 Research
 Open Access
Assessing JPEG2000 encryption with keydependent wavelet packets
 Dominik Engel^{1, 2},
 Thomas Stütz^{1, 2}Email author and
 Andreas Uhl^{2}
https://doi.org/10.1186/1687417X20122
© Engel et al; licensee Springer. 2012
Received: 3 June 2011
Accepted: 13 April 2012
Published: 13 April 2012
Abstract
We analyze and discuss encryption schemes for JPEG2000 based on the wavelet packet transform with a keydependent subband structure. These schemes have been assumed to reduce the runtime complexity of encryption and compression. In addition to this "lightweight" nature, other advantages like encrypted domain signal processing have been reported. We systematically analyze encryption approaches based on keydependent subband structures in terms of their impact on compression performance, their computational complexity and the level of security they provide as compared to more classical techniques. Furthermore, we analyze the prerequisites and settings in which the previously reported advantages actually hold and in which settings no advantages can be observed. As a final outcome it has to be stated that the compression integrated encryption approach based on the idea of secret wavelet packets can not be recommended.
Keywords
1. Introduction
For securing multimedia datalike any other type of datafull encryption with a stateoftheart cipher, is the most secure option. However, in the area of multimedia many applications do not require the level of security this option provides, and seek a tradeoff in security to enable other requirements, including low processing demands, retaining bitstream compliance and scalability, and the support for increased functionality, such as, transparent encryption [1] and region of interest (ROI) encryption, or signal processing in the encrypted domain (adaptation, searching, watermarking, ...).
JPEG2000 is the most recent and comprehensive suite of standards for scalable coding of visual data [2, 3]. JPEG2000 filled areas of application that JPEG could not provide for, especially where applications require a scalable representation of the visual data. Recently JPEG2000 has evolved into the format of choice for many specialized and high end applications. For example, the Digital Cinema Initiative (DCI), an entity created by seven major motion picture studios, has adopted JPEG2000 as the compression standard in their specification for a unified Digital Cinema System [4]. As a second example, in 2002, the Digital Imaging and Communications in Medicine (DICOM) committee approved the final text of DICOM Supplement 61, marking the inclusion of Part 1 of JPEG2000 in DICOM (ISO 12052). Furthermore, in the ISO/IEC 19794 standard on Biometric Data Interchange Formats JPEG2000 is included for lossy compression, in the most recently published version (ISO/IEC FDIS 197946 as of August 2010) as the only format for iris image data.
Security techniques specifically tailored to the needs of scalable representation in general and JPEG2000 in particular have been proposed recently, e.g., [5–10]. An overview and discussion of the proposed approaches in the context of JPEG2000 can be found in [11]. JPEG2000 security is discussed in JPEG2000 Part 8 [12, 13]. This part has the title "Secure JPEG 2000" and is referred to as JPSEC. It "intends to provide tools and solutions in terms of specifications that allow applications to generate, consume, and exchange Secure JPEG 2000 codestreams" (p. vi). JPSEC extends the codestream syntax to allow parts which are created by security tools, e.g., cipher or authentication tools. Encryption is implemented with conventional ciphers, e.g., AES.
The approaches discussed in this article perform encryption by constructing a secret transform domain. The principal idea of such schemes is that without the key the transform coefficients cannot be interpreted or decoded and therefore no access to the source material is possible (or only at a very low quality). Other than with bitstreamoriented methods, which operate on a final coded media bitstream, these methods apply encryption integrated with compression. In terms of applicability they are therefore restricted to scenarios where the final media bitstream is not yet available (video conferencing, live streaming, photo storage, transmission, and storage of surveillance data etc.).
Transparent encryption was introduced in the context of TV broadcasting [1, 14] and denotes encryption schemes for which public access is granted for a preview image, i.e., anyone can decode an image of reduced quality from the encrypted stream, even without the key data. The difference to other media encryption schemes that guarantee a certain degree of distortion is that the preview image has to be of a (specified) minimum quality, i.e., apart from the security requirement, there is also a quality requirement[10, 15]. Broadcasting applications, for example, can benefit from transparent encryption, as they, rather than preventing unauthorized viewers from receiving and watching their content completely, aim at promoting a contract with nonpaying watchers, for whom the availability of a preview version (in lower quality) may serve as an incentive to pay for the full quality version. The reason for considering transparent encryption as target application scenario is that it has been shown [16], that the lowest resolution contained in a JPEG2000 file encrypted using the techniques discussed in this article can always be decoded by an attacker. This makes the approaches suitable for transparent encryption only, but prevents usage in applications requiring a higher degree of confidentiality.
In [17] the authors suggest to use secret Fourierbased transforms for the encryption of visual data. Other proposals in the area of lightweight encryption [18] propose the encryption of the filter choice used for a wavelet decomposition. However, this suggestion remains vague and is not supported by any experiments, while [19, 20] propose encrypting the orthogonal filterbanks used for an nonstationary multiresolution analysis (NSMRA) decomposition. The use of concealed biorthogonal parametrized wavelet filters for lightweight encryption is proposed by [21]. The use of keydependent wavelet packet decompositions is proposed first by [22, 23]. The latter study [23] evaluates encryption based on keydependent subband structures in a zerotreebased wavelet codec. Parametrized wavelet filters have been employed for JPEG2000 lightweight encryption by [24, 25], however, this approach was shown to be insecure in later study [26]. The scrambling of discrete cosine transform (DCT) and discrete wavelet transform (DWT) coefficients is proposed in [27]. Recently secret DCTbased transforms have been proposed for image and video encryption [28].

Compression impact: KDWP encryption reduces compression performance, the actual decrease is evaluated in Section 3..

Computational demand: The main argument for introducing the general concept of secret transform domains in encryption has always been an improved runtime performance, as the conventional encryption step is not needed. This assumption is indepth evaluated and analyzed in Section 4..

Security: The security of KDWP is indepth analyzed in Section 5.. Special care has been taken to employ the suitable notion of security for multimedia encryption.
In Section 6. we combine the individual assessments of the previous sections and indepth discuss other potential advantages of KDWP. Another potential advantage of KDWP encryption is the capability of performing signal processing operations on the protected data ("encrypted domain processing"), which are compared to the operations possible on a conventionally encrypted JPEG2000 bitstream. Final conclusion are drawn in Section 7..
2. Wavelet packets
The wavelet packet (WP) transform [32] generalizes the pyramidal wavelet transform. In the WP transform, apart from the approximation subband also the detail subband can be decomposed; an example is shown in Figure 1. The WP can be adapted to take the properties of the image to be transformed into account, for example by using the best basis algorithm [32–36]. In this article we refer to each such a WP basis by the terms "WP subband structure", "decomposition structure" or briefly "WP".
The anisotropic WP transform is a generalization of the isotropic case: whereas in the latter, horizontal and vertical wavelet decomposition are always applied in pairs for each subband to be decomposed, this restriction is lifted for the anisotropic WP transform (for an example see Figure 1c).
Note that for the isotropic WP transform a single decomposition refers to both horizontal and vertical filtering and downsampling. For the anisotropic WP transform, a decompositions refers to filtering and downsampling in one direction (horizontal or vertical). Therefore, a decomposition depth of 2 g in the anisotropic case is comparable to a decomposition depth of g in the isotropic case.
2..1 WP in JPEG2000
Part 2 of the JPEG2000 standard [29] allows more decomposition structures than Part 1. Every subband resulting from a highpass filtering can be decomposed at most two more times (either horizontally, vertically or in both directions).
In order to maximize keyspace size for the proposed encryption scheme, we have implemented full support for arbitrary isotropic and anisotropic wavelet decomposition structures in JPEG2000, based on the JJ2000 reference implementation.^{a} The source code for the implementation underlying all results in this paper can be downloaded from http://www.wavelab.at/sources.
2..2 Randomized generation of isotropic WPs
A WP can be derived by a sequence of random binary decomposition decisions. The seed s for the employed secure random number generator is the main parameter for the randomized generation of WP, i.e., comparable to the key in a symmetric crpyto system.
2..2.1 Uniform distribution
Subsequently, this distribution is referred to isouni.
2..2.2 Compressionoriented distribution
Only the parameters, b, c and the seed of the random number generator need to be encrypted. Subsequently, these distributions are denoted/abbreviated by iso and further classified into a constrained (the LL is always further decomposed) and an unconstrained case.
2..3 Randomized generation of anisotropic WPs
Even more than in the case of isotropic WPs, there are anisotropic WP decompositions that are illsuited for energy compaction. The compressionoriented selection method tries to eliminate these subband structures.
2..3.1 Uniform distribution
We use the case distinction introduced by [37] to construct a uniform distribution for the selection of a random subband structure: the probability for any case to be chosen is the ratio of the number of subband structures contained in the case to the total number of subband structures. Subsequently, this distribution is often denoted/abbreviated by anisouni.
2..3.2 Compressionoriented distribution
where h and v are the decomposition depths in horizontal and vertical direction, respectively. If at any node during the randomized generation of an anisotropic WP subband structure, decomposition of the subband at this node in the randomly chosen direction would result in the degree of anisotropy exceeding the maximum degree of anisotropy, the direction of the decomposition is changed. The degree of anisotropy for the approximation and detail subbands influence both, compression performance and keyspace size.
The other parameters are used in the same way as in the isotropic case: The base value b sets the basic probability of decomposition, the change factor c alters this base probability depending on the current decomposition level l.
3. Evaluation of compression performance
In previous study [16, 30] parameter settings for the compressionoriented distribution have been determined for a small number of test images. For the isotropic wavelet packet transform it is proposed to force a maximum decomposition depth for the approximation subband. For the anisotropic wavelet packet transform, in addition to forcing a maximum decomposition depth, the maximum degree of anisotropy for the approximation subband needs to be restricted to preserve compression performance.
The parameters proposed by [16, 30] were obtained empirically by a number of experiments. A large number of different parameter settings were used, but only on three test images. The parameters that were obtained for these three test images are a base value b of 0.25 and a change factor c of 0.1. For the isotropic case the global decomposition depth g has been set to 5. For the anisotropic case g has been set to 10 and the maximal degree of anisotropy of the approximation subband q has been set to 1. Recall that we follow the convention to give the decomposition depth of the isotropic wavelet packet transform in pairs of (horizontal and vertical) decompositions, whereas in the anisotropic case each (horizontal or vertical) decomposition step is counted separately.
We use these parameters in our experiments and evaluate their performance for a larger set of test images. We verify the compression performance of the compressionoriented selection method by an empirical study based on an extended set of images. For this purpose we use a set of 100 grayscale images of 512 × 512 pixels (taken with four different camera models).

Pyramidal (1 subband structure, level 5),

Isouni: random isotropic wavelet packets drawn according to the uniform distribution with g = 5 (100 randomly selected subband structures),

Iso (constrained): random isotropic wavelet packets drawn according to the compressionoriented distribution with g = 5, b = 0.25, and c = 0.1 (100 randomly selected subband structures),

Anisouni: random anisotropic wavelet packets drawn according to the uniform distribution with g = 10 (100 randomly selected subband structures), and

Aniso (constrained): random isotropic wavelet packets drawn according to the compressionoriented distribution with g = 10, b = 0.25, c = 0.1, and q = 1 (100 randomly selected subband structures).
Ratio of header data to packet data for different compression rates (16 quality layers)
Rate  pyramidal  iso  aniso 

0.125  15.9%  32.8%  20.8% 
0.25  10.0%  20.8%  15.1% 
0.5  6.1%  13.6%  9.6% 
1  3.7%  9.2%  5.9% 
2  2.5%  6.4%  3.9% 
As regards the difference between uniform and compressionoriented selection, it can be seen that the compression performance of the latter is above the compression performance of the former. The difference is more evident for the anisotropic case, for which a predominant decomposition of the approximation subband in a single direction, which leads to inferior energy compaction for natural images, is possible. Restricting the maximum degree of anisotropy for the approximation subband in the compressionoriented selection leads to a compression performance that is closer to the pyramidal decomposition.
From an compression performance point of view, the compressionoriented distributions are favorable, although, even these distributions considerably decrease compression efficiency.
4. Computational complexity
In order to compare the computational complexity of KDWP encryption compared to conventional compression and encryption it is sufficient to determine the difference. The main difference between KDWP encryption and conventional compression and encryption is the transform stage, a random WP is chosen for KDWP encryption, while in a conventional encryption approach the pyramidal decomposition is employed. In the commonly applied PCRDO coding approach (postcompressionratedistortion optimization) the complexity of the remaining compression (labeled "RCompression" in Figure 2) is almost completely independent of the bitrate and thus the difference in complexity is almost completely due to the difference in the transform stage. In the conventional approach there is an additional encryption process after compression, which is linearly bitratedependent. First we present experimental performance results in Section 4..1 and complement the findings with a theoretical performance analysis in Section 4..2.
4..1 Experimental performance evaluation
Average compression complexity with our implementation
structure  avg. t  avg. fps 

pyramidal  0.643 s  1.55 fps 
iso  1.005 s  1 fps 
isouni  1.147 s  0.87 fps 
aniso  0.726 s  1.34 fps 
anisouni  1.891 s  0.53 fps 
Runtime performance of encryption routines
AES encryption  

Method  throughput  codestreams  with 2bpp 
AES OFB  42.71 MB/s  0.0015 s  683.36 fps 
The final question is whether it is at all possible to gain any performance benefits with KDWP (even with the most optimized implementation) compared to AES encryption. This question can only be answered with a theoretical analysis.
4..2 Theoretical performance analysis
We determine the computational complexities on a random access machine with an instruction set of basic operations, such as ^, &, +, *, and %.
As we are only interested in the difference of the complexities it is sufficient to determine the complexity of the pyramidal wavelet transform, the average WP transform, and AES encryption.
The wavelet transform stage consists of iterated filter operations, the number of filter operations depends on the WP and the number of pixels. For every input pixel and decomposition level two filter operations are required for the isotropic case. A single filter operation with an nlength filter consists of n multiplications, n additions, n MemReads and a single MemWrite, which yields 25 operations for n = 8 (JPEG2000's 9/7 irreversible filter).
The expected average decomposition depth for isotropic WP distributions (b = 1/ 4, c = 0)
Iso. g  D _{ g }  ${D}_{g}^{c}$  ${D}_{g}^{f}$ 

1  0.5  0.857  1 
2  1.41176  1.6406  1.90625 
3  2.41174  2.3105  2.70703 
4  3.41174  2.89673  3.40967 
5  4.41174  3.40946  4.02496 
The expected average decomposition depth for anisotropic WP distributions (b = 1/ 4, c = 0)
Aniso. g  ${D}_{g}^{a}$  ${D}_{g}^{ac}$ 

2  1.55556  1.64063 
4  3.53125  2.89673 
6  5.53020  3.85843 
8  7.53014  4.59474 
10  9.53014  5.15847 
According to [38] and backedup by our own analysis 352.625 operations are necessary for the encryption of a single byte with AES with a 128bit key in CTRmode.
5. Security evaluation
In order to assess the security of a multimedia encryption approach, we first need to define "security" of a multimedia cryptosystem more precisely. Conventional notions of security for cryptosystems require that the ciphertext does not leak any information (informationtheoretic approach [39]) or any efficiently computable information (the approach of modern cryptography [40]) of the plaintext. This kind of security notions are also referred to as MPsecurity (message privacy) [41]. However this type of security is rarely met nor targeted by multimedia encryption schemes. Thus multimedia encryption is often analyzed with respect to a full message recovery, i.e., how hard is it for an attacker to reconstruct the (image) data. This type of security notion is referred to as MRsecurity (message recovery) [41]. However, a reconstruction of a multimedia datum (on the basis of the ciphertext) may have excellent quality and even be perceived as identical by a human observer, while the perfect recovery of the entire message remains impossible.
Thus in the context of multimedia encryption it is required to take the quality of a reconstruction (by an adversary on the basis of the ciphertext) into account. An adversary, who tries to break a multimedia encryption system, is successful if she can efficiently compute a "high quality" reconstruction of the original multimedia datum. Which quality constitutes a security threat highly depends on the targeted application scenario [11]. In [42] this multimediaspecific security notion is termed MQsecurity (message quality), similar concepts can be found in the multimedia encryption literature [43, 44].
5..1 Usual security analysis: key space
Commonly a multimedia encryption security analysis consists of counting the key space. The common conclusion is that if the key space is large enough the proposed approach is believed to be secure. The number of possible WP is huge even for moderate maximum decomposition depths (e.g., 2^{261}^{.}^{6} for isotropic g = 5, and 2^{1321}^{.}^{9} for an anisotropic depth of 10). Thus the common conclusion would be to consider the scheme secure. Additionally the quality of a reconstruction with a wrong key is often analyzed in literature. However, if we try to decode a JPEG2000 codestream with a wrong WP the decoder will not be able to decode an image, as the WP is required for the decoding. Information, such as the subband/codeblock size and the number of codeblocks in a subband, which is required to make sense of the coded data, is missing. Thus a naive attacker would need to test half of the possible WP before the correct is identified (on average). A futile approach given the number of WP!
5..2 Improved security analysis: entropy
Entropy of the isotropic compressionoriented distribution (b = 1/ 4, c = 0) and the uniform distribution
g  Entropy (iso)  Entropy (isouni) 

2  2.4  4.1 
3  9.1  16.3 
4  32.4  65.4 
5  114.0  261.6 
6  399.5  1046.4 
Still the numbers are sufficiently large such that we have to conclude that KDWP in JPEG2000 are secure. Since entropy values in excess of stateoftheart ciphers key set sizes (128 bit) can be considered secure, security is sufficient for g > 5 in the compression oriented case and for g > 4 in the case of uniform WP distribution.
Since the number of anisotropic WP is by far greater than the number of isotropic WP (for comparable maximum decomposition depths, remember an isotropic decomposition depth g corresponds to an anisotropic decomposition depth of 2 g), the anisotropic case has to be considered secure as well.
Note that this analysis holds for every scheme that strongly requires the WP for decoding. However, the key question now is whether this requirement is strong for JPEG2000 or whether it can be weakened by exploiting specifically tailored attacks.
In the following we will argue that such specificallytailored attacks can be designed for KDWP with JPEG2000 and that they will be even more effective if only the permissible WP of JPEG2000 Part 2 are employed.
5..3 Specific attacks against KDWP in JPEG2000
Due to JPEG2000 coding the LL subband can be reconstructed by an attacker with a specialpurpose decoder. The LL subband can be decoded, because the coded LL subband data is located at the start of JPEG2000 file. Thus an attacker can decode a lowresolution image, a fact that has already been highlighted in previous study [16]. The previous conclusion has been that KDWP in JPEG2000 are only suitable for the application scenario of transparent encryption.
In this study we pose a new question: can an attacker go further and decode subsequent resolutions, i.e., images with an higher quality than targeted? In fact resolution only loosely corresponds to perceived quality, but as every subsequent improvement an attacker can achieve (by image enhancement operations) starts from a previously deciphered lower resolution it makes sense to use resolution as sole quality indicator within the scope of this work. The quality of an attacked image is determined as the fraction of original resolution divided by the obtained resolution (given in the number of pixels).
Therefore, we first need to answer whether the next resolution can be decoded from the codestream (independently of the higher resolutions). This is the case in the coding framework of JPEG2000. A resolution can be decoded independently from the remaining higher resolutions (definitely for resolution progression and at least at the lowest quality for layer progression and always if SOP and EPH markers are employed which signal packet borders).
The next question is whether it is decidable that the employed subband decomposition structure is the correct one. It is also highly likely that it can be decided whether the correct decomposition structure has been employed in the decoding of a resolution: Firstly, the wavelet resolutions are not independent, i.e., statistical crossresolution dependencies are highly likely to identify the correct decomposition. Secondly, the codestream syntax and semantics must also be met while decoding with a subband decomposition structure, i.e., decoding errors clearly indicate an incorrect decomposition structure.
Thus the decomposition structure of a resolution can be determined independently of the higher resolutions in JPEG2000.
Now, how hard is it for an adversary to decode a certain resolution? We first discuss the standardized case of JPEG2000 Part 2.
5..3.1 JPEG2000 Part 2
In JPEG2000 Part 2 a high frequency subband may only be decomposed two more times (either isotropic or anisotropic). In the case of isotropic WP, every next resolution may only be secured by ${Q}_{2}^{3}=1{7}^{3}=4913$ possibilities. In the case of anisotropic WP, a subband has only R_{2} = 18 possibilities to be further decomposed. Thus the restrictions of Part 2 and both distinct cases do not offer security for each subsequent resolution. An attacker would need only 4913/2 or 18/2 trials on average to decode each subsequent resolution.

Three subbands (HL, LH, HH), which leads to 287^{3} = 23639903 possible WP.

One vertical subband, which has 287 possible WP.

One horizontal subband, which has 287 possible WP.
In summary there are about 2^{24}^{.}^{49} possible WP for a subsequent resolution in Part 2. Though less than 2^{24} checks are quite an effort, this number of checks is still computationally feasible.
Thus the application of a JPEG2000 Part 2 encoder and KDWP can not be considered secure (neither MRsecure nor MQsecure) due to the restrictions in terms of admissible WP. For nonstandard WP security is expected to be increased.
5..3.2 Nonstandard WP
Entropy for the distribution on resolutions of the isotropic compressionoriented distribution (g = 5, b = 1/ 4, c = 0) and the uniform distribution
Res. at level  Quality  Entropy (iso)  Entropy (isouni) 

0  1  114.0  261.6 
1  1/4  28.7  65.4 
2  1/16  7.5  16.3 
3  1/64  2.2  4.1 
4  1/256  0.8  1.0 
5  1/1024  0.0  0.0 
Entropy for the distribution of decomposition structures of a resolution as induced by the anisotropic uniform distribution (g = 10, b = 1/ 4, c = 0)
Res. at level  Quality  Entropy (anisouni) 

0  1  1321.9 
1  1/2  660.5 
2  1/4  329.7 
3  1/8  164.4 
4  1/16  81.7 
5  1/32  40.3 
6  1/64  19.7 
7  1/128  9.3 
8  1/256  4.2 
9  1/512  1.5 
10  1/1024  0.0 
In the isotropic case the compression oriented distribution all obtained entropy values are below stateoftheart cipher keylength (AES has at least 128 bit). The uniform distribution also results in entropy values below AES minimum key length, except for the highest resolution. The anisotropic case with uniform distribution shows higher entropy values, but at a quality of 1/16 security drops far below AES minimum key length. The anisotropic case with the compression oriented distribution is expected to be far below the security levels of the uniform distribution. In general an attacker can compute lower resolutions, while only the higher resolutions remain wellprotected.
6. Discussion

Compression impact: The KDWP approach obviously reduces compression performance. If all possible subband structures are equally likely, the approach is not be suitable for application due to the high variance of obtained compression results. The approach of pruning the set of all subband structures (compression oriented distribution) improves the compression performance, however, still with this technique, a significant loss exists. On the other hand, conventional encryption of course does not at all influence compression performance.

Computational demand: Although the additional complexity of encryption with KDWP in a compression framework, such as JPEG2000, seems to be negligible at a first glance, our careful analysis and our evaluation results clearly show that runtime advantages can not be achieved compared to stateoftheart cryptographic ciphers (AES). In general we advise to carefully reconsider statements about an improved runtime performance of transform based image and video encryption schemes.

Security: From a security point of view, stateoftheart cryptographic ciphers are superior to KDWP. The KDWP schemes cannot prevent access to lower resolutions and are thus less secure in terms MQsecurity. Within the standardized framework of JPEG2000 Part 2, the KDWP approach is completely insecure, i.e., an attacker is able to reconstruct the entire image.
From the proposed KDWP schemes, the anisotropic compression oriented scheme is best performing. However, even this approach is overall outperformed by conventional compression and encryption, which shows no decrease in compression performance, better runtime performance and cryptographic security.
Apart from an improved runtime performance, image encryption schemes can offer an improved functionality. On the one hand improved functionality could be suitability for specific applications, such as transparent encryption. However, transparent encryption can also be implemented in the conventional approach [10], the small resolution portion of the JPEG2000 file is simply left in plaintext. A standardized tool, namely JPSEC, can be employed to signal all the metadata (keys, plaintext parts). Even completely JPEG2000compliant implementations are possible [10], which do not require special software for decoding, a great advantage for realworld deployment of transparent encryption. Transparent encryption with KDWP requires a special decoder, i.e., an attacker's decoder, that can cope with wrong or missing WP structure information, while JPSECbased approaches require a JPSECcapable decoder. Thus, the support of transparent encryption by KDWP is not an advantage compared to other approaches, which are more flexible. The conventional approach offers full confidentiality/cryptographic security (MPsecurity).
Image encryption schemes may also allow special encrypted domain processing, which may justify their application although there are disadvantages in compression, computational demand and security. A good example is encrypted content that can still be robustly watermarked; this feature would outweigh many disadvantages.
6..1 Encrypted domain processing
In fact for KDWP in JPEG2000, the encrypted domain is a scrambled JPEG2000 bitstream and only signal processing operations that can be conducted in this domain are possible. In fact the only possible processing is the truncation of the scrambled bitstream, which results in an almost ratedistortion optimal rate adaptation (in case the underlying bitstream is in quality progression order). However, truncation of an encrypted bitstream is also possible with conventional block ciphers if the are used in the proper mode, e.g., counter mode.
Signal processing operations which rely on transform coefficient data cannot be applied to KDWP protected data as these data can not be decoded. Given the results of our evaluation and analysis (decreased compression performance, increased computational complexity, decreased security, no other advantages) it has to be stated that hardly any sensible realistic application scenarios can be identified for KDWPbased encryption at the present time.
7. Conclusion
A primary argument for proposing KDWPbased encryption has always been its "lightweight" nature, introduced by shifting complexity from encryption into the compression pipeline. When comparing JPEG2000 encryption with keydependent wavelet packets (KDWP) to the conventional approach (AES encryption), we have assessed an overall increase in computational complexity through additional complexity introduced in the compression step.
Signal processing in the encrypted domain, often used as a second argument favoring transformbased image encryption schemes, does not offer advantages compared to the appropriate application of conventional ciphers. The security of JPEG2000 encryption with KDWP has been analyzed in depth and has been found to be less secure than conventional encryption, as smaller resolution images remain accessible. Security can not be achieved at all if only permissible subband structures of JPEG2000 Part 2 are employed.
All these facts taken together with a slight decrease in compression efficiency as compared to classical (pyramidal) JPEG2000 make KDWPbased encryption approaches not suitable for most application scenarios.
The presented assessment should serve as guideline in the future development of image and video encryption schemes. The following general conclusions may be drawn: Computational complexity will need to be carefully (re)considered for transformbased image and video encryption schemes. Security has to be analyzed with respect to a multimedia specific security notion (MQsecurity). Image and video encryption schemes have to provide conclusive evidence for improved functionality, such as encrypted domain signal processing.
Appendix 1: Details on the entropy computation of WP distributions
A WP (wavelet packet subband structure) is derived by the following randomized algorithm (see Section 2..2.2): every subband at depth l is further decomposed with a certain probability p_{ l }, which may only depend on the depth l.
Game tree
However, as the number of leaves at depth g is Q(g) the computation of the entropy of the distribution on WPs on the basis of this formula is soon infeasible with growing g.
Cumulative game tree
The nodes can be uniquely identified by the path from the root, i.e., by tuple of edge labels. A node at depth g is a gtuple of edge labels (x_{1},..., x_{ g }). The set of all nodes at depth g is given by {(x_{1},...,x_{ g })x_{1} ∈ {0,1},x_{ i }_{+1} ≤ 4x_{ i }}.
Appendix 2: Details on the entropy computation of distributions of decomposition structures on resolutions
Endnotes
^{a}http://jj2000.epfl.ch/. ^{b}Linux binaries in version 6.3.1 from http://www.kakadusoftware.com.
Declarations
Acknowledgements
This work was funded by the Austrian Science Fund under Project 19159. The authors thank Stefan Huber and Rade Kutil, and Roland Kwitt for the valuable discussions and contributions.
Authors’ Affiliations
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