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On Abelian group representability of finite groups
Concatenations of the hidden weighted bit function and their cryptographic properties
1. | Temasek Laboratories, National University of Singapore, 117411, Singapore, Singapore |
2. | Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, United States |
References:
[1] |
R. E. Bryant, On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication, IEEE Trans. Comput., 40 (1991), 205-213.
doi: 10.1109/12.73590. |
[2] |
C. Carlet, On the higher order nonlinearities of algebraic immune functions, in Advances in Cryptology - CRYPTO 2006, Springer-Verlag, 2006, 584-601.
doi: 10.1007/11818175_35. |
[3] |
C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge Univ. Press, 2010, 257-397. |
[4] |
C. Carlet, D. K. Dalai, K. C. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: analysis and construction, IEEE Trans. Inf. Theory, 52 (2006), 3105-3121.
doi: 10.1109/TIT.2006.876253. |
[5] |
C. Carlet and K. Feng, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, in Advances in Cryptology - ASIACRYPT 2008, Springer-Verlag, 2008, 425-440.
doi: 10.1007/978-3-540-89255-7_26. |
[6] |
C. Carlet and K. Feng, An infinite class of balanced vectorial Boolean functions with optimum algebraic immunity and good nonlinearity, in IWCC 2009, Springer-Verlag, 2009, 1-11.
doi: 10.1007/978-3-642-01877-0_1. |
[7] |
N. Courtois, Fast algebraic attacks on stream ciphers with linear feedback, in Advances in Cryptology - CRYPTO 2003, Springer-Verlag, 2003, 176-194.
doi: 10.1007/978-3-540-45146-4_11. |
[8] |
N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in Advances in Cryptology - EUROCRYPT 2003, Springer-Verlag, 2003, 345-359.
doi: 10.1007/3-540-39200-9_21. |
[9] |
T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Elsevier-Academic Press, New York, 2009. |
[10] |
D. K. Dalai, K. C. Maitra and S. Maitra, Cryptographically significant Boolean functions: Construction and analysis in terms of algebraic immunity, in Proceedings of FSE 2005, Springer-Verlag, 2005, 98-111. |
[11] |
D. K. Dalai, S. Maitra and S. Sarkar, Basic theory in construction of Boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 40 (2006), 41-58.
doi: 10.1007/s10623-005-6300-x. |
[12] |
P. Hawkes and G. G. Rose, Rewriting variables: the complexity of fast algebraic attacks on stream ciphers, in Advances in Cryptology - CRYPTO 2004, Springer-Verlag, 2004, 390-406.
doi: 10.1007/978-3-540-28628-8_24. |
[13] |
D. E. Knuth, The Art of Computer Programming: Bitwise Tricks & Techniques; Binary Decision Diagrams, Addison-Wesley Professional, Boston, 2009. |
[14] |
M. Krause, BDD-based cryptanalysis of keystream generators, in Advances in Cryptology - EUROCRYPT 2002, Springer-Verlag, 2002, 222-237.
doi: 10.1007/3-540-46035-7_15. |
[15] |
N. Li and W. F. Qi, Construction and analysis of Boolean functions of $2t+1$ variables with maximum algebraic immunity, in Advances in Cryptology - ASIACRYPT 2006, Springer-Verlag, 2006, 84-98.
doi: 10.1007/11935230_6. |
[16] |
N. Li, L. Qu, W. Qi, G. Feng, C. Li and D. Xie, On the construction of Boolean functions with optimal algebraic immunity, IEEE Trans. Inf. Theory, 54 (2008), 1330-1334.
doi: 10.1109/TIT.2007.915914. |
[17] |
M. S. Lobanov, Exact relation between nonlinearity and algebraic immunity, Discrete Math. Appl., 16 (2006), 453-460.
doi: 10.1515/156939206779238418. |
[18] |
M. S. Lobanov, Exact relations between nonlinearity and algebraic immunity, J. Appl. Ind. Math., 3 (2009), 367-376.
doi: 10.1134/S1990478909030077. |
[19] |
W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions, in Advances in Cryptology - EUROCRYPT 2004, Springer-Verlag, 2004, 474-491.
doi: 10.1007/978-3-540-24676-3_28. |
[20] |
W. Meier and O. Staffelbach, Fast correlation attacks on stream ciphers, in Advances in Cryptology - EUROCRYPT '88, Springer-Verlag, 1988, 301-314. |
[21] |
S. Mesnager, Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity, IEEE Trans. Inf. Theory, 54 (2008), 3656-3662.
doi: 10.1109/TIT.2008.926360. |
[22] |
E. Pasalic, Almost fully optimized infinite classes of Boolean functions resistant to (fast) algebraic cryptanalysis, in Proceedings of ICISC 2008, Springer-Verlag, 2009, 399-414.
doi: 10.1007/978-3-642-00730-9_25. |
[23] |
P. Rizomiliotis, On the resistance of Boolean functions against algebraic attacks using univariate polynomial representation, IEEE Trans. Inf. Theory, 56 (2010), 4014-4024.
doi: 10.1109/TIT.2010.2050801. |
[24] |
O. S. Rothaus, On bent functions, J. Comb. Theory Ser. A, 20 (1976), 300-305. |
[25] |
C. Tan and S. Goh, Several classes of even-variable balanced Boolean functions with optimal algebraic immunity, IEICE Trans. Fund., E94.A (2011), 165-171. |
[26] |
D. Tang, C. Carlet and X. Tang, Highly nonlinear Boolean functions with optimal algebraic immunity and good behavior against fast algebraic attacks, IEEE Trans. Inf. Theory, 59 (2013), 653-664.
doi: 10.1109/TIT.2012.2217476. |
[27] |
Z. Tu and Y. Deng, A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity, Des. Codes Cryptogr., 60 (2011), 1-14.
doi: 10.1007/s10623-010-9413-9. |
[28] |
Q. Wang, C. Carlet, P. Stănică and C. Tan, Cryptographic properties of the hidden weighted bit function, Discrete Appl. Math., to appear.
doi: 10.1016/j.dam.2014.01.010. |
[29] |
Q. Wang, T. Johansson and H. Kan, Some results on fast algebraic attacks and higher-order non-linearities, IET Inform. Secur., 6 (2012), 41-46. |
[30] |
Q. Wang, J. Peng, H. Kan and X. Xue, Constructions of cryptographically significant Boolean functions using primitive polynomials, IEEE Trans. Inf. Theory, 56 (2010), 3048-3053.
doi: 10.1109/TIT.2010.2046195. |
[31] |
Q. Wang and C. H. Tan, A new method to construct Boolean functions with good cryptographic properties, Inform. Proc. Lett., 113 (2013), 567-571.
doi: 10.1016/j.ipl.2013.04.017. |
[32] |
Q. Wang and C. H. Tan, Balanced Boolean functions with optimum algebraic degree, optimum algebraic immunity and very high nonlinearity, Discrete Appl. Math., 1673 (2014), 25-32.
doi: 10.1016/j.dam.2013.11.014. |
[33] |
A. F. Webster and S. E. Tavares, On the design of S-boxes, in Advances in Cryptology - CRYPTO '85, Springer-Verlag, 1985, 523-534. |
[34] |
X. Zeng, C. Carlet, J. Shan and L. Hu, More balanced Boolean functions with optimal algebraic immunity, and good nonlinearity and resistance to fast algebraic attacks, IEEE Trans. Inf. Theory, 57 (2011), 6310-6320.
doi: 10.1109/TIT.2011.2109935. |
show all references
References:
[1] |
R. E. Bryant, On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication, IEEE Trans. Comput., 40 (1991), 205-213.
doi: 10.1109/12.73590. |
[2] |
C. Carlet, On the higher order nonlinearities of algebraic immune functions, in Advances in Cryptology - CRYPTO 2006, Springer-Verlag, 2006, 584-601.
doi: 10.1007/11818175_35. |
[3] |
C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge Univ. Press, 2010, 257-397. |
[4] |
C. Carlet, D. K. Dalai, K. C. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: analysis and construction, IEEE Trans. Inf. Theory, 52 (2006), 3105-3121.
doi: 10.1109/TIT.2006.876253. |
[5] |
C. Carlet and K. Feng, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, in Advances in Cryptology - ASIACRYPT 2008, Springer-Verlag, 2008, 425-440.
doi: 10.1007/978-3-540-89255-7_26. |
[6] |
C. Carlet and K. Feng, An infinite class of balanced vectorial Boolean functions with optimum algebraic immunity and good nonlinearity, in IWCC 2009, Springer-Verlag, 2009, 1-11.
doi: 10.1007/978-3-642-01877-0_1. |
[7] |
N. Courtois, Fast algebraic attacks on stream ciphers with linear feedback, in Advances in Cryptology - CRYPTO 2003, Springer-Verlag, 2003, 176-194.
doi: 10.1007/978-3-540-45146-4_11. |
[8] |
N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in Advances in Cryptology - EUROCRYPT 2003, Springer-Verlag, 2003, 345-359.
doi: 10.1007/3-540-39200-9_21. |
[9] |
T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Elsevier-Academic Press, New York, 2009. |
[10] |
D. K. Dalai, K. C. Maitra and S. Maitra, Cryptographically significant Boolean functions: Construction and analysis in terms of algebraic immunity, in Proceedings of FSE 2005, Springer-Verlag, 2005, 98-111. |
[11] |
D. K. Dalai, S. Maitra and S. Sarkar, Basic theory in construction of Boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 40 (2006), 41-58.
doi: 10.1007/s10623-005-6300-x. |
[12] |
P. Hawkes and G. G. Rose, Rewriting variables: the complexity of fast algebraic attacks on stream ciphers, in Advances in Cryptology - CRYPTO 2004, Springer-Verlag, 2004, 390-406.
doi: 10.1007/978-3-540-28628-8_24. |
[13] |
D. E. Knuth, The Art of Computer Programming: Bitwise Tricks & Techniques; Binary Decision Diagrams, Addison-Wesley Professional, Boston, 2009. |
[14] |
M. Krause, BDD-based cryptanalysis of keystream generators, in Advances in Cryptology - EUROCRYPT 2002, Springer-Verlag, 2002, 222-237.
doi: 10.1007/3-540-46035-7_15. |
[15] |
N. Li and W. F. Qi, Construction and analysis of Boolean functions of $2t+1$ variables with maximum algebraic immunity, in Advances in Cryptology - ASIACRYPT 2006, Springer-Verlag, 2006, 84-98.
doi: 10.1007/11935230_6. |
[16] |
N. Li, L. Qu, W. Qi, G. Feng, C. Li and D. Xie, On the construction of Boolean functions with optimal algebraic immunity, IEEE Trans. Inf. Theory, 54 (2008), 1330-1334.
doi: 10.1109/TIT.2007.915914. |
[17] |
M. S. Lobanov, Exact relation between nonlinearity and algebraic immunity, Discrete Math. Appl., 16 (2006), 453-460.
doi: 10.1515/156939206779238418. |
[18] |
M. S. Lobanov, Exact relations between nonlinearity and algebraic immunity, J. Appl. Ind. Math., 3 (2009), 367-376.
doi: 10.1134/S1990478909030077. |
[19] |
W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions, in Advances in Cryptology - EUROCRYPT 2004, Springer-Verlag, 2004, 474-491.
doi: 10.1007/978-3-540-24676-3_28. |
[20] |
W. Meier and O. Staffelbach, Fast correlation attacks on stream ciphers, in Advances in Cryptology - EUROCRYPT '88, Springer-Verlag, 1988, 301-314. |
[21] |
S. Mesnager, Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity, IEEE Trans. Inf. Theory, 54 (2008), 3656-3662.
doi: 10.1109/TIT.2008.926360. |
[22] |
E. Pasalic, Almost fully optimized infinite classes of Boolean functions resistant to (fast) algebraic cryptanalysis, in Proceedings of ICISC 2008, Springer-Verlag, 2009, 399-414.
doi: 10.1007/978-3-642-00730-9_25. |
[23] |
P. Rizomiliotis, On the resistance of Boolean functions against algebraic attacks using univariate polynomial representation, IEEE Trans. Inf. Theory, 56 (2010), 4014-4024.
doi: 10.1109/TIT.2010.2050801. |
[24] |
O. S. Rothaus, On bent functions, J. Comb. Theory Ser. A, 20 (1976), 300-305. |
[25] |
C. Tan and S. Goh, Several classes of even-variable balanced Boolean functions with optimal algebraic immunity, IEICE Trans. Fund., E94.A (2011), 165-171. |
[26] |
D. Tang, C. Carlet and X. Tang, Highly nonlinear Boolean functions with optimal algebraic immunity and good behavior against fast algebraic attacks, IEEE Trans. Inf. Theory, 59 (2013), 653-664.
doi: 10.1109/TIT.2012.2217476. |
[27] |
Z. Tu and Y. Deng, A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity, Des. Codes Cryptogr., 60 (2011), 1-14.
doi: 10.1007/s10623-010-9413-9. |
[28] |
Q. Wang, C. Carlet, P. Stănică and C. Tan, Cryptographic properties of the hidden weighted bit function, Discrete Appl. Math., to appear.
doi: 10.1016/j.dam.2014.01.010. |
[29] |
Q. Wang, T. Johansson and H. Kan, Some results on fast algebraic attacks and higher-order non-linearities, IET Inform. Secur., 6 (2012), 41-46. |
[30] |
Q. Wang, J. Peng, H. Kan and X. Xue, Constructions of cryptographically significant Boolean functions using primitive polynomials, IEEE Trans. Inf. Theory, 56 (2010), 3048-3053.
doi: 10.1109/TIT.2010.2046195. |
[31] |
Q. Wang and C. H. Tan, A new method to construct Boolean functions with good cryptographic properties, Inform. Proc. Lett., 113 (2013), 567-571.
doi: 10.1016/j.ipl.2013.04.017. |
[32] |
Q. Wang and C. H. Tan, Balanced Boolean functions with optimum algebraic degree, optimum algebraic immunity and very high nonlinearity, Discrete Appl. Math., 1673 (2014), 25-32.
doi: 10.1016/j.dam.2013.11.014. |
[33] |
A. F. Webster and S. E. Tavares, On the design of S-boxes, in Advances in Cryptology - CRYPTO '85, Springer-Verlag, 1985, 523-534. |
[34] |
X. Zeng, C. Carlet, J. Shan and L. Hu, More balanced Boolean functions with optimal algebraic immunity, and good nonlinearity and resistance to fast algebraic attacks, IEEE Trans. Inf. Theory, 57 (2011), 6310-6320.
doi: 10.1109/TIT.2011.2109935. |
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