New approach to image encryption
Trees-Juen ChuangJa-Chen Lin*
National Chiao Tung University
Department of Computer and Information Science Hsinchu, Taiwan 30050, Republic of China
Abstract. A new cryptographic method for encrypting still images is proposed. This method is designed on the basis of the base-switching lossless image compression algorithm. The method there-fore simultaneously possesses both image encryption and lossless compression abilities. A given image is first partitioned into nonover-lapping fixed-size subimages, and each subimage will then have its own base value. These subimages are then encoded and encrypted one by one according to the base values. By choosing the function to encrypt the base value, there are (128!)t [or (128!)3t] possible ways to encrypt a gray-scaled (color) image if t layers are used in the encryption system. The theoretical analysis needed to support the proposed encryption method is provided, and the experimental results are also presented. © 1998 SPIE and IS&T.
[S1017-9909(98)01202-1] 1 Introduction
Some special images need encryption before transmission or storage. Examples include military and medical images. There are three types of data encryption, namely, block~or transposition!,1 stream ~or substitution!,2,3 and product cipher.1 Various data encryption techniques such as the DES cipher4,5~a kind of block cipher! have been applied to image encryption. We review below some recent image encryption techniques. References 6–8 are all based on SCAN language9,10 and Ref. 11 is a kind of stream cipher based on the quasi-m array,12 Gold code array,12 and fast Fourier transform~FFT!. In optical systems, the work pro-posed recently by Refrigior and Javidi13is based on random phase encoding. Note that if the random phase mask is not band limited the speckle noise can destroy the optical re-construction of the encrypted image. The speckle-free kinoform encrypted by smoothed random phase masks is thus required.14 In TV signal encryption, the scramble method15 is one of the most commonly used encryption methods, and it encrypts each arbitrary area by means of a kind of block cipher. The tool for image encryption~TIE! system developed by Koch~see Ref. 16 for a short review of the TIE! and the multiresolution image encryption pro-posed by Macq and Quisquater17,18are other new methods in digital TV broadcasting. TIE allows hierarchical access to image data and can create a ‘‘demoversion’’ of an image which can be distributed in low resolution. The
recon-structed image in high resolution can be delivered and re-stored by the correct password. The methods proposed by Macq and Quisquater are used in mobile image transmis-sion, and the image is encrypted by permuting details of the multiresolution transform coefficients. Note that such per-mutation should be executed in order to obtain a ciphered correlated image. Also note that all these kinds of TV sig-nal encryption methods must have at least two properties: transparency and transcodability~Ref. 18 describes these in detail!.
For still image encryption, we propose here a new method which can have the function of both lossless image compression and encryption simultaneously. Although Refs. 7 and 11 also use image compression, Ref. 7 is only moderately secure for encrypting quite important images ~only 24log npossibilities to encrypt an image of n3n
reso-lution! and the other is a kind of lossy image compression. Our proposed method provides not only lossless compres-sion but also a high level of security. There are (128!)t possible ways to encrypt a gray-scaled image, and (128!)3t possible ways to encrypt a color image. Here, the positive integer t represents the number of layers that we used, as will be explained in the last paragraph of Sec. 3.2. In gen-eral, 1<t<(m3n/333) if the image is m3n in size. Note that the number 128! is large as compared with the 1016of DES.5
Our method is a kind of stream cipher based on the base-switching ~BS! algorithm19 which is a new lossless image compression algorithm. Because of the compression property, the proposed method can save storage space while increasing security. The remaining part of the paper is organized as follows. In Sec. 2, the overview of the pro-posed system is presented. In Sec. 3, a review of the BS algorithm is given in Sec. 3.1, followed by the proposed encryption/decryption techniques ~Secs. 3.2 and 3.3, re-spectively!. Experimental results, a time complexity analy-sis, and a comparison with some recent works reported be-tween 1992 and 1995 are illustrated in Sec. 4. The conclusions are stated briefly in Sec. 4.
2 Overview of the Proposed System
As shown in Fig. 1, we first divide the input image ~gray-scaled data! into subimages, i.e., blocks, k3k in size. ~Throughout this paper, the subimage size that we use is 333 for efficiency of the compression ratio.! The subim-ages are then processed one by one. In the part of the
trans-*Author to whom all correspondence should be sent.
Paper 96-057 received Oct. 28, 1996; revised manuscript received Sept. 30, 1997; accepted for publication Oct. 7, 1997.
mitter, for each subimage, we use the BS algorithm to en-code the subimage and then encrypt it using the proposed encryption technique. The encoded and encrypted subim-age will then be transmitted to the recipient. The subimsubim-ages are recovered in the recipient end one by one. For each subimage, the recipient first decrypts the subimage and then decodes it using the BS decoding algorithm. Of course, the recipient must know in advance the encrypted keys before decrypting and decoding the subimage received.
In the next section, we sequentially present the three main parts shown in Fig. 1: BS encoding, encryption, and decryption.~An explanation of BS decoding is omitted be-cause it is not essential.! Some definitions and notations are also described and discussed in the next section.
3 The Proposed Method
3.1 A Review of the BS Transform
The base-switching transform was originally introduced to compress images.19 For the reader’s benefit, we review BS in this subsection. Let A be a given 333 subimage, whose nine gray values are g0,g1,... ,g8 ~see Fig. 2!. Define the
‘‘minimum’’ m, ‘‘base’’ b, and the ‘‘value-reduced subim-age’’ A
8
~see Fig. 3! of subimage A bym5 min 0<i<8 gi, ~1! b5 max 0<i<8 gi2 min 0<i<8 gi11, ~2! and A3
8
335A3332m3I333, ~3!respectively. Here, each of the nine elements of I333 is 1.
Note that~3! means that
gi
8
5gi2m for all i50,1,2,... ,8. ~4!Also note that min 0<i<8 gi
8
50 and max 0<i<8 gi8
5b21. ~5!Therefore, the nine-dimensional vector A
8
5(g08
,g18
,... ,g88
) can be treated as a nine-digit number(g0
8
g18
¯g88
)b in the base-b number system. Forconve-nience, let V333 be the collection of all 333 subimages
A
8
, and B be the base set$1,2,3,... ,256%. Then we define an integer-value function f :V3333B→$non-negative decimal integers% byFig. 1 The system overview.
Fig. 2 An arbitrarily given subimageA.
f~A
8
,b!5the decimal integer equivalent to the base-b number ~g0
8
g18
¯g88
!b 5(
i50 8 gi8
3bi, ~6! 5$¯@~g08
3b1g18
!3b1g28
#3b1¯%3b1g88
. ~7!The following two properties are both proved in Ref. 19. Property 1. The inequality f (A
8
,b),b9 always holds. Property 2. For each base b, and for each given integerl satisfying b21<l<(i850(b21)3bi5b921, we can find a unique 333 A8
such that f (A8
,b)5l.By Property 1, the number of bits needed to store the deci-mal integer f (A
8
,b) using a binary number is therefore at mostZb5dlog2 b9e. ~8!
When we want to reconstruct A
8
5(g08
,g18
,... ,g88
), all we have to do is to switch that binary ~base-2! number to a base-b number (g08
g18
¯g88
)b. Once A8
is known, Eq. ~4!implies that A can be obtained quickly by adding m to each pixel value of A
8
.3.1.1 Several possible ways to encode subimage A according to the value of base b
As stated in ~5!, for each value-reduced subimage A
8
5(g08
,g18
,... ,g88
), we always havemin$g0
8
,g18
,... ,g88
%50, ~9!and
max$g0
8
,g18
,... ,g88
%5b21. ~10! Therefore, at least one of the nine pixels of A8
has a gray value of 0, and at least one of the nine pixels of A8
has a gray value of b21. As a result, there are at least two ways to encode A. The first way is to store directly$b;m% and a binary equivalent of ~g0
8
g18
¯g88
!b. ~11!The second way is to store
$b;m;imin;imax%
and
$giuiÞimin,iÞimax%. ~12!
@Here, iminP$0,1,... ,8% is such that gi
8
min50, and imaxP$0,1,... ,8% is such that gi
max
8
5b21. If more than one i in $0,1,... ,8% have their gi8
value being 0, say, g28
5g38
5g58
50, then use the smallest i as imin ~hence, imin52 in this
case!. An analogous statement makes imaxunique.# Because
there are 938572 possible combinations of the pair (imin,imax), we can use dlog(938)e57 bits to indicate the
positions of the pair (imin,imax) in the storage system ~12!.
Equation~12! can therefore be simplified to
$b;m; P~imin,imax!%, ~13!
and a binary equivalent of the seven-digit base-b number (giuiÞimin,iÞimax)b.
Here, P(imin,imax) is a 7-bit index used to get the pair
imin and imax. To know when the storage system ~13! can
save more bits than~11! does, note that first both ~13! and ~11! need to store b and m; second, ~11! needs dlog2b9e bits
to represent a nine-digit number g0
8
g18
¯g88
in the base-b number system, whereas ~13! needs 7 bits to indicate the location of the (imin,imax) pair, and another@log2b7# bits toencode a seven-digit number g0
8
g18
¯g88
~with gimin
8
andgi
max
8
taken away! in the base-b number system. ~gimin
8
andgi
max
8
need no storage if we know the position of imin andimax ~see Fig. 4!, because gi
min
8
50 and gi8
max5b21 alwayshold according to~9! and ~10!.! Property 3 is used to com-pare the storage system ~11! and ~13!. The proof of this property used the fact that 71log2b7.log2b9if and only if
b,23.5511.314. The details of this are omitted.
Property 3. Using the storage system ~13! is better than using the storage system~11! if and only if b.11.314.
3.1.2 Formats [to encode subimage A5(g0,... ,g8)] Rule 1: If bP$1,2,...,11%, then the coding format is
Rule 2: If bP$12,13,... ,127%, then the coding format is Fig. 4 If the position ofimin andimaxare known, then only seven
gray values are needed to be encoded. In this exampleimin53 and
Rule 3: If bP$128,129,... ,256%, then we use a ‘‘dummy’’ base b5128 and the coding format is
In the above, Rule 3 is needed because we cannot use 7 bits to distinguish all 256 possible values of b. Some read-ers might suggest the use of an 8-bit number to store b so that rule 3 is no longer needed because all b that are greater than 11 can then use rule 2. But we found this would in fact increase the storage space, and the reason is that for most of the images, more than 99% of the subimages have b less than 128~thus, wasting one more bit in both Rules 1 and 2 is not of merit!.
3.2 Encryption
After the encoding stage ~using Rules 1–3!, we could get each encoded subimage and then encrypt it. Our encryption approach is to cipher the base value b for each encoded subimage. Note that unless a cipher ‘‘stealer’’ can decipher the base value b, he cannot even know how many of the subsequent bits should be used as the data for the current subimage ~see Rules 1–3!, not to mention recovery of the subimage. The base value b is in the range 1<b<128; we therefore require that the encrypted value f (b) also stays in the range 1–128 for simplicity. Many reported methods could be applied to the encryption of b. The simplest one is the so-called Caesar cipher20
f~b![@~b211k!mod 128#11, ~14!
where k is the key. Beside Eq. ~14!, there are many other b→ f (b) mapping functions that can be applied to this en-cryption system. Note that all these mapping functions are bijective ~one to one and onto!, and both b and f (b) (PZ1) are in the range of@1–128#. For example,
f~b![@~k1~b21!1k0!mod 128#11, ~15!
where k0and k1are the keys~affine transformation!.20
An-other example is to let
f~b![@~k01k1~b21!1k2~b21!21¯1km~b
21!m!mod 128#11, ~16!
where k0– kmare the keys~polynomial transformation20 of
degree m!. Some may also use the one-time pad or the Vernam cipher,21 i.e., let
f~b!5@~b21!%k#11, ~17!
where k5(k0k1,¯k6)2 is a 7-bit key. At any rate, we en-crypt an image by mapping the base values b. Because b P$1 – 128% ~we use 7 bits to denote b!, and because there are 128! possible functions f that map $1–128% onto $1– 128% in a one-to-one manner, our system has 128! possible ways to encrypt a two-dimensional~2D! gray-scaled image. The security level can even be higher using the method stated below. To encrypt the base values$b1,... ,bp,...%of
many consecutive subimages by a more powerful function F, we may define
H
F~b1![ f1~b1!F~bp![$@ f1~bp!1 f2~bp21!#mod 128%11, ~18!
where f1 and f2 are any mapping functions chosen from,
say, Eqs.~14!–~17!. Note that b1 is the base value of the
first subimage, whereas bp ( p.1) is the base value of the
pth subimage. The key will be K5$kf1,kf2% if kf1 and kf2
are the keys of mapping functions f1 and f2, respectively.
It is easy to see that there are (128!)2 ways to encrypt an image because each f1 and f2 has 128! possible choices.
@The two-layer scheme ~18! can be improved further to a t-layer scheme F~bp![
FS
(
q51 p fq~bp2q11!D
mod 128G
11 for p,t, ~19! F~bp![FS
(
q51 t fq~bp2q11!D
mod 128G
11 for p>t.Of course, t should be smaller than the total number of subimages and the key. K5$kf1,kf2,¯ ,kft% is such that
each kfq is the key of corresponding fq. Equation ~18!
could be regarded as a special case of Eq.~19! with t52.# If Eq.~19! is used in our encryption system, then there are (128!)t possible ways to encrypt a gray-scaled image. The security level is high. Also note that if the image is color our encryption system could be applied three times to the three color components, respectively. We therefore have (128!)3t possible ways to encrypt a color image.
3.3 Decryption
Without the loss of generality, we show below how to de-crypt the first subimage and the pth subimage ( p.1) of an image which has been encoded and encrypted earlier using the techniques introduced in Secs. 3.1 and 3.2. ~The re-maining subimages can be decrypted similarly.!
We first check the first 7 bits of the received data to get the value of F(b1) where b1 is the base value of the first
subimage. b15F21@F(b1)# can then be known ~as
ex-plained in the next paragraph!, and the BS decoding can then use the base value b1 to reconstruct~decode! the first
subimage.
Because F(b1)5@ f1(b1)mod 128#11 according to Eq.
~19!, b1 could be solved easily using b15@ f121(F(b1)
21)#mod 128 where f1is a given mapping function whose
~14!–~17!.# Note that, because f1 is bijective, f121 always
exists. We discuss below how to obtain bp by Eq.~19! if
p.1.
Assume that the values of F(bp) and b1– bp21 ~which
have been known by decryption! are all given. We wish to find the value of bp using the given key K 5$kf
1,kf2,... ,kft%. Note that~19! reads F~bp![
FS
(
q51 t fq~bp2q11!D
mod 128G
11 [$@ f1~bp!1 f2~bp21!1 f3~bp22!1¯ 1 f1~bp2t11!#mod 128%11. ~20!Because b1– bp21have been decrypted, we can use the key
K to compute the value of C5 f2(bp21)1 f3(bp22)1¯
1 ft(bp2t11). Therefore, Eq. ~20! can be rewritten as
F(bp)[@ f1(bp)1C#mod 12811. As a result, f1(bp)
[@F(bp)212C#mod 128, i.e.,
bp[ f121$@F~bp!212C#mod 128%. ~21!
Once the value of bpis known, the BS decoding system can then be used to reconstruct the pth subimage. Details of the decoding are trivial and, hence, are omitted.
In this section, we present the experimental results ob-tained by applying the proposed encryption method to two 256-level gray scale images ~the ‘‘Lena’’ and the ‘‘Jet’’
shown in Fig. 5!. Figures 6 and 7 show the incorrectly decrypted images of Fig. 5, and Fig. 8 shows the success-fully decrypted results. Note that the successsuccess-fully recon-structed images in Fig. 8 are lossless. Also note that we purposely shifted the decrypted base values b by 1, i.e., b→b11, to get Figs. 6~a! and 6~b!, and it can be seen that the resulting images are very unrecognizable. We can also see that the image size of the images shown in Figs. 6 and 7 is not even equal to that of the original images shown in Fig. 5 ~the last several lines of the incorrectly decrypted images in Figs. 6 and 7 are filled with black!. The reason is as explained in the first paragraph of Sec. 3.2.
Table 1 compares the possible decryption ways of some recently published encryption methods and ours. In this pa-per, the size of each test image is 5123512. Proof that ~5123512!!!~128!!b512/3c3b512/3c
is given in the Appendix. Also note that our method has the extra advantage of lossless compression, and the bit rates of the compressed and encrypted images@gray-scaled images ~8 bits/pixel! and color images ~24 bits/pixel!# are illus-trated in Table 2.
We then discuss in this paragraph the time complexity of the proposed method. In compression, 3.44–4.55 clock cycles ~the average is 3.94 clock cycles! are required to encode a pixel~a more detailed analysis is presented in Ref. 19!. In encryption, if a one-layer ~two-layer! encryption scheme is used, three additions/subtractions and one modu-lus~five additions/subtractions, one multiplication, and one Fig. 5 The original images: (a) Lena and (b) Jet.
Fig. 6 The incorrectly decrypted images of Figs. 5(a) and 5(b) with t51. [Thefis that used in Eq. (14).]
Fig. 7 The incorrectly decrypted images of Figs. 5(a) and 5(b) with t52. [Thef1andf2were used, respectively, in Eqs. (14) and (15).]
Fig. 8 The successfully decrypted images of Figs. 5(a) Lena and
modulus! are required to encrypt a 333-on line subimage, and 7/9 clock cycles ~10/9 clock cycles! are therefore needed to encrypt a pixel @the definition of clock cycle under modern very large scale integrated ~VLSI! technol-ogy can be found in Chap. 2 of Ref. 22#. In other words, the proposed method using a one-layer~two-layer! encryption scheme takes a total of 4.72~5.05! clock cycles to compress and encrypt a pixel. In general, processing all 5123512 pixels of an image required about 9.29 ~9.41! s in a SUN Sparc 10 workstation.
4 Concluding Remarks
A new method for image encryption was proposed in this paper. The method is based on the BS lossless image com-pression scheme; therefore, both image encryption and lossless compression can be obtained simultaneously. There are (128!)t @or (128!)3t# possible ways to encrypt a gray-scaled~color! image. The fact that decryption without the correct decrypt keys would even fail in knowing the exact image size increases the difficulty for image stealers. Also note that the term (128!)t @or (128!)3t# can also be raised up to be (256!)t@or (256!)3t# if we discard Rule 3 in Sec. 3.1.2 and extend Rule 2 to cover all bP$12,... ,256%. Of course, this increase in the possible ways to encrypt an image would reduce the image compression ratio some-what, and the reason for this is given at the end of Sec. 3.1.2.
Note that readers may also combine the proposed method with other image encryption methods based on
SCAN language, such as Refs. 6 or 8, to improve security further. Details of this are omitted here.
Appendix ~The proof of ~5123512!!!~128!!b512/3c3b512/3c! ~5123512!!5~262144!! 5~10326213114!! 5
S
)
j51 10 jD
3S
)
j511 20 jD
••••••S
)
j5262121 262130 jDS
)
j5262131 262144 jD
5S
)
i51 26213)
j5~i21!31011 i310 jD
S
)
j5262131 262144 jD
~128!!b512/3c3b512/3c5~128!!1703170 5~128!!28900 5~128!!2621312687 5S
)
i51 26213 128!D
~128!!2687 since)
j5262131 262144 j,1076!128!!~128!!2687 and)
j5~i21!31011 i310 j,)
j5262131 262144 j!~128!!for each i51,2,¯ ,26213, we have ~5123512!!!~128!!b512/3c3b512/3c. Acknowledgments
This work was supported by the National Science Council, Republic of China, under Contract No. NSC 86-2213-E009-108. The authors thank the referees for the helpful suggestions and clear guidelines for revising the paper.
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Table 2 The bit rates (bits/pixel) of the proposed method.
Image
Scale
Gray scaled Color
Lena 5.29 14.72
Jet 5.00 13.11
Table 1 The comparisons of some encryption methods reported
recently.
Method
Possible decrypted
waysa (lossless/lossy)Compression
(94’ ICS)b 249 Losslessf
(93’ JEI)c 3070083(511)3(511) Lossy
(92’ PR)dand (95’ JEI)e (5123512)! No compression
Our method
S
t5b
512 3c
3b
512 3c
D
(128!)b512/3c3b512/3c LosslessaThe test image is gray scaled and 5123512 in size. bReference 7.
cReference 11. dReference 6. eReference 8.
fThe bit rate of (94’ICS) is about 7.93 bits/pixel for a gray-scaled
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Trees-Juen Chuang received his BS
de-gree from Soochow University, Taiwan, in 1992. Since 1993 he has been studying toward his PhD degree and is currently a PhD candidate in the Computer and Infor-mation Science Department of National Chiao Tung University. His recent re-search interests include pattern recogni-tion, image processing, and image encryp-tion.
Ja-Chen Lin received his BS degree in
computer science in 1977 and his MS de-gree in applied mathematics in 1979, both from National Chiao Tung University, Tai-wan. In 1988 he received his PhD degree in mathematics from Purdue University. From 1981 to 1982 he was an instructor at National Chiao Tung University and from 1984 to 1988 he was a graduate instructor at Purdue University. He joined the De-partment of Computer and Information Science at National Chiao Tung University in August 1988, and is currently a Professor there. His recent research interests include pattern recognition, image processing, and parallel computing. Dr. Lin is a member of the Phi-Tau-Phi Scholastic Honor Society, the Image Processing and Pattern Recognition Society, and the IEEE Computer Society.
