a bitwise approach for chaotic image...

POSTER 2018, PRAGUE MAY 10 1

A Bitwise Approach for Chaotic Image Encryption

Jakub ORAVEC

Dept. of Electronics and Multimedia Communications, Technical University Kosice,Bozeny Nemcovej 32, 040 01 Kosice, Slovakia

[email protected]

Abstract. This paper describes image encryption algo-rithm that utilizes logistic map for generation of binary se-quences. The sequences are then used for rearrangement ofimage pixels and for calculation of their new intensities. Asbinary sequences use generated values more effectively thanusually employed quantization, the proposed solution doesnot require so many iterations of the map. Also, the pre-sented approach establishes dependencies between pixel in-tensities from individual color planes. Paper also evaluatesperformance of the proposed algorithm by commonly usedmeasures as key sensitivity and resistance against statisticaland differential attacks.

Keywordsbitwise operations, confusion, diffusion, image encryp-tion, logistic map.

1. IntroductionImage encryption algorithms based on chaotic maps

have been designed since late 1990s [1]. Multiple problemssuch as small key space or key diffusion have been alreadysolved [2–4]. However, there are still some issues that relateto chaotic image encryption. Values generated by chaoticmaps are usually floating point numbers. In order to per-form operations with integer intensities of image pixels, thegenerated values need to be processed.

There are multiple ways of the processing, includingvarious numerical operations or quantization. This step usu-ally suppresses dynamics of chaotic maps and also takessome time [5]. Approach presented in this paper tries tosolve this problem by utilizing generated values as binarysequences. Elements of these sequences are then combinedto binary words used during encryption or decryption.

The rest of this paper is organized as follows: chap-ter 2 describes techniques used in proposed algorithm. Thirdchapter explains steps of encryption and decryption algo-rithms. Fourth chapter analyses experimental results. Chap-ter 5 concludes the paper with summary of advantages anddrawbacks of proposed solution.

2. Used Techniques2.1. Logistic Map

Logistic map (LM) is one dimensional chaotic map thatmaps initial value x0 ∈ [0; 1] to iterates xn from the sameinterval [6]. LM uses a parameter r ∈ [0; 4] and is given as:

xn+1 = r · xn · (1− xn) (1)

where n denotes sequential number of iteration.

Dynamics of LM could be observed on bifurcation di-agram shown in Fig. 1. The initial value x0 was set as 0.5.

0 0.5 1 1.5 2 2.5 3 3.5 4r

0

0.2

0.4

0.6

0.8

1

nx

Fig. 1. Bifurcation diagram of logistic map.

Chaotic regime of LM starts at point where r >3.56995 [7]. However, there are some regions with pre-dictable values of xn also after this point. The most notableis area around r ∼ 3.85 which is called island of stability.

In order to achieve desired chaotic behavior of gener-ated sequences, some of the first generated values are dis-carded. These are contained in a so-called transient period.Usual size of transient period is 1,000 iterations of the map.

2.2. Double Precision Data Type

Proposed algorithm uses double precision numbers ac-cording to IEEE 754 standard [8]. The numbers are repre-sented by 64 bits in following manner: first bit denotes thesign (0 for + and 1 for −), following 11 bits denote the inte-ger part and remaining 52 bits represent the fractional part.

As LM produces iterates xn ∈ [0; 1], a sequence of 52bits could be obtained form each double precision iterate.

2 J. ORAVEC, A BITWISE APPROACH FOR IMAGE ENCRYPTION

3. Proposed SolutionProposed image encryption algorithm utilizes plaintext

images with arbitrary resolution and color depth of 8 or 24bits per pixel as its input. Operations done during encryptionor decryption could be divided into five stages:

• rearrangement of image pixels to binary matrix,• generation of pseudo-random values (based on key),• confusion (permutation of bits in binary matrix),• diffusion (calculation of new bit values) and• rearrangement of binary matrix to image.

Encryption of images uses the operations in the sameorder as the list above. Decryption changes the order of op-erations corresponding to confusion and diffusion – the dif-fusion is removed prior to removal of confusion. Used oper-ations are described in detail in following subsections.

3.1. Image Rearrangements

As the operations done during encryption or decryptionwork with binary data, the images need to be rearranged tobinary matrices before the processing. Also, these bits needto be converted back to images after the processing.

The decomposition to binary matrix is done in twosteps. If the processed image is true color, every column isrearranged to three columns (from red, green and blue colorplane) in the binary matrix. In the case that processed imageis grayscale, it is simply copied in the first step.

Second step decomposes the pixel intensities to 8 bitsby little-endian ordering scheme. These are then stored inrows of the binary matrix. Therefore each row of processedimage is represented by 8 rows in the binary matrix.

An example of image with height h and width w de-composed to the binary matrix is shown in Fig. 2. Inverse re-arrangement is done after processing of this matrix for com-putation of pixel intensities of the processed image.

3.2. Operations with Pseudo-Random Values

Pseudo-random (PR) values are generated by iteratingof the LM (1). At first, parameters par1, par2, par3 andpar4 are determined from inserted key K. As the key K has16 bytes, each parameter is given by corresponding four hex-adecimal characters – key K is divided to four portions K1,K2, K3 and K4. After the conversion to decimal notation,the parameters are quantized to interval [3.9999; 4) by (2):

pari = 3.9999 + 10−4 · Ki

1 + 232(2)

where i = 1, 2, 3, 4 and constants are used for preserving therequired interval of pari.

R[1; 1]bit 1

G[1; 1]bit 1

B[1; 1]bit 1

R[1; 2]bit 1

B[1; w]bit 1

R[1; 1]bit 2

G[1; 1]bit 2

B[1; 1]bit 2

R[1; 2]bit 2

B[1; w]bit 2

R[1; 1]bit 8

G[1; 1]bit 8

B[1; 1]bit 8

R[1; 2]bit 8

B[1; w]bit 8

R[h; 1]bit 8

G[h; 1]bit 8

B[h; 1]bit 8

R[h; 2]bit 8

B[h; w]bit 8

Fig. 2. An illustration of image decomposed to binary matrix.

When values of all four parameters are known, theLM (1) is iterated for 1,000 times for each parameter. Thefirst computation uses initial value x0 set to 0.5. After 1,000iterations with par4, additional four iterates x4,001 to x4,004are computed. These are used for calculation of four param-eters r1 to r4 by (3):

ri = 3.9999 + 10−4 · x4,000+i (3)

where i = 1, 2, 3, 4 and the constant is used for preservingthe required interval of ri.

The parameters r1 to r4 are used for computation ofPR sequences that are utilized during confusion and diffu-sion. Amounts of required iterates numitCol, numitRow andnumitDif are computed by set of equations (4):

numitCol =⌈wbin · dlog2(hbin)e

52

⌉numitRow =

⌈hbin · dlog2(wbin)e52

⌉numitDif =

⌈wbin · hbin52

⌉ (4)

where hbin and wbin are height and width of binary matrix.

Next step generates the PR sequences. All four se-quences seqcol, seqrow, seqdif1 and seqdif2 are produced bythe LM (1) with initial values x0 set to 0.5 and transient pe-riod with length of 1,000 iterates. The sequences use param-eters r1 to r4 according to their order. Amount of sequenceelements is determined as follows: seqcol has numitCol ele-ments, seqrow has numitRow elements and both seqdif1 andseqdif2 have numitDif elements.

Generated sequence elements are then rearranged tobits. As the iterates are double precision numbers, fractionalpart of each one is decomposed to 52 bits. Because maximal


possible shifts during confusion are represented by multiplebits, the bits of sequences seqcol and seqrow are combined todlog2(hbin)e or dlog2(wbin)e bit words. This step producesshift values for multiple columns or rows from each iterate,so it increases efficiency of the proposed solution.

3.3. Confusion

Rearrangement of bits representing image pixel inten-sities is done in two steps. First one shifts bits in columnsof the binary matrix and then the bits in rows of the matrixare shuffled. The amount of shifting for performed circularshifts is given by elements of sequences seqcol and seqrow.

Removal of confusion also uses these two steps, buttheir order is reversed. At first, the shuffling in rows is re-moved by circular shifts with negative sizes of elements fromseqrow. Then, the bits in columns are rearranged back bycircular shifts with sizes of negative elements from seqcol.

3.4. Diffusion

Diffusion stage breaks remaining relations between bitsin binary matrix. This task is done by two operations – com-bination with PR bit and ciphertext chaining. Both these op-erations are performed as eXclusive OR (XOR) of two bits.

In total, there are two iterations of bit scanning dur-ing diffusion. This is done for establishing dependencies be-tween all bits from matrix independently on their location. Ifonly one iteration of scanning was used, the actually scannedbit could not affect already processed bits.

The PR bits for combinations are obtained from se-quences seqdif1 and seqdif2. Ciphertext chaining uses valueof previous bit except for the first scanned bit, which usesvalue of the last bit in both diffusion iterations. This solutionresults in dependencies between successive bits.

Diffusion is removed by performing the same opera-tions as effects of XOR could be reversed by usage of an-other XOR. Therefore, only the order of iterations is in-verted, the removal of second iteration of diffusion is doneprior to removal of the first iteration.

4. Experimental ResultsA set of images used for verification of results obtained

by the proposed algorithm is shown in Fig. 3. First image,called lena has resolution of 512x512 pixels and color depthof 24 bits per pixel. Second image, called black has resolu-tion of 32x16 pixels and color depth of 8 bits per pixel. Thisimage is magnified by 32 times in all following figures forproviding detailed view on intensities of individual pixels.

lena black

Fig. 3. A set of testing images.

Hexadecimal notation of keys used for encryption anddecryption is shown in Table 1. The differences betweenindividual keys are highlighted by bold characters.

key value

K1 0x746869736973617365637265746b6579

K2 0x756869736973617365637265746b6579

K3 0x00000000000000000000000000000000

Table 1. A set of keys utilized during experiments.

4.1. Size of Key Space and Key Sensitivity

As the proposed image encryption algorithm uses keyswith length of 16 bytes, the total size of key space is givenas 25616 = 28·16 = 2128. A key space with this size couldbe considered as big enough against brute-force attacks.

Resulting encrypted images are illustrated in Fig. 4.The key sensitivity of the proposed algorithm could be ex-amined in Fig. 5. The image black was encrypted by keyK1. Then it was decrypted by keys K1 and K2. It couldbe viewed that even small modification in decryption key re-sults in incorrectly decrypted image.

4.2. Resistance against Statistical Attacks

Statistical attacks try to reveal some properties of en-cryption algorithms by comparing corresponding pairs ofplaintext and encrypted images [9]. Resistance against themis usually examined by comparison of histograms and corre-lation plots and by values of correlation coefficients ρ. Thehistograms for red color planes of image lena and its ver-sion encrypted by key K1 are shown in Fig. 6. It could beconcluded that peaks present in histogram of plaintext imageare suppressed by encryption, therefore also the relations be-tween pixel intensities of these two images are decreased.


lena encrypted by K 1 black encrypted by K 1

black encrypted by K 2 black encrypted by K 3

Fig. 4. Encrypted versions of testing images.

black encrypted by K 1and decrypted by K1

black encrypted by K 1and decrypted by K2

Fig. 5. Demonstration of key sensitivity.

Correlation plots use intensities of two adjacent pixelsas coordinates of plotted points. A plot for 1,000 randomlychosen horizontally adjacent pixel pairs from red color planeof image lena and its version encrypted by key K1 is shownin Fig. 7. While the first plot shows signs of correlation, thesecond plot has more uniform distribution of points.

Correlation coefficients for horizontally (ρhor), verti-cally (ρver) and diagonally (ρdiag) adjacent pixel pairs arecomputed by (5):

ρ =

∑numpp

i=1 (in1(i)− in1) · (in2(i)− in2)√∑numpp

i=1 (in1(i)− in1)2 ·∑numpp

i=1 (in2(i)− in2)2[-]

(5)where vectors in1 and in2 contain intensities of adjacent im-age pixels, i = 1, 2, . . . , numpp represents index of pixelpair and in denotes mean of vector in.

Obtained values of correlation coefficients ρ are in-cluded in Table 2. These values are means of 100 repeatedmeasurements with 1,000 randomly chosen pixel pairs (only50 pixel pairs for image black because of its resolution).

pixel intensity

frequen

cy0

500

1000

1500

0 50 100 150 200 250

pixel intensity

frequen

cy

0 50 100 150 200 2500

500

1000

1500

2000

2500

3000

3500

4000

lena

lena encrypted by K1

Fig. 6. Comparison of histograms before and after encryption.

intensity of pixel [l, k, 1]

inte

nsi

ty o

f p

ixe

l [l,

k+1

, 1

]

0 50 100 150 200 2500

50

100

150

200

250

intensity of pixel [l, k, 1]

inte

nsi

ty o

f p

ixe

l [l,

k+

1,

1]

0

50

100

150

200

250

0 50 100 150 200 250

lena

lena encrypted by K1

Fig. 7. Correlation plots before and after encryption.


The results from Table 2 show that the encryption sig-nificantly decreases absolute values of correlation coeffi-cients ρ for all adjacencies. Values computed for differentadjacencies are quite consistent. Also, the values producedby encryption with different keys are quite similar. Further-more, the values of correlation coefficients ρ for color planesof encrypted images do not depend on values of correlationcoefficients ρ for color planes of plaintext images.

Values of correlation coefficients ρ obtained for imageblack are not as good. This situation is caused by intensitiesof image pixels, which are all the same.

4.3. Resistance against Differential Attacks

Differential attacks introduce small changes in plain-text images and investigate the differences made after en-cryption [9]. The modified plaintext image P2 is an exactcopy of original plaintext image P1, only an intensity of onepixel is incremented or decremented by one intensity level.Then, these images are encrypted with the same setting andproduce corresponding images E1 and E2.

Resistance against differential attacks is evaluated bytwo parameters – Number of Pixel Change Ratio (NPCR)and Unified Average Changing Intensity (UACI) [10]. Theyare computed separately for each color plane by (6) and (7):

NPCR =100

h · w

h∑l=1

w∑k=1

Diff(l, k) [%]

Diff(l, k) =

{1 if E1(l, k) 6= E2(l, k)

0 otherwise

(6)

UACI =100

h · w

h∑l=1

w∑k=1

|E1(l, k)− E2(l, k)|2L − 1

[%] (7)

where h and w are height and width of color plane, l and kstand for row and vector indices and L denotes color depth.

While NPCR only counts the amount of changes be-tween images E1 and E2, UACI also takes into account in-tensities of differences between these images.

Calculated values of NPCR and UACI are illustratedin Table 2. These values are means of 100 repeated mea-surements, each one had randomly chosen location of thepixel with modified intensity. The text ”plain” in key col-umn stands for plaintext version of image (not encrypted).

The resulting values of NPCR and UACI are quite con-sistent for all color planes of image lena encrypted with dif-ferent keys. However, encryption of image black producesmore variable values. This is caused by lower resolution ofthe image, where even small amount of changes in encryptedimages significantly affects the computed values.

4.4. Computational Complexity

The computational complexity of the proposed imageencryption algorithm could be evaluated by complexitiesof its parts. Rearrangements utilize image data as vectors,which is more effective than element wise processing.

Elements of PR sequences are generated in two stages.Firstly, 4,004 iterations of the LM (1) are used for producingparameters par1 to par4. Then, simple operations whichdetermine number of required iterations are computed andthe PR sequences are generated. Finally, elements of thesesequences are decomposed to bits. The last mentioned stephelps to decrease amount of required iterations.

The confusion stage is performed on vectors of pixelintensities. This solution is more effective by means of com-putational complexity than scanning and rearrangement ofeach image pixel.

However, the complexity of diffusion stage is the sameas the one of many other algorithms. Two iterations of diffu-sion are necessary for establishing dependencies between al-ready scanned and successive pixel intensities. Element wiseprocessing is used because of ciphertext chaining with previ-ously computed values. In the case that ciphertext chainingwould be performed for multiple values at the same time (ina parallel way), there could be some values that do not de-pend on all other values.

Therefore, the computational complexity of the mostdifficult part of the proposed algorithm is equal to O(2n),where n is the number of bits representing all pixels in theprocessed image.

5. ConclusionThis paper presented an image encryption algorithm

which processes images as matrices of binary values. Theproposed solution has several advantages – as the decompo-sition to bits is done prior to confusion stage, the confusionalso helps to introduce dependencies between the pixel in-tensities by rearrangement of their bits. Also, the processingdone in diffusion stage uses all bits from fractional part ofiterates of the LM (1) which is more effective than quantiza-tion or other operations done with the iterates.

Probably the biggest drawback of the presented ap-proach is the fact that diffusion stage is performed elementwise. This is done for establishing dependencies between in-tensities of all image pixels. In the future, the computationalcomplexity could be decreased by other approaches used inthe diffusion stage. It seems that blocks of pixel intensi-ties could be processed in a parallel way. Then each blockwould be combined with all other blocks. There are someunanswered questions regarding this solution – what is theoptimal size and amount of the blocks and which operationused for the combination reaches the best numerical results.


image color plane key ρhor [-] ρver [-] ρdiag [-] NPCR [%] UACI [%]

lena

redplain 0.9727 0.9739 0.9559 not computed for a plaintext image

K1 -0.0015 -0.005 -0.0048 99.622 33.5866

K2 0.0035 -0.0028 -0.005 99.6235 33.5736

greenplain 0.9672 0.973 0.9508 not computed for a plaintext image

K1 -0.0012 0.0013 -0.0023 99.6346 33.5894

K2 0.001 -0.0034 0.0012 99.6208 33.5606

blueplain 0.9501 0.9547 0.9314 not computed for a plaintext image

K1 0.0084 0.0015 -0.0028 99.6384 33.5778

K2 0.0064 0.0047 -0.0026 99.646 33.5607

black grayscaleplain ∼ 1 ∼ 1 ∼ 1 not computed for a plaintext image

K1 -0.0614 0.0639 -0.0327 98.4375 34.3229

K2 0.0629 -0.0532 0.0492 99.2188 34.6201

Table 2. Obtained values of numerical parameters.

AcknowledgementsResearch described in the paper was supervised by As-

soc. Prof. L’. Ovsenık, DEMC FEEI TU Kosice and sup-ported by research grants KEGA 023TUKE-4/2017, VEGA1/0772/17 and APVV-0025-12.

References[1] FRIDRICH, J. Symmetric ciphers based on two-dimensional

chaotic maps. International Journal of Bifurcation andChaos, 1998, vol. 8, no. 6, p. 1259-1284. ISSN: 0218-1274.DOI: 10.1142/S021812749800098X.

[2] MAO, Y., CHEN G., LIAN, S. A novel fast image encryption schemebased on 3D chaotic baker maps. International Journal of Bifurcationand Chaos, 2004, vol. 14, no. 10, p. 3613-3624. ISSN: 0218-1274.DOI: 10.1142/S021812740401151X.

[3] GUAN, Z.-H., HUANG, F., GUAN, W. Chaos-based image encryp-tion algorithm. Physics Letters A, 2005, vol. 346, no. 1-3, p. 153-157.ISSN: 0375-9601. DOI: 10.1016/j.physleta.2005.08.006.

[4] YE, R. A novel chaos-based image encryption scheme withan efficient permutation-diffusion mechanism. Optics Communica-tions, 2011, vol. 284, no. 22, p. 5290–5298. ISSN: 0030–4018.DOI: 10.1016/j.optcom.2011.07.070.

[5] LI, S., CHEN, G., MOU, X. On the dynamical degradation of digi-tal piecewise linear chaotic maps. International Journal of Bifurcationand Chaos, 2005, vol. 15, no. 10, p. 3119-3151. ISSN: 0218-1274.DOI: 10.1142/S0218127405014052.

[6] MAY, R. Simple mathematical models with very complicated dynam-ics. Nature, 1976, vol. 261, no. 5560, p. 459-467. ISSN: 0028-0836.DOI: 10.1038/261459a0.

[7] GLEICK, J. Chaos: making a new science. London: Vintage Books,1998, 352 p. ISBN: 978-07-4938-606-1.

[8] IEEE Computer Society. IEEE standard for floating-point arithmetic, 2008, 58 p. ISBN: 978-07-3815-752-8.DOI: 10.1109/IEEESTD.2008.4610935.

[9] LEVICKY, D. Aplikovana kryptografia (in Slovak). Kosice: Elfa,2018, 437 p. ISBN: 978-80-8086-265-7.

[10] WU, Y., NOONAN, J., AGAIAN, S. NPCR and UACI randomnesstests for image encryption. Journal of Selected Areas in Telecommuni-cations, 2011, vol. 2, no. 4, p. 31-38. ISSN: 1925-2676.

About Authors. . .

Jakub ORAVEC was born in Kosice, Slovakia. Since 2015,he is a PhD student at Dept. of Electronics and MultimediaCommunications, FEEI TU Kosice. His research interestsinclude image encryption techniques, steganography and im-age processing algorithms.

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