Cryptanalyzing a class of image encryption schemes based on Chinese remainder theorem

doi:10.1016/j.image.2014.06.011

Signal Processing: Image Communication

Volume 29, Issue 8, September 2014, Pages 914-920

https://doi.org/10.1016/j.image.2014.06.011 Get rights and content

Highlights

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A special property of Chinese Remainder Theorem (CRT) was found and proven.
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A class of image encryption schemes based on Chinese Remainder Theorem (CECRT) was proved vulnerable to an efficient chosen-plaintext attack.
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Some defects of CECRT, such as invalid compression function and low sensitivity to plain-image, were reported.

Abstract

As a fundamental theorem in number theory, the Chinese Reminder Theorem (CRT) is widely used to construct cryptographic primitives. This paper investigates the security of a class of image encryption schemes based on CRT, referred to as CECRT. Making use of some properties of CRT, the equivalent secret key of CECRT can be recovered efficiently. The required number of pairs of chosen plaintext and the corresponding ciphertext is only $(1 + ⌈ (\log_{2} L) / l ⌉)$ , the attack complexity is only O(L), where L is the plaintext length and l is the number of bits representing a plaintext symbol. In addition, other defects of CECRT, such as invalid compression function and low sensitivity to plaintext, are reported. The work in this paper will help clarify positive role of CRT in cryptology.

Introduction

Both the transmission and the storage of digital data have dual requirements of high operating efficiency and security, which lead to the joint operations of compression and encryption. According to the order of the operations, joint compression and encryption schemes can be categorized into three classes: encryption on compressed data [1], [2], [3]; simultaneous compression and encryption [4], [5], [6], [7], [8]; compression on encrypted data [9], [10], [11]. Recently, Chinese Remainder Theorem (CRT) is used in constructing simultaneous compression and encryption schemes or the basis of some efficient encryption schemes.

The earliest known example of CRT can be found in the book, The Mathematical Classic of Sunzi, written by Chinese mathematician Sun Tzu in the fifth century. In 1247, another Chinese mathematician Jiushao Qin generalized it into a statement about simultaneous congruences and provided the complete solution in Mathematical Treatise in Nine Sections [12]. Antiquity of Chinese mathematicians’ study on the remainder problem (and maybe sparsity of Chinese mathematicians’ contribution to classic mathematics) made the complete form of the statement be called Chinese Remainder Theorem. As a fundamental theorem in number theory, it has been widely used in various fields of information security, e.g. speed up implementation of the RSA algorithm [13], [14], secret sharing [15], and secure code [16]. For a comprehensive survey of the cryptographic applications of CRT and chaos-based cryptanalysis, refer to [17] and [18], respectively.

As reviewed in [19, Sec. 4.3.2], CRT supports the modular representation of a large number (dividend) as a set of numbers (remainders) in some given small domains. It converts the addition, subtraction, and multiplication of large numbers into very simple operations on small numbers. In addition, the conversion provides simultaneous operations on different moduli for parallel computing. Considering these benefits, a number of symmetric encryption schemes based on CRT have been proposed since 2001. The schemes designed in [20], [21], [22], [23] all consider the gray level of some plain-image pixels as remainders, and the summing divisor of CRT as the cipher-element, where the moduli sequences are considered as the secret key or key stream. Conversely, the scheme proposed in [24] combines the gray levels of some plain-image pixels into a big divisor and stores the smaller remainder as the cipher-elements. Reference [25] follows this idea and further encrypts the remainders in a stream cipher mode, using two pseudo-random number sequences (PRNS). In 2013, an image encryption scheme, called CECRT in this paper, was proposed [26]. It first permutes the pixels of the plain-image and then performs the CRT operations as reported in [20], [21], [22], [23]. In [22], [26], the authors claimed that their schemes possess the feature of simultaneous compression and encryption. As shown in [27], cryptanalysis is an integral work to evaluate security level of any encryption scheme, it is important to analyze security properties of the encryption schemes based on CRT.

As CECRT is a typical example of the class of symmetric encryption schemes based on CRT and almost all security defects of other schemes can be found in it, we will focus on breaking CECRT. We found a property of CRT on the relationship among the product of some moduli, the divisor corresponding to a special set of remainders, and the divisor. To the best of our knowledge, this is the first time that the property of CRT is reported. Based on it, we prove that the diffusion part of CECRT can be compromised efficiently using only a pair of chosen-plaintext and the corresponding ciphertext. Then, the permutation part of CECRT can be broken using the existing standard cryptanalysis methods. In addition, the following security defects of CECRT are also reported: (1) the compression performance of CECRT is marginal and even negative; (2) the ciphertext is not sensitive to changes in the plaintext; (3) the moduli of CRT are not suitable to be used as sub-key.

The rest of this paper is organized as follows. In Section 2, CECRT is briefly described. Then, the comprehensive cryptanalyses on CECRT are presented in Section 3, together with detailed experimental results. The last section concludes the paper.

Section snippets

Description of CECRT

The kernel of CECRT relies on the Chinese Remainder Theorem, which states that the system of linear congruences ${x \equiv q_{i} (\mod m_{i})}_{i = 1}^{t}$ has unique solution $x \equiv \sum_{i = 1}^{t} e_{i} {\tilde{m}}_{i} q_{i} (\mod m),$ when $m_{1}, m_{2}, \dots, m_{t}$ are coprime integers, where ${\tilde{m}}_{i} = m / m_{i}$ , $m = \prod_{i = 1}^{t} m_{i}$ , $(e_{i} {\tilde{m}}_{i}) \equiv 1 (\mod m_{i})$ , ${q_{i}}_{i = 1}^{t} \subset Z$ , and t is an integer larger than or equal to one. Let $P = {p_{i}}_{i = 1}^{L}$ and $C = {c_{i}}_{i = 1}^{L / k}$ denote the plaintext and the corresponding ciphertext, respectively, where k is the number of plaintext symbols encrypted at one time. Without loss of

Cryptanalysis

To carry out an efficient chosen-plaintext attack on CECRT, some properties of Chinese Remainder Theorem are introduced first.

Conclusion

The security of a class of encryption schemes using the Chinese Remainder Theorem has been analyzed in detail. Based on some properties of CRT, the sub-key used in the confusion part of CECRT can be easily recovered with only one pair of chosen plaintext and the corresponding ciphertext. Then, the permutation part can be broken with the reported general method. In addition, other defects of CECRT are reported to facilitate the proper use of CRT in cryptography. The work in this paper may be

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 61100216), Hunan Provincial Innovation Foundation For Postgraduate (No. CX2014B277), and the Alexander von Humboldt Foundation of Germany.

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View full text

Cryptanalyzing a class of image encryption schemes based on Chinese remainder theorem

Highlights

Abstract

Introduction

Section snippets

Description of CECRT

Cryptanalysis

Conclusion

Acknowledgements

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IEEE Signal Process. Lett.

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