Optical encryption attacks to Double Random Phase Encryption

Double Random Phase Encryption (DRPE) technique has been criticised recently for poor security and low cryptography resistance because of its linearity. Recently has the security of DRP started to be thoroughly analysed and a few weaknesses were reported [1,2,3].

Double Random Phase Encryption technique

As it shortly described in [4], the image to be encrypted P is immediately followed by a first random phase mask, which is the first key X. Both the image and the mask are located in the object focal plane of a first lens (see Fig. 1).



In the image focal plane of this lens is therefore obtained the Fourier transform (FT) of the product $ P\cdot X$ . This product is then multiplied by another random phase mask that is the second key Y. Lastly, another FT is performed by a second lens to return to the spatial domain. Since the last FT does not add anything to the security of the system, we will perform all our analyses in the Fourier plane. The ciphered image C is then:

$\displaystyle C = Y \cdot \mathcal{F}(P\cdot X)$ (1)

where F stands for the Fourier transform operation. In most of the paper, we will assume that P is a grey-level image.

Attacks to the DRPE

Several attacks are proposed against the double random phase encryption scheme. Of source, as it mentioned in Javidi's article [4], brute force attack is useless due to huge amount of keys to be tested.

Reducing the number of combinations

More wise attack is the use of approximate version of the phase mask, especially to binary phase mask. Binarization of the phase mask reduces possible combinations dramatically. Of course, the fewer phase levels, the more noise is introduced in the reconstructed image.

In order to reduce the combinations of decryption keys further it is advisable [1,5] to decode with partial window of second key Y.

Plain-text attacks

The main idea of the plain-attack is to compromise an encryption system by specific known images. In Javidi's paper [4] is mentioned Dirac's delta function, uniform (spatially constant) image.

These attacks are demonstrated on computer-generated ciphered images, and the article [4] gives a comprehensive survey of attacks to DRPE. The scheme is shown to be resistant against brute force attacks but susceptible to chosen and known plaintext attacks. A technique to recover the exact keys with only two known plain images is described. Comparison of such technique to other attacks proposed in the literature is provided.

To sum up, with at most three chosen plain-ciphered image pairs, it is possible to recover the two encryption keys and break the system. But it is only theoretical review, no experimental works were provided. Also, there is no quantitative analysis of decryptability: only ``fuzzy'' visual estimations are presented (like ``the image is still recognizable'' [4] on page 6).

Personal remarks

In other terms, the plain image is entirely black except for a single pixel. It can be argued that such a plain image can look suspicious to the authorized user that is to encrypt it.

Why such image is suspicious? There is an example: you are going to encrypt an image printed on a paper. You are attaching the printed piece of paper on a pin upon the input scene and illuminating the input scene. A bright reflection from the input scene gives you an exact encryption key.

Related works: a little survey

As an example of cryptographical analysis and optical encryption cryptoresistance testing, Nauton's article is interesting [6], and iterative attempt to decrypt DRPE images is proposed in [7]. Another successful attempt to crack DRPE encryption method is reported in [7]. More detailed analysis of phase encoding's attack and quantization influence is covered in [8].

Moreover, Nauton published plain-text cryptographic attack method in [2]. The Fourier plane encryption algorithm is subjected to a known-plaintext attack, - he writes in article. So that Fourier plane encryption algorithm is found to be susceptible to a known-plaintext heuristic attack. Nauton applied a SA algorithm [9] to find a phase mask which would approximately decrypt the ciphertext. He successfully decrypted DRPE-coded $ 64\times 64$ image.

Bibliography

1
A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells.
Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys.
Optics Letters, 30:1644-1646, 2005.
2
Thomas J. Naughton Unnikrishnan Gopinathan, David S. Monaghan and John T. Sheridan.
A known-plaintext heuristic attack on the fourier plane encryption algorithm.
Optics Express, Vol. 14, No. 8:3181-3186, 2006.
3
X. Peng, P. Zhang, H. Wei, and B. Yu.
Known-plaintext attack on optical encryption based on double random phase keys.
Optics Letters, 31:1044-1046, 2006.
4
Yann Frauel, Albertina Castro, Thomas J. Naughton, and Bahram Javidi.
Resistance of the double random phase encryption against various attacks.
Optics Express, Vol. 15, No. 16:10253-10265, 6 August 2007.
5
X. Peng, H. Wei, and P. Zhang.
Chosen-plaintext attack on lensless double-random phase encoding in the fresnel domain.
Optics Letters, 31:3261-3263, 2006.
6
David S. Monaghan, Unnikrishnan Gopinathan, Thomas J. Naughton, and John T. Sheridan.
Key-space analysis of double random phase encryption technique.
Applied Optics, Vol. 46, No. 26:6641-6647, 10 September 2007.
7
Guohai Situ, Unnikrishnan Gopinathan, David S. Monaghan, and John T. Sheridan.
Cryptanalysis of optical security systems with significant output images.
Applied Optics, Vol. 46, No. 22:5257-5262, 1 August 2007.
8
David S. Monaghan, Guohai Situ, Unnikrishnan Gopinathan, Thomas J. Naughton, and John T. Sheridan.
Role of phase key in the double random phase encoding technique: an error analysis.
Applied Optics, Vol. 47, No. 21:3808-3816, 20 July 2008.
9
S. Kirkpatrick, C.D. Gellatt, and M.P. Vecchi.
Optimization by simulated annealing.
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