Elsevier

Optics Communications

Volume 285, Issue 6, 15 March 2012, Pages 1078-1081
Optics Communications

A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms

https://doi.org/10.1016/j.optcom.2011.12.017Get rights and content

Abstract

We analyze the security of a recently proposed asymmetric cryptosystem that based on the phase-truncated Fourier transforms (PTFTs), and describe a specific attack method to break the cryptosystem. This specific attack, which is based on a two-step iterative amplitude retrieval approach and works by using the public keys and ciphertexts, would allow an attacker to reveal the encrypted information and the decryption keys that generated in the encryption procedure. The numerical simulation results, which are given to show the effectiveness of the proposed specific attack, imply that some appropriate measurements should be made to enhance the resistance of the PTFT-based cryptosystem against the specific attack when it is used as a public-key cryptosystem.

Introduction

In 1995, Refregier and Javidi proposed a double random phase encoding (DRPE) technique to transform an input image into a stationary white noise [1]. Based on their creating research works, a number of subsequent proposals for optical security and encryption have been proposed in the past decades [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. In practice, however, the DRPE encryption scheme presents some weakness against attacks [12], [13], [14], [15], [16]. Its main weakness lies in its linearity and the most dangerous attack only requires two known plain images [16]. Recently, Qin and Peng have proposed an asymmetric cryptosystem based on the PTFTs to remove the linearity of DRPE by the nonlinear operation of phase truncation [17]. In the PTFT-based cryptosystem, the two phase distributions in the Fourier plane and output plane are kept as decrypting keys and the keys for encryption will be not used for decryption. One of the benefits is that the two encryption keys can be used as public keys to encode different images. Since almost all reported optical encryption techniques belong to the category of symmetric cryptosystems, from the perspective of cryptology, the proposed PTFT-based asymmetric cryptosystem has great practical significance. However, if the encryption keys are used as two public keys and applied to encode different plaintexts, the cryptosystem will be placed into a more exposed and vulnerable position. As we know, a cryptosystem could be described as robust and secure if only it were able to resist various attacks. Therefore, it is no doubt that making attacks on a new cryptosystem is an important and meaningful thing.

In this paper, we propose a specific attack method that based on a two-step iterative amplitude retrieval approach to try to reveal the encrypted information. Unfortunately, the results show that the cryptosystem is vulnerable to this attack. In the attack, discussed in this paper, the two RPMs used for encryption and the ciphertext are known by the attacker and the initial random phase codes used in the iteration process are just the two encryption keys, which mean the attack is very special to some extent. That is why we refer to this as a specific attack, which can lead to an unpredictable decryption. The simulation results also imply that some appropriate measurements should be made to enhance the resistance of the PTFT-based cryptosystem against the specific attack when it is used as a public-key cryptosystem. In this following, we will show how the PTFT scheme is vulnerable to the specific attack and how an opponent can decrypt the ciphertext.

Section snippets

The asymmetric cryptosystem based on the PTFTs

In the asymmetric cryptosystem that based on the PTFTs [17], two statistically independent RPMs R(x) and R′(u) are placed at the image and Fourier planes, respectively. With the help of the RPM R(x), the input image I(x) can be transformed into:g(u)=PTFTI(x)R(x),where the operators PT{} and FT[] denote the phase truncation and Fourier transform, respectively. Note that the phase truncation of the complex amplitude just leads to amplitude reservation. By using the RPM R′(u), the primary image

A specific attack on the cryptosystem based on the PTFTs

The attack process can be completed by a two-step approach which can be described as follows: the first step is to access g′(u) which is an estimate of g(u) by using R′(u) and the ciphertext E0(x), and the second step is to achieve an estimate of primary image by using g′(u) and R(x). In the following, we will explain the two steps in detail.

An iteration process is used to achieve the aim of the first step. In the kth(k = 0,1,2,3, …) iteration, the amplitude distribution at the Fourier plane, the

Numerical simulations and discussion

Numerical simulations have been performed to test the effectiveness of the proposed attack technique by using two types of normalized images. A gray-scale image with a size of 256 × 256 pixels, as shown in Fig. 1(a), is taken as the input image. The two public keys R(x) and R′(u) with the sizes of 256 × 256 pixels are shown in Fig. 1(b) and Fig. 1(c) respectively. Fig. 1(d)–(e) shows the two decryption keys generated in the encryption procedure. In the first step of our attack, the RPM R(x) is used

Conclusions

In conclusion, we have demonstrated that the PTFT-based asymmetric cryptosystem is vulnerable to a specific attack, which is based on a two-step iterative amplitude retrieval approach. In this attack, the public keys are used as the initial random phase codes during the iteration process. We tested the approach by decrypting a gray-scale image Lena and a binary image that encoded with the same two public keys. The effectiveness of the proposed specific attack has been demonstrated by the

Acknowledgment

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (R1090168), the National Natural Science Foundation of China (NSFC) (11074219 and 10874150), the Excellent Young Teacher Item Fund of Zhejiang Education Department (Z. J. Edu. GKC (2010) No.175) and the Program for Innovative Research Team of Young Teachers in Zhejiang A & F University (2009RC01).

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