American Institute of Mathematical Sciences

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

Wave-shaped round functions and primitive groups

Riccardo Aragona 1, , Marco Calderini 2, , Roberto Civino 1,, , Massimiliano Sala 3, and  Ilaria Zappatore 4,

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.

Keywords: Cryptosystems, non-invertible S-boxes, almost perfect non-linearity, groups generated by round functions, primitive groups.
Mathematics Subject Classification: 20B15, 20B35, 94A60.
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:
[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

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

Figure 1.  Round function of an SPN and of an FN
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Wave functions
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  A 4x5 APN S-box
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Feistel structure of wave ciphers
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Feistel to SPN reduction
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  An example of 40 × 32 proper diffusion layer with parallel kernel, where each "·" represents 0
Figure Options

Download full-size image

Download as PowerPoint slide

Fig. 6.">Figure 7.  Diffusion properties of the matrix λ of Fig. 6.
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  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 $
Table Options

Download as excel

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 $
[1]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715
[2]

Benjamin Hellouin de Menibus, Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2335-2346. doi: 10.3934/dcds.2020116
[3]

Or Landesberg. Horospherically invariant measures and finitely generated Kleinian groups. Journal of Modern Dynamics, 2021, 17: 337-352. doi: 10.3934/jmd.2021012
[4]

A. Yu. Ol'shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. Electronic Research Announcements, 2001, 7: 63-71.

[5]

Yang Yang, Xiaohu Tang, Guang Gong. New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property. Advances in Mathematics of Communications, 2017, 11 (1) : 67-76. doi: 10.3934/amc.2017002
[6]

Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2391-2402. doi: 10.3934/dcdss.2019150
[7]

Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323
[8]

Peter Müller, Gábor P. Nagy. On the non-existence of sharply transitive sets of permutations in certain finite permutation groups. Advances in Mathematics of Communications, 2011, 5 (2) : 303-308. doi: 10.3934/amc.2011.5.303
[9]

Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002
[10]

Raf Cluckers, Julia Gordon, Immanuel Halupczok. Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups. Electronic Research Announcements, 2014, 21: 137-152. doi: 10.3934/era.2014.21.137
[11]

Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012
[12]

Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39.

[13]

Olof Heden, Denis S. Krotov. On the structure of non-full-rank perfect $q$-ary codes. Advances in Mathematics of Communications, 2011, 5 (2) : 149-156. doi: 10.3934/amc.2011.5.149
[14]

Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $ S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007
[15]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The Fueter primitive of biaxially monogenic functions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 657-672. doi: 10.3934/cpaa.2014.13.657
[16]

Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems & Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[17]

Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467
[18]

Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005
[19]

Sergei V. Ivanov. On aspherical presentations of groups. Electronic Research Announcements, 1998, 4: 109-114.

[20]

Benjamin Weiss. Entropy and actions of sofic groups. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (303)
  • HTML views (355)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy