- Email: [email protected]
Reply to: “Comment on: ‘Image encryption with chaotically coupled chaotic maps’ [Physica D 2010]”
Physica D 239 (2010) 1001
Contents lists available at ScienceDirect
Physica D journal homepage: www.elsevier.com/locate/physd
Reply to: ‘‘Comment on: ‘Image encryption with chaotically coupled chaotic maps’ [Physica D 2010]’’ A.N. Pisarchik a,∗ , M. Zanin b a
Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
b
Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain
article
info
Article history: Received 26 January 2010 Accepted 19 February 2010 Available online 26 February 2010 Communicated by V. Rom-Kedar
abstract We respond to the comment by Arroyo et al. on our paper ‘‘Image encryption with chaotically coupled chaotic maps’’. © 2010 Elsevier B.V. All rights reserved.
Keywords: Chaotic cryptosystem Logistic map Image cipher
In their comment on our paper ‘‘Image encryption with chaotically coupled chaotic maps’’ [1], Arroyo et al. [2] correctly stated that Eq. (7) of Ref. [1] is wrong because it does not ensure that all values resulting from iteration of the logistic map Eq. (6) fall into the phase space of the chaotic attractor. This is obviously a misprint. In fact, the original algorithm which we used in our numerical simulations was free from this error; it contained the correct formula (Eq. (5) of Ref. [2]) and hence this misprint appeared only in the text and did not affect the results of the paper [1]. Another problem raised by the authors of the comment [2] is the existence of some pixels that are not encrypted by the algorithm, meaning that the information of some pixels may not be diffused through whole image; even so, this does not represent an important degradation of the algorithm security. This inconvenience appears due to the wrong direction of the encryption process. The original algorithm [1] encrypts every image pixel pi in a sequential fashion. Through a chaotic map, another pixel pk is chosen and its value is updated according to Steps 4, 5 and 6 described in Section 2.3 of Ref. [1]. Therefore, the updated pixel is the one chosen via the chaotic map and hence there is no way to ensure that all pixels will be mixed in the encryption process. To improve the algorithm, the direction of the mixing step can be inverted, that is Step 6 of (i) the algorithm [1] can be modified so that x(i) = xn + x(j) and thus all pixels will be encrypted. As regards the timing attack, this is clearly an interesting point, not usually taken into account when new encryption schemes are
∗
Corresponding author. E-mail address: [email protected] (A.N. Pisarchik). URL: http://www.cio.mx (A.N. Pisarchik).
0167-2789/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2010.02.011
proposed. By calculating the encryption time of a given image, it is possible to infer some information about two of the secret keys of the system, explicitly n and R. Some strategies can be developed to dodge this problem; for example, always performing a certain number of iterations of the logistic map n0 n, but taking into account only the result after n iterations; in this way, the time needed to encrypt the image is independent of the key value. On the other hand, any solution addressed to masking these two secret keys always results in increasing computational cost, which is not desirable in the context of image encryption. We believe that even if the attacker knows both keys, we can rely on the security given by the initial conditions and parameters of the logistic maps. It is easy to check that the security does not substantially decrease; according to the example given in footnote 8 of Ref. [1], the new key space dimension will be of about 10110 . Summing up, we agree with Arroyo et al. [2] that some improvements can be made to enhance the security and performance of the original algorithm [1], and we are grateful for their interest and ideas. References [1] A.N. Pisarchik, M. Zanin, Image encryption with chaotically coupled chaotic maps, Physica D 237 (2008) 2638–2648. [2] D. Arroyo, S. Lib, J.M. Amigó, G. Alvarez, R. Rhouma, Comment on ‘‘Image encryption with chaotically coupled chaotic maps’’, Physica D 239 (12) (2010) 1002–1006.
Contents lists available at ScienceDirect
Physica D journal homepage: www.elsevier.com/locate/physd
Reply to: ‘‘Comment on: ‘Image encryption with chaotically coupled chaotic maps’ [Physica D 2010]’’ A.N. Pisarchik a,∗ , M. Zanin b a
Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico
b
Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain
article
info
Article history: Received 26 January 2010 Accepted 19 February 2010 Available online 26 February 2010 Communicated by V. Rom-Kedar
abstract We respond to the comment by Arroyo et al. on our paper ‘‘Image encryption with chaotically coupled chaotic maps’’. © 2010 Elsevier B.V. All rights reserved.
Keywords: Chaotic cryptosystem Logistic map Image cipher
In their comment on our paper ‘‘Image encryption with chaotically coupled chaotic maps’’ [1], Arroyo et al. [2] correctly stated that Eq. (7) of Ref. [1] is wrong because it does not ensure that all values resulting from iteration of the logistic map Eq. (6) fall into the phase space of the chaotic attractor. This is obviously a misprint. In fact, the original algorithm which we used in our numerical simulations was free from this error; it contained the correct formula (Eq. (5) of Ref. [2]) and hence this misprint appeared only in the text and did not affect the results of the paper [1]. Another problem raised by the authors of the comment [2] is the existence of some pixels that are not encrypted by the algorithm, meaning that the information of some pixels may not be diffused through whole image; even so, this does not represent an important degradation of the algorithm security. This inconvenience appears due to the wrong direction of the encryption process. The original algorithm [1] encrypts every image pixel pi in a sequential fashion. Through a chaotic map, another pixel pk is chosen and its value is updated according to Steps 4, 5 and 6 described in Section 2.3 of Ref. [1]. Therefore, the updated pixel is the one chosen via the chaotic map and hence there is no way to ensure that all pixels will be mixed in the encryption process. To improve the algorithm, the direction of the mixing step can be inverted, that is Step 6 of (i) the algorithm [1] can be modified so that x(i) = xn + x(j) and thus all pixels will be encrypted. As regards the timing attack, this is clearly an interesting point, not usually taken into account when new encryption schemes are
∗
Corresponding author. E-mail address: [email protected] (A.N. Pisarchik). URL: http://www.cio.mx (A.N. Pisarchik).
0167-2789/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2010.02.011
proposed. By calculating the encryption time of a given image, it is possible to infer some information about two of the secret keys of the system, explicitly n and R. Some strategies can be developed to dodge this problem; for example, always performing a certain number of iterations of the logistic map n0 n, but taking into account only the result after n iterations; in this way, the time needed to encrypt the image is independent of the key value. On the other hand, any solution addressed to masking these two secret keys always results in increasing computational cost, which is not desirable in the context of image encryption. We believe that even if the attacker knows both keys, we can rely on the security given by the initial conditions and parameters of the logistic maps. It is easy to check that the security does not substantially decrease; according to the example given in footnote 8 of Ref. [1], the new key space dimension will be of about 10110 . Summing up, we agree with Arroyo et al. [2] that some improvements can be made to enhance the security and performance of the original algorithm [1], and we are grateful for their interest and ideas. References [1] A.N. Pisarchik, M. Zanin, Image encryption with chaotically coupled chaotic maps, Physica D 237 (2008) 2638–2648. [2] D. Arroyo, S. Lib, J.M. Amigó, G. Alvarez, R. Rhouma, Comment on ‘‘Image encryption with chaotically coupled chaotic maps’’, Physica D 239 (12) (2010) 1002–1006.