Lessons learnt from the cryptanalysis of chaos-based ciphers

Studies in Computational Intelligence

Volumn 354, Issue , 2011, Pages 257-295

  Alvarez, Gonzalo ^a   Amigó, José María ^b   Arroyo, David ^a   Li, Shujun ^c  

^a CSIC   (Spain)

^b MIGUEL HERNÁNDEZ UNIVERSITY   (Spain)

^c UNIVERSITY OF KONSTANZ   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79961099164     PISSN: 1860949X     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-20542-2_8     Document Type: Article

References (117)

1. Addabbo, T., Alioto, M., Fort, A., Pasini, A., Rocchi, S., Vignoli, V.: A class of maximum-period nonlinear congruential generators derived from the reńyi chaotic map. IEEE Transactions on Circuits and Systems-I: Regular Papers 54(4), 816-828 (2007) (Pubitemid 46621573)
2. Alvarez, G.: Security problems with a chaos-based deniable authentication scheme. Chaos, Solitons & Fractals 26(1), 7-11 (2005) (Pubitemid 40556362)
3. Alvarez, G., Arroyo, D., Nunez, J.: Application of Gray code to the cryptanal-ysis of chaotic cryptosystems. In: 3rd International IEEE Scientific Conference on Physics and Control (PhysCon 2007), September 3-7, Potsdam, Germany (2007), http://lib.physcon.ru/?item=1358
4. Alvarez, G., Li, S.: Estimating short-time period to break different types of chaoitc modulation based secure communications. arxiv:nlin.CD/0406039 (2004), http://arxiv.org/abs/nlin/0406039
5. Alvarez, G., Li, S.: Breaking an encryption scheme based on chaotic baker map. Physics Letters A 352(1-2), 78-82 (2006) (Pubitemid 43286853)
6. Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos-based cryptosystems. International Journal of Bifurcation and Chaos 16(8), 2129-2151 (2006) (Pubitemid 44671739)
7. Alvarez, G., Li, S., Hernandez, L.: Analysis of security problems in a medical image encryption system. Computers in Biology and Medicine 37(3), 424-427 (2007) (Pubitemid 46048792)
8. Alvarez, G., Li, S., Montoya, F., Romera, M., Pastor, G.: Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos, Solitons & Fractals 24(3), 775-783 (2005) (Pubitemid 40114334)
9. Alvarez, G., Montoya, F., Pastor, G.: Cryptanalysis of a discrete chaotic cryp-tosystem using external key. Physics Letters A 319, 334-339 (2003)
10. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Cryptanalysis of a chaotic encryption system. Physics Letters A 276, 191-196 (2000)
11. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Cryptanalysis of an ergodic chaotic cipher. Physics Letters A 311, 172-179 (2003)
12. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Breaking parameter modu-lated chaotic secure communication system. Chaos, Solitons & Fractals 21(4), 793-797 (2004)
13. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Cryptanalysis ofdynamic look-up table based chaotic cryptosystems. Physics Letters A 326, 211-218 (2004)
14. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Keystream cryptanalysis of a chaotic cryptographic method. Computer Physics Communications 156, 205-207 (2004)
15. Amiǵo, J., Kocarev, L., Szczepanski, J.: Theory and practice ofchaotic cryp-tography. Physics Letters A 366(3), 211-216 (2007)
16. Amiǵo, J.M., Szczepanski, J.: Approximations of dynamical systems and their applications to cryptography. International Journal of Bifurcation and Chaos 13, 1937-1948 (2003) (Pubitemid 37208911)
17. Amiǵo, J.M.: Chaos-Based Cryptography. In: Kocarev, L., Galias, Z., Lian, S. (eds.) Intelligent Computing Based on Chaos. Studies in Computational Intelligence, vol. 184, pp. 291-313. Springer, Heidelberg (2009)
18. Amiǵo, J.M., Szczepanski, J., Kocarev, L.: A chaos-based approach to the design of cryptographically secure substitutions. Physics Letters A 343, 55-60 (2005) (Pubitemid 40941628)
19. Argenti, F., Benzi, S., Re, E.D., Genesio, R.: Stream cipher system based on chaotic maps. In: Proceedings of SPIE Mathematics and Applications of Data/Image Coding, Compression, and Encryption III, vol. 4122, pp. 10-17. SPIE (2001)
20. Ariffin, M., Noorani, M.: Modified Baptista type chaotic cryptosystem via matrix secret key. Physics Letters A 372, 5427-5430 (2008)
21. Arroyo, D.: Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems. Ph.D. thesis, ETSIA of the Polytechnic University ofMadrid, Madrid, Spain (2009)
22. http://digital.csic.es/handle/10261/15668
23. Arroyo, D., Alvarez, G., Amiǵo, J.M.: Estimation of the control parameter from symbolic sequences: Unimodal maps with variable critical point. Chaos: An Interdisciplinary Journal of Nonlinear Science 19 (2009), Art. no. 023125
24. Arroyo, D., Alvarez, G., Amiǵo, J.M., Li, S.: Cryptanalysis of a family of self-synchronizing chaotic stream ciphers. Communications in Nonlinear Science and Numerical Simulation 16(2), 805-813 (2011)
25. Arroyo, D., Alvarez, G., Li, S.: Some hints for the design of digital chaos-basedcryptosystems: lessons learned from cryptanalysis. In: Second IFAC Con-ference on Analysis and Control of Chaotic Systems, Queen Mary, University of London (2009)
26. Arroyo, D., Alvarez, G., Li, S., Li, C., Fernandez, V.: Cryptanalysis of a new chaotic cryptosystem based on ergodicity. International Journal of Modern Physics B 23(5), 651-659 (2009)
27. Arroyo, D., Alvarez, G., Li, S., Li, C., Nunez, J.: Cryptanalysis of a discrete-time synchronous chaotic encryption system. Physics Letter A 372(7), 1034-1039 (2008)
28. Arroyo, D., Li, C., Li, S., Alvarez, G.: Cryptanalysis of a computer cryptog-raphy scheme based on a filter bank. Chaos, Solitons & Fractals 41, 410-413 (2009)
29. Arroyo, D., Li, C., Li, S., Alvarez, G., Halang, W.A.: Cryptanalysis of an image encryption scheme based on a new total shuffling algorithm. Chaos, Solitons & Fractals 41(5), 2613-2616 (2009)
30. Arroyo, D., Rhouma, R., Alvarez, G., Li, S., Fernandez, V.: On the security of a new image encryption scheme based on chaotic map lattices. Chaos: An Interdisciplinary Journal of Nonlinear Science 18 (2008), Art. no. 033112
31. Banerjee, S., Yorke, J.A., Grebogi, C.: Robust chaos. Physical Review Letters 80, 14 (1998)
32. Baptista, M.S.: Cryptography with chaos. Physics Letters A 240(1-2), 50-54 (1998) (Pubitemid 128390514)
33. Beth, T., Lazic, D.E., Mathias, A.: Cryptanalysis of cryptosystems based on remote chaos replication. In: EUROCRYPT 1994. LNCS, vol. 950, pp. 318- 331. Springer, Heidelberg (1994)
34. Brumley, D., Boneh, D.: Remote timing attacks are practical. In: Proceedings of the 12th USENIX Security Symposium, pp. 1-14. USENIX Association (2003)
35. Chee, C.Y., Xu, D.: Chaotic encryption using dicrete-time synchronous chaos. Physics Letters A 348(3-6), 284-292 (2006) (Pubitemid 41803022)
36. Chen, G., Ueta, T.: Yet another chaotic attactor. International Journal of Bifurcation and Chaos 9(7), 1465-1466 (1999)
37. Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic theory. Springer, New York (1982)
38. Cuomo, K., Oppenheim, A.V., Strogatz, S.: Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing 40(10), 626-633 (1993)
39. Dedieu, H., Kennedy, M., Hasler, M.: Chaos shift keying: modulation and de-modulation of a chaotic carrier using self-synchronizing Chua's circuits. IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Process-ing 40, 634-641 (1993)
40. Dedieu, H., Ogorzalek, M.J.: Identifiability and identification of chaotic sys-tems based on adaptive synchronization. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 44(10), 948-962 (1997) (Pubitemid 127768197)
41. Feki, M.: An adaptive chaos synchronization scheme applied to secure com-munication. Chaos, Solitons & Fractals 18(1), 141-148 (2003)
42. Feldmann, U., Hasler, M., Schwarz, W.: Communication by chaotic signals: the inverse system approach. International Journal of Circuit Theory and Applications 24, 551-579 (1996) (Pubitemid 126625501)
43. Fradkov, A.L., Markov, A.Y.: Adaptive synchronization of chaotic systems based on speed gradient method and passification. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 44, 905-912 (1997) (Pubitemid 127768192)
44. Fridrich, J.: Symmetric ciphers based on two-dimensional chaotic maps. In-ternational Journal of Bifurcation and Chaos 8, 1259-1284 (1998)
45. Gao, T., Chen, Z.: Image encryption based on a new total shuffling algorithm. Chaos, Solitons & Fractals 38(1), 213-220 (2008) (Pubitemid 351503987)
46. Gonźalez-Miranda, J.: Synchronization and control of chaos. Imperial College Press, London (2004)
47. Habutsu, T., Nishio, Y., Sasase, I., Mori, S.: A secret key cryptosystem by iterating a chaotic map. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 127-140. Springer, Heidelberg (1991)
48. Hasler, M.: Synchronization of chaotic systems and transmission of informa-tion. International Journal of Bifurcation and Chaos 8(4), 647-659 (1998) (Pubitemid 128341688)
49. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (1961)
50. Hirsch, M.W., Smale, S.: Differential equations, dynamical systems, and linear algebra. Academic Press, Inc., San Diego (1974)
51. Hu, G., Feng, Z., Meng, R.: Chosen ciphertext attack on chaos communication based on chaos synchronization. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 50(2), 275-279 (2003)
52. Huijberts, H., Nijmeijer, H., Willems, R.: System identification in communi-cation with chaotic systems. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 47, 800-808 (2000)
53. Inoue, E., Ushio, T.: Chaos communication using unknown input observers. Electronics and Communications in Japan Part III: Fundamental Electronic Science 84(12), 21-27 (2001) (Pubitemid 32697860)
54. Jakimoski, G., Kocarev, L.: Chaos and cryptography: Block encryption ci-phers based on chaotic maps. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 48(2), 163-169 (2001) (Pubitemid 32254607)
55. Jiang, Z.P.: A note on chaotic secure communication systems. IEEE Transac-tions on Circuits and Systems-I: Fundamental Theory and Applications 49(1), 92-96 (2002)
56. Kocarev, L.: Chaos-based cryptography: A brief over view. IEEE Circuits and Systems Magazine 1(2), 6-21 (2001)
57. Kocarev, L., Halle, K.S., Eckert, K., Chua, L.O., Parlitz, U.: Transimission of digital signals by chaotic synchronization. International Journal of Bifurcation and Chaos 2(4), 973-977 (1992)
58. Kocarev, L., Makraduli, J., Amato, P.: Public-key encryption based on Cheby-shev polynomials. Circuits, Systems, and Signal Processing 24, 497-517 (2005) (Pubitemid 41830142)
59. Kocarev, L., Sterjev, M., Fekete, A., Vattay, G.: Public-key encryption with chaos. Chaos: An Interdisciplinary Journal of Nonlinear Science 14(4), 1078-1082 (2004)
60. Kocarev, L., Szczepanski, J., Amigo, J., Tomovski, I.: Discrete chaos-I: The-ory. IEEE Transactions on Circuits and Systems-I: Regular Papers 53(6), 1300-1309 (2006)
61. Kocher, P.C.: Timing attacks on implementations ofdiffie-hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104-113. Springer, Heidelberg (1996) (Pubitemid 126106234)
62. Kolumban, G., Kennedy, M., Chua, L.O.: The role of synchronization in dig-ital communications using chaos - Part II: Chaotic modulation and chaotic synchronization. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 45(11), 1129-1140 (1998) (Pubitemid 128745132)
63. Letellier, C., Gouesbet, G.: Topological characterization of reconstructed at-tractors modding out symmetries. Journal de Physique II 6(11), 1615-1638 (1996) (Pubitemid 126795496)
64. Li, C., Li, S., Alvarez, G., Chen, G., Lo, K.-T.: Cryptanalysis of two chaotic encryption schemes based on circular bit shift and XOR operations. Physics Letters A 369, 23-30 (2007)
65. Li, S.: Analyses and new designs ofdigital chaotic ciphers. Ph.D. thesis, School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, China (2003), http://www.hooklee.com/pub.html
66. Li, S., Alvarez, G., Chen, G.: Breaking a chaos-based secure communication scheme designed by an improved modulation method. Chaos, Solitons & Fractals 25(1), 109-120 (2005) (Pubitemid 40220065)
67. Li, S., Alvarez, G., Chen, G.: Return-map cryptanalysis revisited. Interna-tional Journal of Bifurcation and Chaos 16(5), 1557-1568 (2006) (Pubitemid 44259857)
68. Li, S., Alvarez, G., Li, Z., Halang, W.: Analog chaos-based secure communica-tions and cryptanalysis: a briefsurvey. In: Kurths, J., Fradkov, A., Chen, G. (eds.) 3rd Int. IEEE Scientific Conference on Physics and Control (PhysCon 2007), Potsdam, Germany, p. 92 (2007), Full edition available at: http://www.hooklee.com/Papers/PhysCon2007.pdf.
69. Li, S., Chen, G., Mou, X.: On the dynamical degradation ofdigital piecewise linear chaotic maps. International Journal of Bifurcation and Chaos 15(10), 3119-3151 (2005) (Pubitemid 41831941)
70. Li, S., Li, C., Chen, G., Bourbakis, N.G., Lo, K.T.: A general quantitative cryptanalysis of permutation-only multimedia ciphers against plaintext at-tacks. Signal Processing: Image Communication 23(3), 212-223 (2008)
71. Li, S., Mou, X., Cai, Y.: Pseudo-random bit generator based on couple chaotic systems and its applications in stream-ciphers cryptography. In: Pandu Ran-gan, C., Ding, C. (eds.) INDOCRYPT 2001. LNCS, vol. 2247, pp. 316-329. Springer, Heidelberg (2001)
72. Li, T.Y., Yorke, J.A.: Period three implies chaos. The American Mathematical Monthly 82, 985-992 (1975)
73. Lian, K.Y., Liu, P.: Synchronization with message embedded for generalized Lorenz chaotic circuits and its error analysis. IEEE Transactions on Cir-cuits and Systems-I: Fundamental Theory and Applications 47(9), 1418-1424 (2000) (Pubitemid 32032534)
74. Ling, B.W.-K., Ho, C.Y.-F., Tam, P.K.-S.: Chaotic filter bank for computer cryptography. Chaos, Solitons & Fractals 34, 817-824 (2007) (Pubitemid 46601918)
75. Liu, L., Wu, X., Hu, H.: Estimating system parameters of Chua's circuit from synchronizing signal. Physics Letters A 324(1), 36-41 (2004)
76. Lorenz, E.: Deterministic nonperiodic flow. Journal of the Atmospheric Sci-ences 20, 130-141 (1963)
77. Manjunath, G., Fournier-Prunaret, D.: A qualitative analysis of decipher-ing errors in chaos shift keying. International Journal of Bifurcation and Chaos 19(6), 2085-2092 (2009)
78. Masuda, N., Jakimoski, G., Aihara, K., Kocarev, L.: Chaotic block ciphers: from theory to practical algorithms. IEEE Transactions on Circuits and Systems-I: Regular Papers 53(6), 1341-1352 (2006) (Pubitemid 44209546)
79. Matthews, R.: On the derivation of a chaotic encryption algorithm. Cryptologia 13, 29-42 (1989)
80. Maze, G.: Algebraic methods for constructing one-way trapdoor functions. Ph.D. thesis, University of Notre Dame (2003)
81. Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptog-raphy. CRC Press, Boca Raton (1997)
82. Milĺerioux, G., Amiǵo, J.M., Daafouz, J.: A connection between chaotic and conventional cryptography. IEEE Transactions on Circuits and Systems-I: Regular Papers 55(6), 1695-1703 (2008)
83. Millerioux, G., Daafouz, J.: Unknown input observers for message-embedded chaos synchronization of discrete-time systems. International Journal of Bi-furcation and Chaos 14(4), 1357-1368 (2004) (Pubitemid 40242160)
84. Millerioux, G., Mira, C.: Coding scheme based on chaos synchronization from noninvertible maps. International Journal of Bifurcation and Chaos 8, 2019-2029 (1998) (Pubitemid 128355282)
85. NIST: A statistical test suite for random and pseudorandom number genera-tors for cryptographic applications. NIST Special Publication 800-22 Revision 1A (2010), http://csrc.nist.gov/rng/rng2.html
86. Orúe, A., Alvarez, G., Pastor, G., Romera, M., Montoya, F., Li, S.: A new parameter determination method for some double-scroll chaotic systems and its applications to chaotic cryptanalysis. Communications in Nonlinear Science and Numerical Simulations 15(11), 3471-3483 (2010)
87. Orúe, A., Fernandez, V., Alvarez, G., Pastor, G., Romera, M., Montoya, F.: Determination of the parameters for a Lorenz system and application to break the security of two-channel chaotic cryptosystems. Physics Letters A 372(34), 5588-5592 (2008)
88. Pareek, N.K., Patidar, V., Sud, K.K.: Discrete chaotic cryptography using external key. Physics Letters A 309, 75-82 (2003)
89. Parker, A., Short, K.M.: Reconstructing the keystream form a chaotic en-crypiton scheme. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 48(5), 624-630 (2001) (Pubitemid 32536507)
90. Parlitz, U., Chua, L.O., Kocarev, L., Halle, K.S., Shang, A.: Transmission of digital signals by chaotic synchronization. International Journal of Bifurcation and Chaos 2(4), 973-977 (1992)
91. Pastor, G., Romera, M., Montoya, F.: A revision of the Lyapunov exponent in 1D quadratic maps. Physica D 107, 17-22 (1997) (Pubitemid 127701493)
92. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Physical Re-view Letters 64(8), 821-824 (1990)
93. Ṕerez, G., Cerdeira, H.A.: Extracting messages masked by chaos. Physical Review Letters 74(11), 1970-1973 (1995)
94. Pisarchik, A.N., Flores-Carmona, N.J., Carpio-Valadez, M.: Encryption and decryption of images with chaotic map lattices. Chaos: An Interdisciplinary Journal of Nonlinear Science 16(3) (2006), Art. no. 033118
95. Rajendra, U., Bhat, S., Kumar, S., Min, L.: Transimission and storage of medical images with patient information. Comput. Biol. Med. 33, 303-310 (2003) (Pubitemid 36638785)
96. Rhouma, R., Solak, E., Arroyo, D., Li, S., Alvarez, G., Belghith, S.: Comment on "modified Baptista type chaotic cryptosystem via matrix secret key". Phys. Lett. A 372, 5427 (2008); Physics Letters A 373(37), 3398-3400 (2009)
97. Shannon, C.: Communication theory of secrecy systems. Bell System Technical Journal 28, 656-715 (1949)
98. Skrobek, A.: Approximation of a chaotic orbit as a cryptanalytical method on Baptista's cipher. Physics Letters A 372(6), 849-859 (2008)
99. Solak, E., Çokal, C., Yildiz, O.T., Biyikǒglu, T.: Cryptanalysis of Fridrich's chaotic image encryption. International Journal of Bifurcation and Chaos 20(5), 1405-1413 (2010)
100. Stamp, M., Low, R.M.: Applied cryptanalysis: breaking ciphers in the real world. John Wiley & Sons, Inc., Hoboken (2007)
101. Stinson, D.R.: Cryptography: Theory and Practice. CRC Press, Boca Raton (1995)
102. Storm, C., Freeman, W.J.: Detection and classification ofnonlinear dynamic switching events. Physical Review E 58, 1159-1162 (2002)
103. Szczepanski, J., Amiǵo, J., Michalek, T., Kocarev, L.: Cryptographically se-cure substitutions based on the approximation of mixing maps. IEEE Trans-actions on Circuits and Systems-I: Regular Papers 52, 443-453 (2005) (Pubitemid 40324800)
104. Tao, C., Du, G.: A new approach to breaking down chaotic secure communica-tion. International Journal of Bifurcation and Chaos 13(9), 2689-2698 (2003) (Pubitemid 37396716)
105. Tao, C., Du, G., Zhang, Y.: Decoding digital information from the cas-caded heterogeneous chaotic systems. International Journal of Bifurcation and Chaos 13(6), 1599-1608 (2003) (Pubitemid 37067812)
106. Vaidya, P.G., Angadi, S.: Decoding chaotic cryptography without access to the superkey. Chaos, Solitons & Fractals 17(2-3), 379-386 (2003)
107. Wang, X., Duan, C., Gu, N.: A new chaotic cryptography based on ergodicity. International Journal ofModern Physics B 22(7), 901-908 (2008)
108. Wang, X., Zhan, M., Lai, C.H., Hu, G.: Error function attack of chaos syn-chronization based encrypiton schemes. Chaos: An Interdisciplinary Journal of Nonlinear Science 14(1), 128-137 (2004) (Pubitemid 38430244)
109. Wu, C.W.: Synchronization in coupled chaotic circuits and systems. World Scientific, New Jersey (2002)
110. Wu, C.W., Chua, L.O.: A simple way to synchronize chaotic systems with applications to secure communications systems. International Journal of Bi-furcation and Chaos 3(6), 1619-1627 (1993)
111. Xiao, D., Liao, X., Wong, K.W.: An efficient entire chaos-based scheme for deniable authentication. Chaos, Solitons & Fractals 23(4), 1327-1331 (2005)
112. Yang, T.: Recovery of digital signals from chaotic switching. International Journal of Circuit Theory and Applications 23(6), 611-615 (1995)
113. Yang, T.: A survey of chaotic secure communication systems. Intenational Journal ofComputational Cognition 2(2), 81-130 (2004)
114. Yang, T., Wu, C.W., Chua, L.O.: Cryptography based on chaotic systems. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Ap-plications 44(5), 469-472 (1997) (Pubitemid 127768156)
115. Yang, T., Yang, L.-B., Yang, C.-M.: Breaking chaotic secure communications using spectrogram. Physics Letters A 247(1-2), 105-111 (1998) (Pubitemid 128398791)
116. Yang, T., Yang, L.-B., Yang, C.-M.: Breaking chaotic switching using gener-alized synchronization. IEEE Transactions on Circuits and Systems-I: Fun-damental Theory and Applications 45(10), 1062-1067 (1998) (Pubitemid 128745933)
117. Zhang, Y., Tao, C., Jiang, J.J.: Theoretical and experimental studies of pa-rameter estimation based on chaos feedback synchronization. Chaos: An In-terdisciplinary Journal of Nonlinear Science 16(4) (2006), Art. no. 043122