conversion method from binary integers to negabinary integers is simpler.We will consider the issue of the digit capacity of the base − integer converted from a known base  integer, that is, there is Theorem 1 as follows.
Proof.First,  ≥  holds obviously, followed by  must be even, since ( Next, we prove  is not greater than  + 2. Denoted by {(   −1 ⋅ ⋅ ⋅  0 ) − } min the minimum of  + 1 digits integers in base − for all satisfying   ̸ = 0.It is obvious that the value of the positive integer as expressed by ( and there is (2) The above inequality shows that (   −1 ⋅ ⋅ ⋅  0 )  cannot be converted to such a base − integer of digit capacity greater than  + 2.
Theorem 1 shows that (   −1 ⋅ ⋅ ⋅  0 )  can be converted to a base − integer with (+2)-digit at most, that is, when   be converted, there may be carries from the nth digit place to the ( + 1)th digit place or the ( + 2)th digit place, but not to the ( + 3)th digit place.To be more exact, there is the following Corollary 2.
When an integer (   −1 ⋅ ⋅ ⋅  0 )  in base  is converted to the corresponding integer (   −1 ⋅ ⋅ ⋅  0 ) − in base −, we suppose the conversion process is from right to left digit-bydigit.Then Corollary 2 shows that, when   being converted, there may be two carries need to be carried to the ( + 1)th digit place and the ( + 2)th digit place if  is even; otherwise, there is only a carry to the ( + 1)th digit place.From Corollary 2, we can get the following conversion rules, namely, Corollary 3.
(i) When  is even, if The left arrow "←" in the above denotes that the value of the expression on the right side is assigned to the variable on the left side.In the following, we shall prove that Corollary 3, in fact, only proves that the converted integer in base − is equal to the original base  integer.
Proof.If  is even and   +   ≥ , there is Corollary 2 indicates 0 ≤   ≤  − 1, and from 0 <   ≤  − 1, we have 0 ≤   +   −  <  − 1, i.e.,   ←   +   − .Corollary 2 tells us again that, when  −1 is converted, there is only a carry from the (−1)th digit place to the kth digit place, and when  −2 is converted, there are two carries at most from the ( − 2)th digit place to the ( − 1)th digit place and the th digit place.That is to say, when   being converted, the initial values  +1 =  +2 = 0. Therefore, it follows from (7) that i.e.,   ←   +   .
So we can obtain Algorithm 1 from Corollary 3.For binary, the conversion process is more concise, and the specific conversion rules are described in the following Corollary 4.

Corollary 4 (conversion rules of binary positive integers to negabinary integers). Suppose (𝑎
the converted negabinary integer, and the initial values   = 0 for  = 0, 1, . . ., +2.Then the following conversion rules hold.(i) When  is odd and (ii) When  is even and (iii) If   = 0, then the value of   keeps changeless.

Scrambling Transform Based on Negative Base
Image scrambling, which can change a certain image into another meaningless image, is a special transform from a subset of natural numbers to the subset itself.Rearranging the position of the pixels in the position space of image by permutation on a finite set of the natural numbers can achieve the purpose of scrambling image, such as Arnold transform, baker map, and knight's tour.Next, we define a new class of scrambling transforms based on the principle of number representation in negative bases.

Negative Base Transform.
Unlike the common image scrambling transform, the scrambling transform studied in this paper is one-dimensional, which is such a transform of a certain subset of the natural numbers to the subset itself.First, we define negative base transform in the following.
Definition 5 (negative base transform).For any  ∈ N, if transform  − satisfies then  − : N → N is referred to as negative base transform with respect to base -, abbreviated as NBT, and  is called transformation parameter of NBT, where   is the th digit place of the converted integer ( From Definition 5, we know that, for computing  − (), the natural number  must be converted to the form of (   −1 ⋅ ⋅ ⋅  0 ) − , and then calculate 1 gives the values of NBT with parameter −2 of the natural numbers from 0 to 15.According to Table 1, we can know that, after the natural numbers between 0 and 15 is transformed by NBT, the order of the transform  −2 () is already different from the original order of independent variable, and some of  −2 () are no longer in the set {0, 1, 2, . . ., 15}, e.g.,  −2 (7) = 27.

Modular Negative Base Transform.
Assume N[, ] = {;  ∈ N ∩ [, ]} denote a subset of the natural numbers from  to , where ,  ∈ N,  ≤ , and N denotes the set of natural numbers.In general, Image scrambling transform is a bijective map from a finite set Unfortunately, NBT is an injective map from the set of natural numbers N to itself, but not surjective, and cannot be used directly in application of image scrambling.Hence we shall make some modifications for NBT so that the modified transform is a bijective map from N[, ] to N[, ].According to the transform NBT, we shall introduce another new transform from N to N[0,   − 1] as follows.
Definition 6 (modular negative base transform).For  ∈ N, if transform  −, satisfies then transform  −, is called modular negative base transform with respect to base − and modulo   , abbreviated as MNBT, where  is an integer greater than 0.
Obviously, MNBT is a surjection from N to N[0,   − 1], but not an injection.In order to make MNBT a bijection, it is necessary to restrict the definition domain of MNBT such that Theorem 7 is true in the following.
Proof.Suppose  −, is an MNBT as defined in (11).In fact, we only need to prove that  −, is injective if  = 0, that is, prove that there is Since any integer in N[0,   − 1] can only be expressed as a -digit integer in base  at most, it follows from Theorem 1 that  1 and  2 only are represented as a ( + 2)-digit integer in base − at most.Without loss of generality, we expressed  1 and  2 as ( + 2)-digit integers in base −; if the number of digits is not enough, we fill 0 in front of them, that is, where for  = 1, 2 and  = 0, 1, . . .,  + 1,  , is possibly equal to 0. Let Assume  −, is not an injection; there is Since  1 ̸ =  2 implies  1 ̸ =  2 , then there must be If we can prove that ( 16) is not true, the fact that  −, is an injective map is proven, i.e., we complete the proof of the theorem.In accordance with (15), there is If This also contradicts  1 ,  2 ∈ N[0,   − 1] and shows that  1, ̸ =  2, is not true.Therefore we have already completed the proof of Theorem 7.
It is noteworthy that MNBT is a bijective map from , but in this case, MNBT will map some points to themselves, i.e., the transformation periods of these points are 1, for example,  −, (0) = 0. Thus MNBT cannot be used directly in the application of image scrambling and need to be modified further.In fact, combining MNBT with translational transform with respect to shift , we can obtain a permutation from N[,  +   − 1] to N[,  +   − 1], which is of course a bijective map.In Definition 8,  −,, is also referred to as LNBT with parameter (, , ).Obviously, if  = 0, LNBT is simplified as MNBT.The following corollary is easily established in accordance with Theorem 7.

Corollary 9. LNBT is a bijective map from N[𝑝, 𝑝 + 𝐵
Corollary 9 shows that LNBT is a scrambling transform from N[, +  −1] to itself.Next, we describe the calculation method of LNBT, shown as the Algorithm 2.
It is obvious that scrambling transform LNBT can map the subset  of N[,  +   − 1] to N[,  +   − 1]; moreover, LNBT can map (scatter) uniformly the entries of the subset  to the set N[, +  −1] after many times of LNBT. Figure 1 is the result after 19 times of permutation by LNBT, and it can be seen that the black pixels have been scattered uniformly onto the entire position space of the images; especially, the effect is better in the case of implementing transformation after the image is scanned into one-dimensional sequence (the form of row vector).Generally, as long as  is even and  is large enough, the scrambling effect is very good.When the pixels are permuted along x-direction, the positional relationship between adjacent pixels at y-direction cannot be changed.Likewise, when permuting the pixels along y-direction, the positional relationship between adjacent pixels at x-direction cannot be changed.Therefore, the pixels shuffled along x-and y-directions separately will result in the black pixels aligned, as shown in Figures 1(b) and 1(f).

Minimum Transformation Period and Inverse Transformation.
In the application of digital watermarking and image encryption, it is often necessary to restore the scrambled image.There are two kinds of methods for restoring scrambled image, that is, by inverse scrambling transform and by minimum transformation period.

Theorem 10 (period existence theorem for LNBT and MNBT).
There exists a positive integer  1 such that  Proof.We only prove that  −,, exists the transformation period and can also use the same method to prove  −, .Since the number of entries in the set N[,  +   −1] is finite, we only prove that, for any  ∈ N[, +  −1], there is   such that    −,, () = ; in fact, the transform period of  −,, is the common multiple of all   .
Looking at Table 3, we can get the conclusion as shown in Proposition 11.Proposition 11.The set N[,  +   − 1] is the definition domain of LNBT, where  is nonnegative.For different  and , there is the following conclusion.
(i) The minimum transformation periods of LNBT are not greater than   .If B=2,4,8 and  is odd, or  is even and  = 1, the minimum transformation periods of LNBT is equal to   .
(ii) If  is even, the difference between   and the minimum transformation period is bigger.
(iii) If is odd, the difference between   and the minimum transformation period is also bigger.Proposition 11 is very important, and in the application of image scrambling, we often take  as an even number and  as an odd number; especially, take  = 1 and  = 2.In general, the effect is better if the difference between   and the minimum transformation period is smaller.In fact,   is often a transformation period of LNBT with parameter (, , ), but is not necessarily the minimum transformation period.For example, LNBT with parameter (−2, 3, 1) is a cycle on the set N [1,8], namely, (1 2 7 4 5 6 3 8), and this shows that the minimum transformation period of LNBT with parameter (−2, 3, 1) is equal to 8. Figure 2 is the experimental results by using minimum transformation period to restore scrambled image, where the parameter values are (2, 9, 1) in (b) and (c).

Inverse Transforms of MNBT and LNBT.
It is not easy to compute the minimum transformation period of MNBT and LNBT for different parameters (, , ); on the other hand, although the minimum transformation period is sometimes known, but the minimum transformation period is possibly large, so that the computational complexity of restoring scrambled image becomes large.For example, if we use LNBT with parameters (2, 18, 1) to shuffle the pixels of an image of size 512 × 512 after scanned into a sequence, then the minimum transformation period is 262144.Therefore, the calculation is sometimes difficult by using the periodicity of LNBT to restore scrambled image, while the calculation of restoring scrambled image is more concise and faster by inverse transform of LNBT.
Figure 3 is the experimental results of image scrambled by LNBT and image restored by inverse LNBT, where the parameter is (2, 9, 1) and the size of image is 512×512.Image (b) is the scrambled image by 19 times of LNBT along x-and y-direction separately and image (c) is the restored image by 19 times of inverse transform of LNBT. =  at x-direction and take  as an odd number.For example, Figure 4(a) is an image of size 216×256; then we can select the parameter (2, 8, 1) at x-direction and (6, 3, 1) at y-direction.Figure 4(e) is another image of size 256×256; then we can select the same parameter (2, 8, 1) both x-and y-direction.From Figure 4, we find that the scrambled pixels are aligned along x-and ydirection, respectively.

Scrambling after Scanned into a Sequence
(i) Scan Line-by-Line.In order to eliminate the alignment phenomenon of pixels as described above after image was scrambled along x-and y-direction separately, the easiest method is that we use LNBT to permute the sequence of pixels after two-dimensional image is scanned into a onedimensional sequence line-by-line.Figure 5(a) is an image of size 216×216, and after being scanned into a one-dimensional vector, the length of the corresponding one-dimension vector is 6 6 ; thus we select the parameter (6, 6, 1) to shuffle the pixels.Figure 5(e) is another image of size 256×256, and we can select the parameter (2, 16, 1) to shuffle the pixels after being scanned into a one-dimension vector.From Figures 4 and 5, we find that the smaller the parameter B is, the better the scrambling effect is.
(2) Scan in Zigzag Order.After scanning a two-dimension image into a sequence of pixels line-by-line, if taking  = 2 we can obtain a good scrambling result by many times of LNBT, as shown in Figure 5(h), but if selecting  = 6, the scrambling effect is not very good.In Figure 6, we first use the zigzag scan technique to change a two-dimension image into the corresponding one-dimension vector and then use LNBT to permute this sequence of pixels; last, we use the same zigzag order to arrange the scrambled sequence and  obtain the corresponding scrambled image.Compared with Figure 5, we find the effect of image scrambling becomes better in Figure 6.

Irregular Region Scrambling of Image.
We consider two cases that the size of image cannot express as the form of 2 and the image region to be shuffled is irregular.Since LNBT is a kind of one-dimensional scrambling transform from N[,  +   − 1] to N[,  +   − 1], if using LNBT to permute two-dimensional image, we must change the image into the corresponding one-dimensional sequence of pixels, and then we permute the sequence of pixels; of course, we can also permute an image along the horizontal and vertical direction separately.Therefore, we can shuffle image region of any shape and images of any size.In the application of part information hiding and encryption of image (e.g., sensitive area scrambling) [30][31][32], unlike Arnold transform, LNBT can shuffle image region of any shape.(2) Case 2: Irregular Shape.Sometimes we require shuffling a certain region of image [30][31][32]; in this case, the region to be shuffled may be irregular with respect to size or shape, and therefore it is difficult for two-dimensional scrambling transform to shuffle such irregular region.Unlike Arnold transform, baker map, John Conway game, knight's tour,   magic square transform, and Peano-Hilbert curve, LNBT can be exploited to shuffle region with irregular shape of image, and specific operations are divided into two steps.First, the pixels of the region of image to be scrambled are scanned as one-dimensional sequence; second, use the above scrambling method for irregular size to shuffle the one-dimensional sequence.In Figure 8, we scan the pixels of image region into one-dimensional sequence and then permute these pixels of the sequence.plane not only makes the scrambled image meaningless but also changes the statistical properties of the original image.Thus this shows that LNBT is also a tool of image encryption.

Digital Watermarking against Cropping
Attack.In the robust watermarking of digital images, image scrambling is a powerful method for resisting cropping attack.LSB (least significant bit) algorithm is a simple digital watermarking technology, but as a large amount of data hiding method, LSB has a significant position.watermarked image (b).Images (e), (g), (i), and (k) are the extracted watermarks from the watermarked imaged after attacked by cropping, and Figure 10 shows that LNBT is effective against cropping attack.

Correlations between Adjacent
Pixels.The goal of image scrambling is to remove the correlation between the pixels of a plaintext image, and therefore the correlation value between adjacent pixels of image is an important index for evaluating the performance of the image scrambling algorithm.Assuming that   and   are two sequences consisted of horizontal or vertical or diagonally adjacent pixels, their correlation value can be calculated by where  is the length of sequences.Obviously, if the correlation value is close to 1, it indicates that there is a higher correlation between adjacent pixels.If the correlation value is close to 0, then the two adjacent pixel sequences have a lower correlation.Therefore, the smaller the correlation value is, the better the performance of the scrambling algorithm is.
To measure the performance of LNBT scrambling algorithm by using the correlation values of neighboring pixel sequences.First, we use other common scrambling algorithms and the proposed algorithm in this paper to perform a 32-round scrambling transformation on the Lena image.Then pick out the largest value of GDD (see Section 5.2) in the 32-round transforms as the last results of each scrambling algorithm for comparison.Finally, we randomly select 5000 pairs of horizontally adjacent pixels, vertically adjacent pixels, and diagonally adjacent pixels and then calculate the correlation values of the neighboring pixel sequences of the three directions in the original image and the correlation values of the three directions in the scrambled images, respectively.Columns 4, 5, and 6 of Table 4 are correlation values of adjacent pixel sequences of the three directions in the original image and in the scrambled images, where LNBT-1 denotes the LNBT scrambling algorithm of permuting separately along x-and y-direction; LNBT-2 denotes the LNBT scrambling algorithm after scanned lineby-line; LNBT-3 denotes the LNBT scrambling algorithm after scanned in zigzag order.We can know from Table 4 that the LNBT scrambling transform can successfully remove the correlation between adjacent pixels.Figures 11(a), 11(b), and 11(c) display the gray-scale distributions of 5000 pairs of horizontally adjacent pixels, of vertically adjacent pixels, and of diagonally adjacent pixels, respectively, which are selected randomly from original image.(d), (e), and (f) are the gray-scale distributions of an adjacent pixel sequence pairs for horizontal, vertical and diagonal directions in the scrambled image using LNBT-1; (g), (h), and (i) show the gray distributions of an adjacent pixel sequence pairs for the three directions in the scrambled image using LNBT-2; (j), (k), and (l) are the gray-scale distributions using LNBT-3.The results of Table 4 and Figure 11 show that LNBT can successfully remove the correlation between adjacent pixels of plaintext image 5.2.Scrambling Degree.Unlike correlation values, scrambling degree can be used to evaluate the correlation of adjacent pixels of a whole scrambled image.If a scrambling algorithm can obtain a large scrambling degree, the algorithm can achieve high confusion and diffusion properties.Image scrambling degree is determined by the gray difference between adjacent pixels [25].Assuming that (, ) is the gray value of the pixel at position (, ), the gray difference of pixel (, ) can be calculated using [23][24][25]  (, ) Assume that () represents the average gray-scale difference of plaintext image and   () represents the average gray-scale of the scrambled image.Then, according to paper [25], image scrambling degree can be calculated as follows: From (32), it can be seen that the value of  is between -1 and 1, and the larger the value of  is, the better the performance of the scrambling algorithm is.The data in the third column of Table 4 is the maximum value in 32round transform for each scrambling algorithm.The remark below Table 4 shows that GDD does not accurately evaluate the performance of a scrambling algorithm, and sometimes there is a large visual difference.

Figure 1 :
Figure 1: Images of size 128 × 128 and black area of size 16 × 16, 19 times of LNBT: (a) black pixels in the center of an image, (e) black pixels in the upper-left of another image; (b) and (f) are the scrambled images of (a) and (e) with parameter (2, 7, 1) along x-and y-direction, respectively, (c) and (g) are the scrambled images of (a) and (e) with parameter (2, 14, 1) after scanned into a sequence line-by-line, and (d) and (h) are the scrambled images of (a) and (e) with parameter (2, 14, 1) after scanned into a sequence in zigzag order.

Figure 2 :
Figure 2: Restoring scrambled images by the minimum transformation period: (a) is an original images of size 512×512; (b) is the scrambled images of (a) using 22 times of LNBT to shuffle the pixels along x-and y-direction separately; (c) is the restored image using 490 times of LNBT to permute the pixels along x-and y-direction separately from (b).
(i) Case 1: Irregular Size.If the size of image cannot be expressed as the form of   1 1 ×   2 size in this paper.Suppose the size of an image is  × , and for any positive integers  1 and  1 satisfying  ̸ =   1 1 or any positive integers  2 and  2 satisfying  ̸ =   2 2 .Without loss of generality, we suppose  cannot be expressed as the form of   , then there exist   ,   , , and   such that  = ×    +   where 1 ≤   <     and thus the set N[1, ] can be divided into  + 1 subsets.For the first X subsets, we select LNBT of parameters (  ,   , ) to shuffle N[    +1, (+1)    ] where  =     +1,  = 0, 1, . . ., −1.For the last subset N[    + 1, ], first, extend the left end of N[    + 1, ] to N[ −     + 1, ]; then shuffle the extended subset N[ −     + 1, ] after the first  subsets were shuffled.

Figure 7
is the scrambling result of the image of size 720×648.First, permute x-coordinate between 1 and 512 as shown in Figure 7(b), and next, permute x-coordinate between 209 and 720 as shown in Figure 7(c).The results of permuting y-coordinates are shown in Figures 7(d) and 7(e).
) and 12(j) are the scrambled images of the third round of 2x2 sampling transform and of the 29th round of LNBT-1, respectively.
If selecting a negative integer − ( > 1) as the base of number representation, we only need  figures to express all nonnegative integers, that is, we can use the symbols {0, 1, 2, ...} to express all nonnegative integers[29].In this paper, for convenience, the notation (   −1 ⋅ ⋅ ⋅  0 )  is used to denote an ( + 1)-digit integer in base , where  is an integer of absolute value greater than 1, and   = 0, 1, . . ., || − 1,  = 0, 1, . . ., .For the convenience of description, we also call   the th digit of (   −1 ⋅ ⋅ ⋅  0 ) 2.1.Principle of NumberRepresentation with Negative Base..The rest of this paper, we always suppose  is an integer greater than 1.Next, we shall discuss the calculation for conversion from nonnegative integers in base  to integers in base −.2.2.Conversion Calculation from Base B Integer to Base −BInteger.Assuming that  is an integer greater than 1, a base − integer can be converted directly to the corresponding decimal integer expressed by the sum of the product which the th power of − is multiplied by the th digit value, as do a base  integer to decimal integer, namely,

Table 1 :
NBT of the integers between 0 and 15 with parameter −2.

Table 2 :
The results of LNBT.

Table 4 :
GDD and correlation values.Remark: visually, the scrambling performance of the third round of 2x2 sampling transform is best, and the 29th round of LNBT-1 has the best scrambling effect.Figures12(i *