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Int. J. Communications, Network and System Sciences, 2011, 4, 452-455 doi:10.4236/ijcns.2011.47054 Published Online July 2011 (http://www.SciRP.org/journal/ijcns) Copyright © 2011 SciRes. IJCNS Chaotic Encryption Scheme Using 1-D Chaotic Map Mina Mishra, Vijay H. Mankar Department of Electronics & Telecommunication, Nagpur University, Maharastra, India Department of Electronics Engineering, Government Polytechnic Nagpur, Maharastra, India E-mail: {minamishraetc, vhmankar}@gmail.com Received February 5, 2011; revised April 3, 2011; accepted May 14, 2011 Abstract This paper proposes three different chaotic encryption methods using 1-D chaotic map known as Logistic map named as Logistic, NLFSR and Modified NLFSR according to the name of chaotic map and non-linear function involved in the scheme. The designed schemes have been crypt analyzed for five different methods for testing its strength. Cryptanalysis has been performed for various texts using various keys selected from domain of key space. Logistic and NLFSR methods are found to resist known plaintext attack for available first two characters of plaintext. Plaintext sensitivity of both methods is within small range along with me- dium key sensitivity. Identifiability for keys of first two of the scheme has not been derived concluding that methods may prove to be weak against brute-force attack. In the last modified scheme avalanche effect found to be improved compared to the previous ones and method is found to resist brute-force attack as it derives the conclusion for identifiability. Keywords: Cryptanalysis, Plaintext Sensitivity, Key Sensitivity, Identifiability, Brute-Force Attack 1. Introduction Chaotic cryptography [1] deals with hiding and recover- ing of secret messages using algorithm which consists of encryption rule that uses chaotic functions (analog or di- gital). Logistic map is one-dimensio nal map that consists of a parameter which is acting as secret key in the de- signed encryption schemes. The proposed methods use non-linear functions like, sinusoidal, non-linear shift re- gister and logistic map to built confusion and diffusion. The sine function describes a smooth repetitive oscil- lation. It’s most basic form as a function of time (t) is: sinyt At where, A, the amplitude is th e pe ak deviation of th e func- tion from its center position. ω, the angular frequency, specifies how many oscillations occur in a unit time in- terval, in radians per second. φ, the phase, specifies where in its cycle the oscillation begins at t = 0. A NLFSR (Non-Linear Feedback Shift-register) is a common component in modern stream ciphers, especially in RFID and smartcard applications. NLFSRs are known to be more resistant to cryptanalytic attacks than Linear Feedback Shift Registers (LFSR’s), although construction of large NLFSRs with guaranteed long periods remains an open problem. A NLFSR, is a shift register whose cu- rrent state is a non-linear function of its previous state. The NLFSR used here is shown in Figure 1 output. The logistic map is a polynomial mapping of degree 2, it takes a point, in a plane and maps it to a new point using followi n g exp ressions: 11 x krxk xk ; where, map depends on the parameter r. From r = 3.57 to r = 4, the map exhibits chaotic behavior which is shown in Figure 2. Cryptanalysis is the study of attacks against crypto- graphic schemes to disclose its possible weakness. Dur- ing cryptanalyzing a ciphering algorithm, the general assumption made is that the cryptanalyst knows exactly the design and working of the cryptosystem under study, i.e., he/she knows everything about the cryptosystem ex- cept the secret key. It is possible to differentiate between different levels of attacks on cryptosystems. They are briefly explained as fol l ows: a) Cipher text-only attack: The attacker possesses a string of cipher text. b) Known plain text: The attacker possesses a string of plain text, p, and the corresponding cipher text, c. c) Chosen plain text: The attacker has obtained tempo- rary access to the encryption machinery. He nce he/she can choose a plain text string, p, and construct the corres- M. MISHRA ET AL. Copyright © 2011 SciRes. IJCNS 453 Figure 1. NLFSR using 8-bit shift registers. Figure 2. Plot of Logistic map for r = 3.99 , x(0) = 0.99, n = 50 00. ponding cipher text string, c. d) Chosen cipher text: The attacker has obtained tem- porary access to the decryption machinery. Hence he/ she can choose a cipher text string, c, and construct the corre- sponding plain text string, p. e) Brute Force Attack: A brute force attack is the me- thod of breaking a cipher by trying every possible key. The brute force attack is the most expensive one, owing to the exhaustive search. In addition to the five general attacks described above, there are some other specialized attacks, like, differential and linear attacks. Differential cryptanalysis is a kind of ch osen-plaintext attack aimed at finding the secret key in a cipher. It ana- lyzes the effect of particular differences in chosen plain- text pairs on the differences of the resultant cipher text pairs. These differences can be used to assign probabili- ties to the possible keys and to locate the most probable key. Linear cryptanalysis is a type of known-plaintext at- tack, whose purpose is to construct a linear approximate expression of the cipher under study. It is a method of finding a linear approximation expression or linear path between plaintext and cipher text bits and then extends it to the entire algorithm and finally reaches a linear appro- ximate expression without intermediate value. 2. Methodology The complete methodology involved during designing is clearly cited with the help of block diagram as shown in Figure 3. The plain text is encrypted by an encryption rule which uses non-linear function and the state generated by the chaotic system in the transmitter [2]. Th e scrambled out- put is inputted further to the chao tic system such that the chaotic dynamics is changed continuously in a very complex way. Then another state variable of the chaotic system [3] in the transmitter is transmitted through the channel. Recovery of the plaintext is done by decrypting the input (ciphertext) using reverse process of en cryption, as used in the transmitter. In the modified NLFSR method traditional encryption method is used along with above method. For Logistic- sinusoidal function and logistic map and for NLFSR- Non-linear shift register and logistic map has been used respectively. 3. Analysis and Results The analysis accomplished on the designed ciphers has been done for key space [4], avalance effect, known- plaintext attack, Identifiability [5,6] and results are cited in tabular form. a) Logistic: Key Space range is from 3.57 to 4.0; b) NLFSR: Key Space range is from 3.57 to 4.0; c) Modified NLFSR: Key space is from 3.57 to 4.0. 4. Conclusions and Future Scope Tables 1 and 2 show that Logistic and NLFSR methods are found to resist known plaintext attack for available first two characters of plaintext but modified NLFSR proves its resistance against the attack for available some numbers of characters of plaintext as given in the column of Table 3. It also proves its resistance against brute- force attack by deriving conclusion for identifiability of selected key. All the three methods are similar to one- time pad type instead more secure. More secure ciphers can be designed and crypt ana- lyzed [7,8] using different non-linear functions and 2-D Figure 3. Methodology used in design of chaotic cryptosys- tem. (a), Transmitter end (encryptor); (b), Receiver end (de- cryptor). (p: plaintext; c: ciphertext; x: state of chaotic fun c- tion; y: Intermediate encrypted plaintext). M. MISHRA ET AL. Copyright © 2011 SciRes. IJCNS 454 Table 1. Analysis table for logistic method. Sl. no. Plaintext Key value Ciphertext Plaintext sensitivity (in %) Key sensitivity (in %) Domain for key with increment = 0.0001 Identifiability of key for iteration value = 1 or 2 Ro b ustness against known plaintext attack for p=[p1 p2] Whether key can act as secret key against Brute Force attack? 1. Hello! How are you? 3.65 BHZOR0Nhw`X Vb?WoM 3.947446.7105(3.57, 3.78) NI R NO 2. Ram scored 98 marks in Maths. 3.71 jPw+4(f}mmWek f8bq9mC|*i: 2.083343.7500(3.67, 3.87) NI R NO 3. Thank you, Sir. 3.87 >3ieVyl1[w5f\ 4.166750 (3.77, 3.97) NI R NO 4. The match was very exciting. 3.88 J?/el?jZl67 H3;L LE~2Je-yeOzY 4.741446.1207(3.78, 3.98) NI R NO 5. I will be leaving at 9 p.m. 3.89 ZDWuQ/9H<r4c* \{>l}s?D4UgX 1.851942.5926(3.79, 3.99) NI R NO NI – Non-Identifiable ; I – Identifi able ; R – Robust; p [p1 p2…pn] – First ‘n’ characters of available plaintext string. Table 2. Analysis table for NLFSR method. Sl.no. Plaintext Key value Ciphertext Plaintext sensitivity (in %) Key sensitivity (in %) Domain for key With increment = 0.0001 Identifiability of key for iteration value =1 or 2 Robustness against known plaintext attack for p = [p1 p2] Whether key can act as secret key against Brute Force attack? 1. I am going to market. 3.7328 ¸èøxè0ø¸OØ¨0v 0.5682 20.4545(3.57,3.77) NI R NO 2. Hello! How are you? 3.7694 ¨78÷øðP¨ø°ý 0.6579 25 (3.57,3.77) NI R NO 3. Ram scored 98 marks in Maths. 3.8551 K¸ÐÈ÷P¨'¸PØÏw ´0Ïv 0.4167 18.3333(3.76,3.96) NI R NO 4. Thank you, Sir. 3.8641 ,w× ø¯6ÐPu 0.8333 30.8333(3.78,3.98) NI R NO 5. The match was very exciting. 3.9065 +§·0ÇïÏo§P§- È/wèu 0.4310 28.0172(3.785,3.985) NI R NO Table 3. Analysis table for modified NLFSR method. Sl. no. Plaintext Key value Ciphertext Plaintext sensitivity (in %) Key sensitivity (in %) Domain for key With increment = 0.0001 Identifiability of key for iteration value = 1 or 2 Robustness against known plaintext attack. Whether key can act as secret key against Brute Force attack? 1. What is your name? 3.6424 -§Án_ªÏ¾X<Ó¬ sÊmÄ 9.2105 16.4474(3.57,3.77) I R for p=[p1 p2 …p5] YES 2. I am going to market. 3.7328 || vfª( ò `(¬ô6ýR2 & 10.795517.0455(3.57,3.77) I R for p=[p1 p2…p19] YES 3. Sita is singing very well. 3.8544 BÛ¼~eÂ=ßRf1 àdØùFî;d*p 6.4815 12.9630(3.66,3.86) I R for p=[p1 p2 …p19] YES 4. Ram scored 98 marks in Maths. 3.8551 Lª ½jdSÜ* ¢æ %<Äc&;$ÍèÅ<- ±& 5.8333 14.5833(3.76,3.96) I R for p=[p1 p2 …p19] YES 5. Jaycee publication. 3.8529 =txý®&Ó`Fà- â$# 14.375023.1250(3.77,3.97) I R for p=[p1 p2 … p15] YES M. MISHRA ET AL. Copyright © 2011 SciRes. IJCNS 455 chaotic maps [9] using different encryption rule. Im- provement in the property of avalanche effect of designed ciphers is required in future. 5. References [1] F. Dachselt and W. Schwarz, “Chaos and Cryptography,” IEEE Transactions on Circuits and Systems I, Vol. 48, No. 6, 2001, pp. 1498-1509. doi:10.1109/TCSI.2001.972857 [2] T. Yang, “A Survey of Chaotic Secure Communication Systems,” International Journal of Computational Cog- nition, Vol. 2, No. 2, 2004, pp. 3473-3749. [3] G. Alvarez, F. Montoya, M. Romera and G. Pastor, “Cha- otic Cryptosystems”, In: L. D. Sanson, Ed., Proceedings of the IEEE 33rd 1999 Annual International Carnahan Conference on Security Technology, Madrid, 5-7 October 1999, pp. 332-338. [4] G. Alvarez and S. Li, “Some Basic Cryptographic Re- quirements for Chaos-Based Cryptosystems,” Interna- tional Journal of Bifurcation and Chaos, Vol. 16, No. 8, 2006, pp. 2129-2151. doi:10.1142/S0218127406015970 [5] S. Nõmm and C. H. Moog, “Identifiability of Discrete- Time Nonlinear Systems,” Proceedings of the 6th IFAC Symposium on Nonlinear Control Systems, Stuttgart, Vol. 1, No. 3, 2004, pp. 477-489. [6] F. Anstett, G. Millerioux and G. Bloch, “Chaotic Cryp- tosystems: Cryptanalysis and Identifiability,” IEEE Tran- sactions on Circuits and Systems, Vol. 53, No. 12, 2008, pp. 2673-2680. [7] E. Solak, “On the Security of a Class of Discrete-Time Chaotic Cryptosystems,” Physics Letters A, Vol. 320, No. 5-6, 2004, pp. 389-395. doi:10.1016/j.physleta.2003.11.008 [8] N. Masuda and K. Aihara, “Cryptosystems with Discre- tized Chaotic Maps,” IEEE Transactions on Circuits and Systems I, Vol. 53, No. 6, 2002, pp. 1341-1352. [9] F. Huang and Z. H. Guan, “Cryptosystem Using Chaotic Keys,” Chaos, Solitons and Fractals, Vol. 23, No. 3, 2005, pp. 851-855.doi:10.1016/j.chaos.2004.05.026 |