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In the design of cryptographic algorithms, S-boxes provide the cryptosystems with the information confusion function. The traditional cryptography indexes of the S-boxes generally include linear deviation, differential characteristics, algebraic immunity, fixed point mumber, snowslide effect, and so on. In 2006, Kocarev et al. (Kocarev L, Szczepanski J, Amigo J M and Tomovski I 2006 IEEE Transactions on Circuits and Systems-I: regular papers 53 6 1300) set up a discrete chaos theory based on the finite set. In light of the theory in this paper, we introduce the definition of the Lyapunov exponent with Hamming distance, calculate and compare the Lyapunov exponent values of the S-boxes in several cryptographic algorithms. In this paper we prove that a map defined on the Euclidean distance has a maximal Lyapunov exponent value of 0. In this paper it is shown that the relationship between the Lyapunov exponent and the snowslide effect of the S-box is the relationship between the butterfly effect in chaos theory and the snowslide effect in cryptography. The definition of the Lyapunov exponent of the proposed S-boxes may be complementary to the traditional cryptography indexes of the S-box.
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Keywords:
- finite set /
- discrete chaos theory /
- S-boxes /
- Lyapunov exponent
[1] Biham E, Shamir A 1991 J. Cryptology 4 3
[2] Mitsuru M 1998 in Advances in Cryptology: EUROCRYPT'93 (Berlin: Springer-Veriag) p386
[3] Hitzl D L, Zele F 1985 Physica D 14 305
[4] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 648 821
[5] Wu C W, Chua L O 1993 Int. J. Bifurcat. Chaos 3 1619
[6] Yang T, Chua L O 1996 Int. J. Bifurcat. Chaos 6 2653
[7] Kocarev L, Szczepanski J, Amigo J M, Tomovski I 2006 IEEE Trans. Circuits Syst. I: Regular Papers 53 1300
[8] Amigo J M, Kocarev L, Szczepanski J 2007 Phys. Lett. A 366 211
[9] Chen G R, Wang X F 2006 Chaotic Theory, Method and Application of Dynamic System (Shanghai: Shanghai Jiaotong University Press) p88 (in Chinese) [陈关荣, 汪小帆 2006 动力系统的混沌化-理论、方法与应用(上海: 上海交通大学出版社) 第88页]
[10] Zhou X Y 2011 Acta Phys. Sin. 60 100503 (in Chinese) [周小勇 2011 60 100503]
[11] Cao G H, Hu K, Tong W 2011 Acta Phys. Sin．60 110508 (in Chinese) [曹光辉, 胡凯, 佟维 2011 60 110508]
[12] Fridrich J 1998 Int. J. Bifurcat. Chaos 8 1259
[13] Wang J, Jiang G P 2011 Acta Phys. Sin. 60 060503 (in Chinese) [王静, 蒋国平 2011 60 060503]
[14] Chirikov B V, Vivaldi F 1999 Physica D 129 223
[15] Lü S W, Fan X B, Wang Z S 2008 Complete Mapping and Application in Cryptography (Hefei: University of Science and Technoloqy of China Press) p244 (in Chinese) [吕述望, 范修斌, 王昭顺 2008 完全映射及其密码学应用(合肥:中国科技大学出版社) 第244页]
[16] Kazlauskas K, Kazlauskas J 2009 Informatica 20 23
[17] Tang G P, Liao X F, Chen Y 2005 Chaos Soliton. Fract. 23 413
[18] Amigó J M, Kocarev L, Szczepanski J 2007 IEEE Trans. Circuits Syst. II: Express Briefs 54 882
[19] Webster A F, Tavares S E 1986 in Advances in Cryptology: Proceedings of CRYPTO'85 (Berlin: Springer-Verlag) p523
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[1] Biham E, Shamir A 1991 J. Cryptology 4 3
[2] Mitsuru M 1998 in Advances in Cryptology: EUROCRYPT'93 (Berlin: Springer-Veriag) p386
[3] Hitzl D L, Zele F 1985 Physica D 14 305
[4] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 648 821
[5] Wu C W, Chua L O 1993 Int. J. Bifurcat. Chaos 3 1619
[6] Yang T, Chua L O 1996 Int. J. Bifurcat. Chaos 6 2653
[7] Kocarev L, Szczepanski J, Amigo J M, Tomovski I 2006 IEEE Trans. Circuits Syst. I: Regular Papers 53 1300
[8] Amigo J M, Kocarev L, Szczepanski J 2007 Phys. Lett. A 366 211
[9] Chen G R, Wang X F 2006 Chaotic Theory, Method and Application of Dynamic System (Shanghai: Shanghai Jiaotong University Press) p88 (in Chinese) [陈关荣, 汪小帆 2006 动力系统的混沌化-理论、方法与应用(上海: 上海交通大学出版社) 第88页]
[10] Zhou X Y 2011 Acta Phys. Sin. 60 100503 (in Chinese) [周小勇 2011 60 100503]
[11] Cao G H, Hu K, Tong W 2011 Acta Phys. Sin．60 110508 (in Chinese) [曹光辉, 胡凯, 佟维 2011 60 110508]
[12] Fridrich J 1998 Int. J. Bifurcat. Chaos 8 1259
[13] Wang J, Jiang G P 2011 Acta Phys. Sin. 60 060503 (in Chinese) [王静, 蒋国平 2011 60 060503]
[14] Chirikov B V, Vivaldi F 1999 Physica D 129 223
[15] Lü S W, Fan X B, Wang Z S 2008 Complete Mapping and Application in Cryptography (Hefei: University of Science and Technoloqy of China Press) p244 (in Chinese) [吕述望, 范修斌, 王昭顺 2008 完全映射及其密码学应用(合肥:中国科技大学出版社) 第244页]
[16] Kazlauskas K, Kazlauskas J 2009 Informatica 20 23
[17] Tang G P, Liao X F, Chen Y 2005 Chaos Soliton. Fract. 23 413
[18] Amigó J M, Kocarev L, Szczepanski J 2007 IEEE Trans. Circuits Syst. II: Express Briefs 54 882
[19] Webster A F, Tavares S E 1986 in Advances in Cryptology: Proceedings of CRYPTO'85 (Berlin: Springer-Verlag) p523
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