# American Institute of Mathematical Sciences

February  2019, 13(1): 67-88. doi: 10.3934/amc.2019004

## Wave-shaped round functions and primitive groups

 1 DISIM, University of L'Aquila, Italy 2 Department of Informatics, University of Bergen, Norway 3 Department of Mathematics, University of Trento, Italy 4 LIRMM of Montpellier, France

* Corresponding author: Roberto Civino

Received  March 2018 Revised  August 2018 Published  December 2018

Fund Project: R. Aragona is member of INdAM-GNSAGA (Italy). R. Civino thankfully acknowledges support by the Department of Mathematics of the University of Trento. R. Aragona, R.Civino, and M. Sala thankfully acknowledge support by MIUR-Italy via PRIN 2015TW9LSR "Group theory and applications".

Round functions used as building blocks for iterated block ciphers, both in the case of Substitution-Permutation Networks (SPN) and Feistel Networks (FN), are often obtained as the composition of different layers. The bijectivity of any encryption function is guaranteed by the use of invertible layers or by the Feistel structure. In this work a new family of ciphers, called wave ciphers, is introduced. In wave ciphers, round functions feature wave functions, which are vectorial Boolean functions obtained as the composition of non-invertible layers, where the confusion layer enlarges the message which returns to its original size after the diffusion layer is applied. Efficient decryption is guaranteed by the use of wave functions in FNs. It is shown how to avoid that the group generated by the round functions acts imprimitively, a serious flaw for the cipher. The primitivity is a consequence of a more general result, which reduce the problem of proving that a given FN generates a primitive group to proving that an SPN, directly related to the given FN, generates a primitive group. Finally, a concrete instance of real-world size wave cipher is proposed as an example, and its resistance against differential and linear cryptanalyses is also established.

Citation: Riccardo Aragona, Marco Calderini, Roberto Civino, Massimiliano Sala, Ilaria Zappatore. Wave-shaped round functions and primitive groups. Advances in Mathematics of Communications, 2019, 13 (1) : 67-88. doi: 10.3934/amc.2019004
##### References:
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Comput., 20 (2009), 339-350.  doi: 10.1007/s00200-009-0100-x.  Google Scholar [19] D. Coppersmith and E. Grossman, Generators for certain alternating groups with applications to cryptography, SIAM J. Appl. Math., 29 (1975), 624-627.  doi: 10.1137/0129051.  Google Scholar [20] J. Daemen and V. Rijmen, The design of Rijndael: AES - the Advanced Encryption Standard, Information Security and Cryptography, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04722-4.  Google Scholar [21] V. Dolmatov, GOST 28147-89: encryption, decryption, and message authentication code (MAC) algorithms, technical report, 2010. Available at http://tools.ietf.org/html/rfc5830. Google Scholar [22] Federal information processing standards publication, Data Encryption Standard and Others, National Bureau of Standards, US Department of Commerce, 1977. Google Scholar [23] E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Ann. Sci. École Norm. Sup., 6 (1889), 9-102.  doi: 10.24033/asens.317.  Google Scholar [24] X.-D. Hou, Affinity of permutations of $\mathbb{F}_{2}^{n}$, Discrete Appl. Math., 154 (2006), 313-325.  doi: 10.1016/j.dam.2005.03.022.  Google Scholar [25] Jr. B. S. Kaliski, R. L. Rivest and A. T. Sherman, Is the Data Encryption Standard a group? (Results of cycling experiments on DES), J. Cryptology, 1 (1988), 3-36.  doi: 10.1007/BF00206323.  Google Scholar [26] M. Matsui, Linear Cryptanalysis Method for DES Cipher, Advances in cryptology - EUROCRYPT '93, Lecture Notes in Comput. Sci., 765 (1994), 386-397. Google Scholar [27] K. Nyberg, Differentially uniform mappings for cryptography, Advances in cryptology - EUROCRYPT '93, Lecture Notes in Comput. Sci., 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6.  Google Scholar [28] K. G. Paterson, Imprimitive permutation groups and trapdoors in iterated block ciphers, Fast Software Encryption, Lecture Notes in Comput. Sci., 1636 (1999), 201-214. Google Scholar [29] J. Petrillo, Goursat's other theorem, The College Mathematics Journal, 40 (2009), 119-124.   Google Scholar [30] G. Piret, T. Roche and C. Carlet, PICARO-a block cipher allowing efficient higher-order sidechannel resistance, Applied Cryptography and Network Security-ACNS2012, Lecture Notes in Comput. Sci., 7341 (2012), 311-328. Google Scholar [31] C. E. Shannon, Communication theory of secrecy systems, Bell System Tech., 28 (1949), 656-715.  doi: 10.1002/j.1538-7305.1949.tb00928.x.  Google Scholar [32] R. Sparr and R. Wernsdorf, Group theoretic properties of Rijndael-like ciphers, Discrete Appl. Math., 156 (2008), 3139-3149.  doi: 10.1016/j.dam.2007.12.011.  Google Scholar [33] R. Wernsdorf, The round functions of RIJNDAEL generate the alternating group, Fast Software Encryption, Lecture Notes in Comput. Sci., 2365 (2002), 143-148. Google Scholar [34] R. Wernsdorf, The one-round functions of the DES generate the alternating group, Advances in Cryptology-EUROCRYPT '92, Lecture Notes in Comput. Sci., 658 (1993), 99-112. doi: 10.1007/3-540-47555-9_9.  Google Scholar [35] R. Wernsdorf, The round functions of SERPENT generate the alternating group, 2000. Available from: http://csrc.nist.gov/archive/aes/round2/comments/20000512-rwernsdorf.pdf. Google Scholar

show all references

##### References:
 [1] C. Adams, The CAST-128 encryption algorithm, 1997, Available from: http://buildbot.tools.ietf.org/html/rfc2144. Google Scholar [2] R. J. Anderson, E. Biham and L. R. Knudsen, SERPENT: A new block cipher proposal, Fast Software Encryption, Lecture Notes in Comput. Sci., 1372 (1998), 222-238. Google Scholar [3] K. Aoki, T. Ichikawa, M. Kanda, M. Matsui, S. Moriai, J. Nakajima and T. Tokita, Camellia: A 128-bit block cipher suitable for multiple platforms-design and analysis, Selected Areas in Cryptography, Lecture Notes in Comput. Sci., 2012 (2000), 39-56. doi: 10.1007/3-540-44983-3_4.  Google Scholar [4] R. Aragona, M. Calderini, A. Tortora and M. Tota, Primitivity of PRESENT and other lightweight ciphers, J. Algebra Appl., 17 (2018), 1850115 (16 pages). doi: 10.1142/S0219498818501153.  Google Scholar [5] R. Aragona, A. Caranti, F. Dalla Volta and M. Sala, On the group generated by the round functions of translation based ciphers over arbitrary fields, Finite Fields Appl., 25 (2014), 293-305.  doi: 10.1016/j.ffa.2013.10.005.  Google Scholar [6] R. Aragona, A. Caranti and M. Sala, The group generated by the round functions of a GOSTlike cipher, Ann. Mat. Pura Appl., 196 (2017), 1-17.  doi: 10.1007/s10231-016-0559-6.  Google Scholar [7] A. Bannier, N. Bodin and E. Filiol, Partition-Based Trapdoor Ciphers, IACR Cryptology ePrint Archive, 2016, Available from: http://eprint.iacr.org/2016/493. Google Scholar [8] E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.  doi: 10.1007/BF00630563.  Google Scholar [9] A. Bogdanov, L. R. Knudsen, G. Leander, C. Paar, A. Poschmann, M. J. B. Robshaw, Y. Seurin and C. Vikkelsoe, PRESENT: An ultra-lightweight block cipher, CHES '07, Lecture Notes in Comput. Sci., 4727 (2007), 450-466. Google Scholar [10] K. A. Browning, J. F. Dillon, M. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, Finite Fields: theory and applications, 518 (2010), 33-42.  doi: 10.1090/conm/518/10194.  Google Scholar [11] M. Calderini, A note on some algebraic trapdoors for block ciphers, Adv. Math. Commun., 12 (2018), 515-524.  doi: 10.3934/amc.2018030.  Google Scholar [12] M. Calderini and M. Sala, Elementary abelian regular subgroups as hidden sums for cryptographic trapdoors, preprint, arXiv: 1702.00581. Google Scholar [13] M. Calderini, I. Villa and M. Sala, A note on APN permutations in even dimension, Finite Fields Appl., 46 (2017), 1-16.  doi: 10.1016/j.ffa.2017.02.001.  Google Scholar [14] P. J. Cameron, Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511623677.  Google Scholar [15] A. Canteaut, S. Duval and L. Perrin, A generalisation of Dillon's APN permutation with the best known differential and nonlinear properties for all fields of size 24k+2, IEEE Trans. Inform. Theory, 63 (2017), 7575-7591.  doi: 10.1109/TIT.2017.2676807.  Google Scholar [16] A. Canteaut and M. Naya-Plasencia, Structural weaknesses of permutations with low differential uniformity and generalized crooked functions, Finite Fields: Theory and Applications Selected Papers from the 9th International Conference Finite Fields and Applications, Contemporary Mathematics, 518 (2010), 55-71. doi: 10.1090/conm/518/10196.  Google Scholar [17] A. Caranti, F. Dalla Volta and M. Sala, An application of the O'Nan-Scott theorem to the group generated by the round functions of an AES-like cipher, Des. Codes Cryptogr., 52 (2009), 293-301.  doi: 10.1007/s10623-009-9283-1.  Google Scholar [18] A. Caranti, F. Dalla Volta and M. Sala, On some block ciphers and imprimitive groups, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 339-350.  doi: 10.1007/s00200-009-0100-x.  Google Scholar [19] D. Coppersmith and E. Grossman, Generators for certain alternating groups with applications to cryptography, SIAM J. Appl. Math., 29 (1975), 624-627.  doi: 10.1137/0129051.  Google Scholar [20] J. Daemen and V. Rijmen, The design of Rijndael: AES - the Advanced Encryption Standard, Information Security and Cryptography, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04722-4.  Google Scholar [21] V. Dolmatov, GOST 28147-89: encryption, decryption, and message authentication code (MAC) algorithms, technical report, 2010. Available at http://tools.ietf.org/html/rfc5830. Google Scholar [22] Federal information processing standards publication, Data Encryption Standard and Others, National Bureau of Standards, US Department of Commerce, 1977. Google Scholar [23] E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Ann. Sci. École Norm. Sup., 6 (1889), 9-102.  doi: 10.24033/asens.317.  Google Scholar [24] X.-D. Hou, Affinity of permutations of $\mathbb{F}_{2}^{n}$, Discrete Appl. Math., 154 (2006), 313-325.  doi: 10.1016/j.dam.2005.03.022.  Google Scholar [25] Jr. B. S. Kaliski, R. L. Rivest and A. T. Sherman, Is the Data Encryption Standard a group? (Results of cycling experiments on DES), J. Cryptology, 1 (1988), 3-36.  doi: 10.1007/BF00206323.  Google Scholar [26] M. Matsui, Linear Cryptanalysis Method for DES Cipher, Advances in cryptology - EUROCRYPT '93, Lecture Notes in Comput. Sci., 765 (1994), 386-397. Google Scholar [27] K. Nyberg, Differentially uniform mappings for cryptography, Advances in cryptology - EUROCRYPT '93, Lecture Notes in Comput. Sci., 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6.  Google Scholar [28] K. G. Paterson, Imprimitive permutation groups and trapdoors in iterated block ciphers, Fast Software Encryption, Lecture Notes in Comput. Sci., 1636 (1999), 201-214. Google Scholar [29] J. Petrillo, Goursat's other theorem, The College Mathematics Journal, 40 (2009), 119-124.   Google Scholar [30] G. Piret, T. Roche and C. Carlet, PICARO-a block cipher allowing efficient higher-order sidechannel resistance, Applied Cryptography and Network Security-ACNS2012, Lecture Notes in Comput. Sci., 7341 (2012), 311-328. Google Scholar [31] C. E. Shannon, Communication theory of secrecy systems, Bell System Tech., 28 (1949), 656-715.  doi: 10.1002/j.1538-7305.1949.tb00928.x.  Google Scholar [32] R. Sparr and R. Wernsdorf, Group theoretic properties of Rijndael-like ciphers, Discrete Appl. Math., 156 (2008), 3139-3149.  doi: 10.1016/j.dam.2007.12.011.  Google Scholar [33] R. Wernsdorf, The round functions of RIJNDAEL generate the alternating group, Fast Software Encryption, Lecture Notes in Comput. Sci., 2365 (2002), 143-148. Google Scholar [34] R. Wernsdorf, The one-round functions of the DES generate the alternating group, Advances in Cryptology-EUROCRYPT '92, Lecture Notes in Comput. Sci., 658 (1993), 99-112. doi: 10.1007/3-540-47555-9_9.  Google Scholar [35] R. Wernsdorf, The round functions of SERPENT generate the alternating group, 2000. Available from: http://csrc.nist.gov/archive/aes/round2/comments/20000512-rwernsdorf.pdf. Google Scholar
Round function of an SPN and of an FN
Wave functions
A 4x5 APN S-box
Feistel structure of wave ciphers
Feistel to SPN reduction
An example of 40 × 32 proper diffusion layer with parallel kernel, where each "·" represents 0
Diffusion properties of the matrix λ of Fig. 6.
Difference distribution table of the S-box $\gamma_1$ defined in Section 3.1
 0x 1x 2x 3x 4x 5x 6x 7x 8x 9x Ax Bx Cx Dx Ex Fx 10x 11x 12x 13x 14x 15x 16x 17x 18x 19x 1Ax 1Bx 1Cx 1Dx 1Ex 1Fx 0x 16 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 1x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2x $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 3x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 4x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 5x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 6x $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 7x $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 8x $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ 2 9x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 Ax $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ Bx $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ Cx $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ Dx $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ Ex $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 Fx $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 2 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$
 0x 1x 2x 3x 4x 5x 6x 7x 8x 9x Ax Bx Cx Dx Ex Fx 10x 11x 12x 13x 14x 15x 16x 17x 18x 19x 1Ax 1Bx 1Cx 1Dx 1Ex 1Fx 0x 16 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 1x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2x $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 3x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 4x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 5x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 6x $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 7x $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 8x $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ 2 9x $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 Ax $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ Bx $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ Cx $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ Dx $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ Ex $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 Fx $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ 2 2 2 $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ 2 $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$ $\cdot$
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