# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020125

## On ideal $t$-tuple distribution of orthogonal functions in filtering de bruijn generators

 Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, CANADA

* Corresponding author

Received  June 2019 Revised  November 2020 Published  December 2020

Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal $t$-tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period $2^n$ and a filtering function in $m$ variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called invariant under the WG transform, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve $t$-tuple uniformity in output sequences, up to $t = (n-m+1)$. The conjecture of Mandal et al. in [29] about the ideal tuple distribution on the WG transformation is proved. It is also shown that the Gold functions and quadratic functions can guarantee $(n-m+1)$-tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.

Citation: Kalikinkar Mandal, Guang Gong. On ideal $t$-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, doi: 10.3934/amc.2020125
##### References:
 [1] S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511804090.  Google Scholar [2] S. Boztas and P.V. Kumar, Binary sequences with Gold-like correlation but larger linear span, IEEE Trans. Inf. Theory, 40 (1994), 532-537.  doi: 10.1109/18.312181.  Google Scholar [3] N. G. de Bruijn, A combinatorial problem, Proc. Koninklijke Nederlandse Akademie v. Wetenschappen, 49 (1946), 758-764.   Google Scholar [4] A. Canteaut, Analysis and Design of Symmetric Ciphers, Habilitation for directing Theses, University of Paris 6, 2006. Available from: https://www.rocq.inria.fr/secret/Anne.Canteaut/canteaut-hdr.pdf. Google Scholar [5] C. Carlet, Boolean functions for cryptography and error correcting codes, Chapter of the monography Boolean models and methods in mathematics, computer science, and engineering, Cambridge University Press, (2010), 257–397. Google Scholar [6] A. H. Chan, R. A. Games and E. L. Key, On the complexities of de Bruijn sequences, Journal of Combinatorial Theory, Series A, 33 (1982), 233-246.  doi: 10.1016/0097-3165(82)90038-3.  Google Scholar [7] A. Chang, P. Gaal, S. W. Golomb, G. Gong, T. Helleseth and P. V. Kumar, On a conjectured ideal autocorrelation sequence and a related triple-error correcting cyclic code, IEEE Trans. Inf. Theory, 46 (2000), 680-687.  doi: 10.1109/18.825842.  Google Scholar [8] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Elsevier/Academic Press, Amsterdam, 2009.   Google Scholar [9] National Institute of Standards and Technology, Digital Signature Standard (DSS), Federal information processing standards publication, FIPS PUB 186-2, Reaffirmed, 2000., Google Scholar [10] J. F. Dillon, Multiplicative difference sets via additive characters, Designs, Codes and Cryptography, 17 (1999), 225-235.  doi: 10.1023/A:1026435428030.  Google Scholar [11] J. F. Dillon and H. Dobbertin, New cyclic difference sets with singer parameters, Finite Fields and Their Applications, 10 (2004), 342-389.  doi: 10.1016/j.ffa.2003.09.003.  Google Scholar [12] L. Ding, C. Jin, J. Guan and Q. Wang, Cryptanalysis of lightweight WG-8 stream cipher, IEEE Trans. Inf. Forensics and Security, 9 (2014), 645-652.  doi: 10.1109/TIFS.2014.2307202.  Google Scholar [13] L. Ding, C. Jin, J. Guan, S. Zhang, T. Cui, D. Han and W. Zhao, Cryptanalysis of WG family of stream ciphers, Computer Journal, 58 (2015), 2677-2685.  doi: 10.1093/comjnl/bxv024.  Google Scholar [14] The eStream project, (2008). Available from: http://www.ecrypt.eu.org/stream/project.html. Google Scholar [15] X. Fan, K. Mandal and G. Gong, WG-8: A lightweight stream cipher for resource-constrained smart devices, 9th International Conference on Quality, Reliability, Security and Robustness in Heterogeneous Networks, Springer Berlin, (2013), 617–632. doi: 10.1007/978-3-642-37949-9_54.  Google Scholar [16] X. Fan, N. Zidaric, M. Aagaard and G. Gong, Efficient hardware implementation of the stream cipher WG-16 with composite field arithmetic, The 2013 ACM Workshop on Trustworthy Embedded Devices (TrustED'13), ACM Press, (2013), 21–34. doi: 10.1145/2517300.2517305.  Google Scholar [17] R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156.  doi: 10.1109/TIT.1968.1054106.  Google Scholar [18] J. Dj Golić, On the security of nonlinear filter generators, in 1996 Proceedings of Fast Software Encryption, Springer, Berlin, Heidelberg, (1996), 173–188. Google Scholar [19] S. W. Golomb, On the classification of balanced binary sequences of period $2^n-1$, IEEE Trans. Inf. Theory, 26 (1980), 730-732.  doi: 10.1109/TIT.1980.1056265.  Google Scholar [20] S. W. Golomb, Shift register sequences, Aegean Park Press, Laguna Hills, CA, (1981).  Google Scholar [21] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For wireless Communication, Cryptography and Radar, Cambridge University Press, New York, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar [22] G. Gong, P. Gaal and S. W. Golomb, A suspected infinity class of cyclic Hadamard difference sets, Proceedings of 1997 IEEE Information Theory Workshop, Longyearbyen, Syalbard, Norway, (1997). Google Scholar [23] G. Gong and A. Youssef, Cryptographic properties of the Welch-Gong transformation sequence generators, IEEE Trans. Inf. Theory, 48 (2002), 2837-2846.  doi: 10.1109/TIT.2002.804043.  Google Scholar [24] B. Gordon, W. H. Mills and L. R. Welch, Some new difference sets, Canadian Journal of Mathematics, 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.  Google Scholar [25] M. Joseph, G. Sekar and R. Balasubramanian, Distinguishing attacks on (ultra-)lightweight WG ciphers, in 5th International Workshop on Lightweight Cryptography for Security and Privacy, LightSec 2016, Springer International Publishing, (2017), 45–59. doi: 10.1007/978-3-319-55714-4_4.  Google Scholar [26] K. Mandal and G. Gong, Cryptographically strong de Bruijn sequences with large periods., in Selected Areas in Cryptography, SAC 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7707 (2012), 104–118. Google Scholar [27] K. Mandal and G. Gong, Feedback reconstruction and implementations of pseudorandom number generators from composited de Bruijn sequences, IEEE Trans. Computers, 65 (2016), 2725-2738.  doi: 10.1109/TC.2015.2506557.  Google Scholar [28] K. Mandal, G. Gong, X. Fan and M. Aagaard, Optimal parameters for the WG stream cipher family, Cryptography and Communications, 6 (2014), 117-135.   Google Scholar [29] K. Mandal, B. Yang, G. Gong and M. Aagaard, On ideal $t$-tuple distribution of filtering de Bruijn sequence generators, Cryptography and Communications, 10 (2018), 629-641.  doi: 10.1007/s12095-017-0248-3.  Google Scholar [30] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, 15 (1969), 122-127.  doi: 10.1109/tit.1969.1054260.  Google Scholar [31] Y. Nawaz and G. Gong, WG: A family of stream ciphers with designed randomness properties, Information Sciences, 178 (2008), 1903-1916.  doi: 10.1016/j.ins.2007.12.002.  Google Scholar [32] Y. Nawaz and G. Gong, The WG stream cipher, (2005). Available from: http://www.ecrypt.eu.org/stream/p2ciphers/wg/wg_p2.pdf., Google Scholar [33] J.-S. No, S. W. Golomb, G. Gong, H. K. Lee and P. Gaal, Binary pseudorandom sequences of period $2^n-1$ with ideal autocorrelation, IEEE Trans. Inform. Theory, 44 (1998), 814-817.  doi: 10.1109/18.661528.  Google Scholar [34] M. A. Orumiehchiha, J. Pieprzyk and R. Steinfeld, Cryptanalysis of WG-7: A lightweight stream cipher, Cryptography Communications, 4 (2012), 277-285.  doi: 10.1007/s12095-012-0070-x.  Google Scholar [35] H. El-Razouk, A. Reyhani-Masoleh and G. Gong, New implementations of the WG stream cipher, IEEE Trans. on VLSI, 22 (2014), 1865-1878.  doi: 10.1109/TVLSI.2013.2280092.  Google Scholar [36] S. RØnjom, Improving algebraic attacks on stream ciphers based on linear feedback shift register over $\mathbb{F}_{2^k}$, Designs Codes Cryptography, 82 (2017), 27-41.  doi: 10.1007/s10623-016-0212-9.  Google Scholar [37] R. A. Rueppel, Analysis and Design of Stream Ciphers, Springer-Verlag, 1986. doi: 10.1007/978-3-642-82865-2.  Google Scholar [38] T. Siegenthaler, R. Forré and A. W. Kleiner, Generation of binary sequences with controllable complexity and ideal $r$-tupel distribution, in Advances in Cryptology–EUROCRYPT 87, Lecture Notes in Comput. Sci, 304 (1987), 15–23. doi: 10.1007/3-540-39118-5_3.  Google Scholar [39] N. Y. Yu and G. Gong, Crosscorrelation properties of binary sequences with ideal two-level autocorrelation, in Proceedings of the 4th International Conference on Sequences and Their Applications (SETA'06), Lecture Notes in Comput. Sci, Springer, Berlin, Heidelberg, 4086 (2006), 104–118. doi: 10.1007/11863854_9.  Google Scholar [40] N. Y. Yu and G. Gong, A new binary sequence family with low correlation and large size, IEEE Trans. Inf. Theory, 52 (2006), 1624-1636.  doi: 10.1109/TIT.2006.871062.  Google Scholar [41] S. V. Smyshlyaev, Perfectly balanced Boolean functions and Golić conjecture, Journal of Cryptology, 25 (2012), 464-483.  doi: 10.1007/s00145-011-9100-7.  Google Scholar [42] G. Yang, X. Fan, M. Aagaard and G. Gong, Design space exploration of the lightweight stream cipher WG-8 for FPGAs and ASICs, Proceedings of the Workshop on Embedded Systems Security, (2013), 1–10. doi: 10.1145/2527317.2527325.  Google Scholar [43] B. Yang, K. Mandal, M. D. Aagaard and G. Gong, Efficient composited de Bruijn sequence generators, IEEE Trans. on Computers, 66 (2017), 1354-1368.  doi: 10.1109/TC.2017.2676763.  Google Scholar

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##### References:
 [1] S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511804090.  Google Scholar [2] S. Boztas and P.V. Kumar, Binary sequences with Gold-like correlation but larger linear span, IEEE Trans. Inf. Theory, 40 (1994), 532-537.  doi: 10.1109/18.312181.  Google Scholar [3] N. G. de Bruijn, A combinatorial problem, Proc. Koninklijke Nederlandse Akademie v. Wetenschappen, 49 (1946), 758-764.   Google Scholar [4] A. Canteaut, Analysis and Design of Symmetric Ciphers, Habilitation for directing Theses, University of Paris 6, 2006. Available from: https://www.rocq.inria.fr/secret/Anne.Canteaut/canteaut-hdr.pdf. Google Scholar [5] C. Carlet, Boolean functions for cryptography and error correcting codes, Chapter of the monography Boolean models and methods in mathematics, computer science, and engineering, Cambridge University Press, (2010), 257–397. Google Scholar [6] A. H. Chan, R. A. Games and E. L. Key, On the complexities of de Bruijn sequences, Journal of Combinatorial Theory, Series A, 33 (1982), 233-246.  doi: 10.1016/0097-3165(82)90038-3.  Google Scholar [7] A. Chang, P. Gaal, S. W. Golomb, G. Gong, T. Helleseth and P. V. Kumar, On a conjectured ideal autocorrelation sequence and a related triple-error correcting cyclic code, IEEE Trans. Inf. Theory, 46 (2000), 680-687.  doi: 10.1109/18.825842.  Google Scholar [8] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Elsevier/Academic Press, Amsterdam, 2009.   Google Scholar [9] National Institute of Standards and Technology, Digital Signature Standard (DSS), Federal information processing standards publication, FIPS PUB 186-2, Reaffirmed, 2000., Google Scholar [10] J. F. Dillon, Multiplicative difference sets via additive characters, Designs, Codes and Cryptography, 17 (1999), 225-235.  doi: 10.1023/A:1026435428030.  Google Scholar [11] J. F. Dillon and H. Dobbertin, New cyclic difference sets with singer parameters, Finite Fields and Their Applications, 10 (2004), 342-389.  doi: 10.1016/j.ffa.2003.09.003.  Google Scholar [12] L. Ding, C. Jin, J. Guan and Q. Wang, Cryptanalysis of lightweight WG-8 stream cipher, IEEE Trans. Inf. Forensics and Security, 9 (2014), 645-652.  doi: 10.1109/TIFS.2014.2307202.  Google Scholar [13] L. Ding, C. Jin, J. Guan, S. Zhang, T. Cui, D. Han and W. Zhao, Cryptanalysis of WG family of stream ciphers, Computer Journal, 58 (2015), 2677-2685.  doi: 10.1093/comjnl/bxv024.  Google Scholar [14] The eStream project, (2008). Available from: http://www.ecrypt.eu.org/stream/project.html. Google Scholar [15] X. Fan, K. Mandal and G. Gong, WG-8: A lightweight stream cipher for resource-constrained smart devices, 9th International Conference on Quality, Reliability, Security and Robustness in Heterogeneous Networks, Springer Berlin, (2013), 617–632. doi: 10.1007/978-3-642-37949-9_54.  Google Scholar [16] X. Fan, N. Zidaric, M. Aagaard and G. Gong, Efficient hardware implementation of the stream cipher WG-16 with composite field arithmetic, The 2013 ACM Workshop on Trustworthy Embedded Devices (TrustED'13), ACM Press, (2013), 21–34. doi: 10.1145/2517300.2517305.  Google Scholar [17] R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156.  doi: 10.1109/TIT.1968.1054106.  Google Scholar [18] J. Dj Golić, On the security of nonlinear filter generators, in 1996 Proceedings of Fast Software Encryption, Springer, Berlin, Heidelberg, (1996), 173–188. Google Scholar [19] S. W. Golomb, On the classification of balanced binary sequences of period $2^n-1$, IEEE Trans. Inf. Theory, 26 (1980), 730-732.  doi: 10.1109/TIT.1980.1056265.  Google Scholar [20] S. W. Golomb, Shift register sequences, Aegean Park Press, Laguna Hills, CA, (1981).  Google Scholar [21] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For wireless Communication, Cryptography and Radar, Cambridge University Press, New York, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar [22] G. Gong, P. Gaal and S. W. Golomb, A suspected infinity class of cyclic Hadamard difference sets, Proceedings of 1997 IEEE Information Theory Workshop, Longyearbyen, Syalbard, Norway, (1997). Google Scholar [23] G. Gong and A. Youssef, Cryptographic properties of the Welch-Gong transformation sequence generators, IEEE Trans. Inf. Theory, 48 (2002), 2837-2846.  doi: 10.1109/TIT.2002.804043.  Google Scholar [24] B. Gordon, W. H. Mills and L. R. Welch, Some new difference sets, Canadian Journal of Mathematics, 14 (1962), 614-625.  doi: 10.4153/CJM-1962-052-2.  Google Scholar [25] M. Joseph, G. Sekar and R. Balasubramanian, Distinguishing attacks on (ultra-)lightweight WG ciphers, in 5th International Workshop on Lightweight Cryptography for Security and Privacy, LightSec 2016, Springer International Publishing, (2017), 45–59. doi: 10.1007/978-3-319-55714-4_4.  Google Scholar [26] K. Mandal and G. Gong, Cryptographically strong de Bruijn sequences with large periods., in Selected Areas in Cryptography, SAC 2012, Lecture Notes in Comput. Sci., Springer, Heidelberg, 7707 (2012), 104–118. Google Scholar [27] K. Mandal and G. Gong, Feedback reconstruction and implementations of pseudorandom number generators from composited de Bruijn sequences, IEEE Trans. Computers, 65 (2016), 2725-2738.  doi: 10.1109/TC.2015.2506557.  Google Scholar [28] K. Mandal, G. Gong, X. Fan and M. Aagaard, Optimal parameters for the WG stream cipher family, Cryptography and Communications, 6 (2014), 117-135.   Google Scholar [29] K. Mandal, B. Yang, G. Gong and M. Aagaard, On ideal $t$-tuple distribution of filtering de Bruijn sequence generators, Cryptography and Communications, 10 (2018), 629-641.  doi: 10.1007/s12095-017-0248-3.  Google Scholar [30] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, 15 (1969), 122-127.  doi: 10.1109/tit.1969.1054260.  Google Scholar [31] Y. Nawaz and G. Gong, WG: A family of stream ciphers with designed randomness properties, Information Sciences, 178 (2008), 1903-1916.  doi: 10.1016/j.ins.2007.12.002.  Google Scholar [32] Y. Nawaz and G. Gong, The WG stream cipher, (2005). Available from: http://www.ecrypt.eu.org/stream/p2ciphers/wg/wg_p2.pdf., Google Scholar [33] J.-S. No, S. W. Golomb, G. Gong, H. K. Lee and P. Gaal, Binary pseudorandom sequences of period $2^n-1$ with ideal autocorrelation, IEEE Trans. Inform. Theory, 44 (1998), 814-817.  doi: 10.1109/18.661528.  Google Scholar [34] M. A. Orumiehchiha, J. Pieprzyk and R. Steinfeld, Cryptanalysis of WG-7: A lightweight stream cipher, Cryptography Communications, 4 (2012), 277-285.  doi: 10.1007/s12095-012-0070-x.  Google Scholar [35] H. El-Razouk, A. Reyhani-Masoleh and G. Gong, New implementations of the WG stream cipher, IEEE Trans. on VLSI, 22 (2014), 1865-1878.  doi: 10.1109/TVLSI.2013.2280092.  Google Scholar [36] S. RØnjom, Improving algebraic attacks on stream ciphers based on linear feedback shift register over $\mathbb{F}_{2^k}$, Designs Codes Cryptography, 82 (2017), 27-41.  doi: 10.1007/s10623-016-0212-9.  Google Scholar [37] R. A. Rueppel, Analysis and Design of Stream Ciphers, Springer-Verlag, 1986. doi: 10.1007/978-3-642-82865-2.  Google Scholar [38] T. Siegenthaler, R. Forré and A. W. Kleiner, Generation of binary sequences with controllable complexity and ideal $r$-tupel distribution, in Advances in Cryptology–EUROCRYPT 87, Lecture Notes in Comput. Sci, 304 (1987), 15–23. doi: 10.1007/3-540-39118-5_3.  Google Scholar [39] N. Y. Yu and G. Gong, Crosscorrelation properties of binary sequences with ideal two-level autocorrelation, in Proceedings of the 4th International Conference on Sequences and Their Applications (SETA'06), Lecture Notes in Comput. Sci, Springer, Berlin, Heidelberg, 4086 (2006), 104–118. doi: 10.1007/11863854_9.  Google Scholar [40] N. Y. Yu and G. Gong, A new binary sequence family with low correlation and large size, IEEE Trans. Inf. Theory, 52 (2006), 1624-1636.  doi: 10.1109/TIT.2006.871062.  Google Scholar [41] S. V. Smyshlyaev, Perfectly balanced Boolean functions and Golić conjecture, Journal of Cryptology, 25 (2012), 464-483.  doi: 10.1007/s00145-011-9100-7.  Google Scholar [42] G. Yang, X. Fan, M. Aagaard and G. Gong, Design space exploration of the lightweight stream cipher WG-8 for FPGAs and ASICs, Proceedings of the Workshop on Embedded Systems Security, (2013), 1–10. doi: 10.1145/2527317.2527325.  Google Scholar [43] B. Yang, K. Mandal, M. D. Aagaard and G. Gong, Efficient composited de Bruijn sequence generators, IEEE Trans. on Computers, 66 (2017), 1354-1368.  doi: 10.1109/TC.2017.2676763.  Google Scholar
Block diagram of a filtering de Bruijn generator (FDBG)
Relations among Hadamard Transform, WG-inv, $x_0$-independence and ideal tuple distribution. $g(\cdot)$ is independent of $x_0$. $\leftrightarrow$ denotes if and only if condition, $\rightarrow$ denotes if condition and $n$ is the NLFSR length
A summary of the Hadamard transform values of three-term functions
 Functions Hadamard Transform values Ref. $w(x) = T3(x^{2^k+1}) = {\rm Tr}(x + x^r + x^{r^2})$ $3$-valued [11] $g(x) = T3(x^{2^k-1}) = {\rm Tr}(x + x^{q_2} + x^{q_2^2})$ at most $5$-valued [39] $WG_{w}(x) = {\rm Tr}(x + (x+1)^{r} + (x+1)^{r^2})$ $3$-valued Lemma 3.7 $WG_{g}(x) = {\rm Tr}(x + (x+1)^{q_2} + (x+1)^{q_2^2})$ at most $5$-valued Lemma 3.7 $WG_{T3}(x^d) = {\rm Tr}(x^d + (x^d+1)^{q1} + (x^d+1)^{q_2})$ at most $5$-valued Theorem 3.8
 Functions Hadamard Transform values Ref. $w(x) = T3(x^{2^k+1}) = {\rm Tr}(x + x^r + x^{r^2})$ $3$-valued [11] $g(x) = T3(x^{2^k-1}) = {\rm Tr}(x + x^{q_2} + x^{q_2^2})$ at most $5$-valued [39] $WG_{w}(x) = {\rm Tr}(x + (x+1)^{r} + (x+1)^{r^2})$ $3$-valued Lemma 3.7 $WG_{g}(x) = {\rm Tr}(x + (x+1)^{q_2} + (x+1)^{q_2^2})$ at most $5$-valued Lemma 3.7 $WG_{T3}(x^d) = {\rm Tr}(x^d + (x^d+1)^{q1} + (x^d+1)^{q_2})$ at most $5$-valued Theorem 3.8
Experimental results for ideal $t$-tuple distributions of the WG transform of the Kasami power functions for $k' = 3, 4$ and $5$. When $k' = 3$, $WG_{R_3}(x^d)$ over ${\mathbb F}_{2^m}$
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