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Applied Optics

Key-length analysis of double random phase encoding

Kazuya Nakano, Masafumi Takeda, Hiroyuki Suzuki

Published May 16, 2017• 10 citations
Publisher

Abstract

Double random phase encoding (DRPE) is a classical optical symmetric-key encryption method. DRPE prevents the key length from being determined because of its redundancy between encryption and decryption, unlike digital symmetric-key cryptographies. In our study, we numerically analyzed the key length of DRPE based on key-space analysis. We estimated the key length of DRPE by calculating the inverse value of the cumulative probability of the normal distribution that was estimated from samples of DRPE and then discuss security against brute-force attacks.

DOI
10.1364/ao.56.004474

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Double random phase encoding (DRPE) is a classical optical symmetric-key encryption method. DRPE prevents the key length from being determined because of its redundancy between encryption and decryption, unlike digital symmetric-key cryptographies. In our study, we numerically analyzed the key length of DRPE based on key-space analysis. We estimated the key length of DRPE by calculating the inverse value of the cumulative probability of the normal distribution that was estimated from samples of DRPE and then discuss security against brute-force attacks.
Although initial research shows that double-random phase encoding (DRPE) is vulnerable to known-plaintext attacks that use phase retrieval algorithms, subsequent research has shown that phase-only DRPE, in which the Fourier amplitude component of an image encrypted with classical DRPE remains constant, is resistant to attacks that apply phase retrieval algorithms. Herein, we numerically analyze the key-space of DRPE and investigate the distribution property of decryption keys for classical and phase-only DRPE. We determine the difference in the distribution property of successful decryption keys for these DRPE techniques from the numerical analysis results and then discuss the security offered by them.
Double Random Phase Encoding (DRPE), which is a typical optical encryption technique, has been reported to be vulnerable to Known Plaintext-Attacks (KPAs) using a Phase Retrieval Algorithm (PRA). But the reported case in which the encryption key is successfully estimated was that the plain image was rather simple such as the image of a character. In addition, although Phase Only DRPE (PO-DRPE) was proposed to achieve more resistance to the KPA than Complex DRPE (C-DRPE) in which both amplitude and phase components are used as an encrypted image, no quantitative results about the relationship between the vulnerability and the plaintext image. In this paper, we show the result of quantitative analysis on KPA by PRA to C-DRPE and PO-DRPE, for the plaintext images of different characteristics. As a result of experiment, KPA to C-DRPE succeeded to estimate the correct key while the probability of success became lower when the number of non-zero pixel increases in the plaintext image. However, KPA to PO-DRPE enabled to estimate only "singular" keys, which are effective for no more than given plaintext/ciphertext image pair and far different from the correct encryption key. We also conducted KPA using two plaintext-ciphertext image pairs for KPA. In the case when two plaintext-ciphertext image pairs were given to KPA, the cryptanalysis succeeded with higher probability than the case of one. Moreover, the probability of success in the KPA was high even in PO-DRPE.
In recent years, various studies have been conducted to illustrate the vulnerability of double random phase encoding (DRPE). In this paper, we propose a novel method via phase reservation and compression to enhance DRPE security. Only a compressed phase distribution is available in the CCD plane, and the amplitude component is not available or requested for optical decryption. Since only noise-like distributions can be obtained by using the correct security keys during optical decryption, a nonlinear correlation algorithm is further applied for authenticating the decrypted image. It is demonstrated that valid conditions for attack algorithms are broken and high security can be achieved for the DRPE system.
In this Letter, we propose an information encryption technique based on the theory of phase-space optics. We show that encoding the plaintext in phase space provides a higher level of security: first, the key-space is significantly enlarged. Second, it is immune to various known-plaintext (cyphertext) attacks to which the double-random phase encryption (DRPE) is vulnerable. Third, the bilinearity of phase-space distributions offers additional security. Theoretical analysis and numerical calculation results show that the proposed technique has significantly different responses to errors added to the cypheretext and the two phase keys in comparison to the classical DRPE.
We demonstrate a method of chosen-plaintext attack on lensless double-random phase encoding (L-DRPE) in the Fresnel domain. With this attack an opponent can access two encryption keys with help of the impulse functions as chosen plaintexts. This shows that a lensless optical encryption scheme based on DRPE is vulnerable to chosen-plaintext attack. Cryptoanalysis also indicates that the security worry originates from the linearity of the encryption and decryption mechanism of the L-DRPE scheme. One of the interesting features of the proposed attack is that the decryption process is lossless. Numerical simulations show good agreement with theoretical analysis.
The principle of ptychography is applied in known plain text attack on the double random phase encoding (DRPE) system. We find that with several pairs of plain texts and cipher texts, the model of attack on DRPE can be converted to the model of ptychographical imaging. Owing to the inherent merits of the ptychographical imaging, the DRPE system can be breached totally in a fast and nearly perfect way, which is unavailable for currently existing attack methods. Further, since the decryption keys can be seen as an object to be imaged from the perspective of imaging, the ptychographical technique may be a kind of new direction to further analysis of the security of other encryption systems based on double random keys.