doi: 10.3934/amc.2020128
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Some results on lightweight stream ciphers Fountain v1 & Lizard

1. 

Indian Institute of Technology Kharagpur, Kharagpur, India

2. 

Indian Statistical Institute, Kolkata, India

3. 

Indian Institute of Technology Madras, Chennai, India

* Corresponding author: Dibyendu Roy

Received  September 2019 Revised  September 2020 Early access December 2020

In this paper, we propose cryptanalytic results on two lightweight stream ciphers: Fountain v1 and Lizard. The main results of this paper are the followings:

$ - $ We propose a zero-sum distinguisher on reduced round Fountain v1. In this context, we study the non-randomness of the cipher with a careful selection of cube variables. Our obtained cube provides a zero-sum on Fountain v1 till $ 188 $ initialization rounds and significant non-randomness till $ 189 $ rounds. This results in a distinguishing attack on Fountain v1 with $ 189 $ initialization rounds.

$ - $ Further, we find that the same cipher has a weakness against conditional Time-Memory-Data-Tradeoff (TMDTO). We show that TMDTO attack using sampling resistance has online complexity $ 2^{110} $ and offline complexity $ 2^{146} $.

$ - $ Finally, we revisit the Time-Memory-Data-Tradeoff attack on Lizard by Maitra et al. (IEEE Transactions on Computers, 2018) and provide our observations on their work. We show that instead of choosing any random string, some particular strings would provide better results in their proposed attack technique.

Citation: Ravi Anand, Dibyendu Roy, Santanu Sarkar. Some results on lightweight stream ciphers Fountain v1 & Lizard. Advances in Mathematics of Communications, doi: 10.3934/amc.2020128
References:
[1]

Lightweight Cryptography Standardization, Available from: https://csrc.nist.gov/projects/lightweight-cryptography. Google Scholar

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F. Armknecht and V. Mikhalev, On lightweight stream ciphers with shorter internal states, International Workshop on Fast Software Encryption (FSE), LNCS, Springer, 9054 (2015), 451–470. doi: 10.1007/978-3-662-48116-5_22.  Google Scholar

[3]

J.-P. Aumasson, I. Dinur, W. Meier and A. Shamir, Cube testers and key recovery attacks on reduced-round MD6 and Trivium, International Workshop on Fast Software Encryption (FSE), LNCS, Springer, 5665 (2009), 1–22. doi: 10.1007/978-3-642-03317-9_1.  Google Scholar

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S. Banik, Some results on Sprout, International Conference on Cryptology in India (Indocrypt), LNCS, Springer, 9462 (2015), 124–139. doi: 10.1007/978-3-319-26617-6_7.  Google Scholar

[6]

S. BanikT. IsobeT. Cui and J. Guo, Some cryptanalytic results on Lizard, IACR Transactions on Symmetric Cryptology, 2017 (2017), 82-98.   Google Scholar

[7]

A. Biryukov and A. Shamir, Cryptanalytic time/memory/data tradeoffs for stream ciphers, International Conference on the Theory and Application of Cryptology and Information Security (Asiacrypt), LNCS, Springer, 1976 (2000), 1–13. doi: 10.1007/3-540-44448-3_1.  Google Scholar

[8]

A. Biryukov, A. Shamir and D. Wagner, Real time cryptanalysis of A5/1 on a PC, International Workshop on Fast Software Encryption (FSE), LNCS, Springer, 1978 (2000), 37–44. doi: 10.1007/3-540-44706-7_1.  Google Scholar

[9]

S. Dey, T. Roy and S. Sarkar, Some results on Fruit, Designs, Codes and Cryptography, Springer, 87 (2019), 349–364. doi: 10.1007/s10623-018-0533-y.  Google Scholar

[10]

I. Dinur and A. Shamir, Cube attacks on tweakable black box polynomials, Annual International Conference on the Theory and Applications of Cryptographic Techniques (Eurocrypt), LNCS, Springer, 5479 (2009), 278–299. doi: 10.1007/978-3-642-01001-9_16.  Google Scholar

[11]

M. F. Esgin and O. Kara, Practical cryptanalysis of full Sprout with TMD tradeoff attacks, International Conference on Selected Areas in Cryptography (SAC), LNCS, Springer, 9566 (2015), 67–85. doi: 10.1007/978-3-319-31301-6_4.  Google Scholar

[12]

V. A. Ghafari and H. Hu, A new chosen IV statistical distinguishing framework to attack symmetric ciphers, and its application to ACORN-v3 and Grain-128a, Journal of Ambient Intelligence and Humanized Computing, Springer, 10 (2019), 2393-2400.   Google Scholar

[13]

V. A. Ghafari, H. Hu and Y. Chen, Fruit-80: A secure ultra-lightweight stream cipher for constrained environments, Entropy, Multidisciplinary Digital Publishing Institute, 20 (2018), 180. Google Scholar

[14]

V. A. Ghafari, H. Hu and M. Alizadeh, Necessary conditions for designing secure stream ciphers with the minimal internal states, IACR Cryptol. ePrint Arch., (2017), 765. Google Scholar

[15]

J. Golić, Cryptanalysis of alleged A5 stream cipher, International Conference on the Theory and Applications of Cryptographic Techniques (Eurocrypt), LNCS, Springer, 1233 (1997), 239–255. Google Scholar

[16]

C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2012. Google Scholar

[17]

M. Hamann, M. Krause, W. Meier and B. Zhang, Design and analysis of small-state Grain-like stream ciphers, Cryptography and Communications, Springer, 10 (2018), 803–834. doi: 10.1007/s12095-017-0261-6.  Google Scholar

[18]

M. HamannM. Krause and W. Meier, LIZARD - A lightweight stream cipher for power-constrained devices, IACR Transactions on Symmetric Cryptology, 2017 (2017), 45-79.   Google Scholar

[19]

M. HellT. Johansson and W. Meier, Grain: A stream cipher for constrained environments, International Journal of Wireless and Mobile Computing, 2 (2007), 86-93.  doi: 10.1504/IJWMC.2007.013798.  Google Scholar

[20]

M. E. Hellman, A cryptanalytic time-memory trade-off, IEEE Transactions on Information Theory, 26 (1980), 401-406.  doi: 10.1109/TIT.1980.1056220.  Google Scholar

[21]

V. Lallemand and M. N. Plasencia, Cryptanalysis of full Sprout, Annual Cryptology Conference (Crypto), LNCS, Springer, 9215 (2015), 663–682. doi: 10.1007/978-3-662-47989-6_32.  Google Scholar

[22]

S. MaitraN. SinhaA. SiddhantiR. Anand and S. Gangopadhyay, A TMDTO attack against Lizard, IEEE Transactions on Computers, 67 (2017), 733-739.  doi: 10.1109/TC.2017.2773062.  Google Scholar

[23]

S. Maitra, S. Sarkar, A. Baksi and P. Dey, Key recovery from state information of Sprout: Application to cryptanalysis and fault attack, IPSI Transactions on Advanced Research, 12 (2016). Google Scholar

[24]

S. MaitraA. Siddhanti and S. Sarkar, A differential fault attack on Plantlet, IEEE Transactions on Computers, 66 (2017), 1804-1808.  doi: 10.1109/TC.2017.2700469.  Google Scholar

[25]

M. J. MihaljevićS. GangopadhyayG. Paul and H. Imai, Internal state recovery of Grain-v1 employing normality order of the filter function, IET Information Security, 6 (2012), 55-64.   Google Scholar

[26]

M. J. MihaljevićS. GangopadhyayG. Paul and H. Imai, Generic cryptographic weakness of $k$-normal Boolean functions in certain stream ciphers and cryptanalysis of Grain-128, Periodica Mathematica Hungarica, 65 (2012), 205-227.  doi: 10.1007/s10998-012-4631-8.  Google Scholar

[27]

V. MikhalevF. Armknecht and C. Müller, On ciphers that continuously access the non-volatile key, IACR Transactions on Symmetric Cryptology, 2016 (2016), 52-79.  doi: 10.46586/tosc.v2016.i2.52-79.  Google Scholar

[28]

R. Posteuca, Related-key differential slide attack against Fountain V1, Proceedings of the Romanian Academy, Series A, 21 (2020), 61–68.  Google Scholar

[29]

S. SarkarS. Maitra and A. Baksi, Observing biases in the state: Case studies with Trivium and Trivia-sc, Designs, Codes and Cryptography, 82 (2017), 351-375.  doi: 10.1007/s10623-016-0211-x.  Google Scholar

[30]

Q. Wang, Y. Hao, Y. Todo, C. Li, T. Isobe and W. Meier, Improved division property based cube attacks exploiting algebraic properties of superpoly, International Cryptology Conference (Crypto), LNCS, Springer, 10991 (2018), 275–305. doi: 10.1007/978-3-319-96884-1_10.  Google Scholar

[31] D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1st Edition, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511813658.  Google Scholar
[32]

B. Zhang, Fountain: A lightweight authenticated cipher (v1), NIST Lightweight Cryptography Competition, (2019), 1, https://csrc.nist.gov/CSRC/media/Projects/Lightweight-Cryptography/documents/round-1/spec-doc/fountain-spec.pdf. Google Scholar

[33]

B. Zhang and X. Gong, Another tradeoff attack on Sprout-like stream ciphers, International Conference on the Theory and Application of Cryptology and Information Security (Asiacrypt), LNCS, Springer, 9453 (2015), 561–585. doi: 10.1007/978-3-662-48800-3_23.  Google Scholar

[34]

B. ZhangX. Gong and W. Meier, Fast correlation attacks on Grain-like small state stream ciphers, IACR Transactions on Symmetric Cryptology, 2017 (2017), 58-81.   Google Scholar

show all references

References:
[1]

Lightweight Cryptography Standardization, Available from: https://csrc.nist.gov/projects/lightweight-cryptography. Google Scholar

[2]

F. Armknecht and V. Mikhalev, On lightweight stream ciphers with shorter internal states, International Workshop on Fast Software Encryption (FSE), LNCS, Springer, 9054 (2015), 451–470. doi: 10.1007/978-3-662-48116-5_22.  Google Scholar

[3]

J.-P. Aumasson, I. Dinur, W. Meier and A. Shamir, Cube testers and key recovery attacks on reduced-round MD6 and Trivium, International Workshop on Fast Software Encryption (FSE), LNCS, Springer, 5665 (2009), 1–22. doi: 10.1007/978-3-642-03317-9_1.  Google Scholar

[4]

S. H. Babbage, Improved "exhaustive search" attacks on stream ciphers, European Convention on Security and Detection, IET, (1995), 161–166. doi: 10.1049/cp:19950490.  Google Scholar

[5]

S. Banik, Some results on Sprout, International Conference on Cryptology in India (Indocrypt), LNCS, Springer, 9462 (2015), 124–139. doi: 10.1007/978-3-319-26617-6_7.  Google Scholar

[6]

S. BanikT. IsobeT. Cui and J. Guo, Some cryptanalytic results on Lizard, IACR Transactions on Symmetric Cryptology, 2017 (2017), 82-98.   Google Scholar

[7]

A. Biryukov and A. Shamir, Cryptanalytic time/memory/data tradeoffs for stream ciphers, International Conference on the Theory and Application of Cryptology and Information Security (Asiacrypt), LNCS, Springer, 1976 (2000), 1–13. doi: 10.1007/3-540-44448-3_1.  Google Scholar

[8]

A. Biryukov, A. Shamir and D. Wagner, Real time cryptanalysis of A5/1 on a PC, International Workshop on Fast Software Encryption (FSE), LNCS, Springer, 1978 (2000), 37–44. doi: 10.1007/3-540-44706-7_1.  Google Scholar

[9]

S. Dey, T. Roy and S. Sarkar, Some results on Fruit, Designs, Codes and Cryptography, Springer, 87 (2019), 349–364. doi: 10.1007/s10623-018-0533-y.  Google Scholar

[10]

I. Dinur and A. Shamir, Cube attacks on tweakable black box polynomials, Annual International Conference on the Theory and Applications of Cryptographic Techniques (Eurocrypt), LNCS, Springer, 5479 (2009), 278–299. doi: 10.1007/978-3-642-01001-9_16.  Google Scholar

[11]

M. F. Esgin and O. Kara, Practical cryptanalysis of full Sprout with TMD tradeoff attacks, International Conference on Selected Areas in Cryptography (SAC), LNCS, Springer, 9566 (2015), 67–85. doi: 10.1007/978-3-319-31301-6_4.  Google Scholar

[12]

V. A. Ghafari and H. Hu, A new chosen IV statistical distinguishing framework to attack symmetric ciphers, and its application to ACORN-v3 and Grain-128a, Journal of Ambient Intelligence and Humanized Computing, Springer, 10 (2019), 2393-2400.   Google Scholar

[13]

V. A. Ghafari, H. Hu and Y. Chen, Fruit-80: A secure ultra-lightweight stream cipher for constrained environments, Entropy, Multidisciplinary Digital Publishing Institute, 20 (2018), 180. Google Scholar

[14]

V. A. Ghafari, H. Hu and M. Alizadeh, Necessary conditions for designing secure stream ciphers with the minimal internal states, IACR Cryptol. ePrint Arch., (2017), 765. Google Scholar

[15]

J. Golić, Cryptanalysis of alleged A5 stream cipher, International Conference on the Theory and Applications of Cryptographic Techniques (Eurocrypt), LNCS, Springer, 1233 (1997), 239–255. Google Scholar

[16]

C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 2012. Google Scholar

[17]

M. Hamann, M. Krause, W. Meier and B. Zhang, Design and analysis of small-state Grain-like stream ciphers, Cryptography and Communications, Springer, 10 (2018), 803–834. doi: 10.1007/s12095-017-0261-6.  Google Scholar

[18]

M. HamannM. Krause and W. Meier, LIZARD - A lightweight stream cipher for power-constrained devices, IACR Transactions on Symmetric Cryptology, 2017 (2017), 45-79.   Google Scholar

[19]

M. HellT. Johansson and W. Meier, Grain: A stream cipher for constrained environments, International Journal of Wireless and Mobile Computing, 2 (2007), 86-93.  doi: 10.1504/IJWMC.2007.013798.  Google Scholar

[20]

M. E. Hellman, A cryptanalytic time-memory trade-off, IEEE Transactions on Information Theory, 26 (1980), 401-406.  doi: 10.1109/TIT.1980.1056220.  Google Scholar

[21]

V. Lallemand and M. N. Plasencia, Cryptanalysis of full Sprout, Annual Cryptology Conference (Crypto), LNCS, Springer, 9215 (2015), 663–682. doi: 10.1007/978-3-662-47989-6_32.  Google Scholar

[22]

S. MaitraN. SinhaA. SiddhantiR. Anand and S. Gangopadhyay, A TMDTO attack against Lizard, IEEE Transactions on Computers, 67 (2017), 733-739.  doi: 10.1109/TC.2017.2773062.  Google Scholar

[23]

S. Maitra, S. Sarkar, A. Baksi and P. Dey, Key recovery from state information of Sprout: Application to cryptanalysis and fault attack, IPSI Transactions on Advanced Research, 12 (2016). Google Scholar

[24]

S. MaitraA. Siddhanti and S. Sarkar, A differential fault attack on Plantlet, IEEE Transactions on Computers, 66 (2017), 1804-1808.  doi: 10.1109/TC.2017.2700469.  Google Scholar

[25]

M. J. MihaljevićS. GangopadhyayG. Paul and H. Imai, Internal state recovery of Grain-v1 employing normality order of the filter function, IET Information Security, 6 (2012), 55-64.   Google Scholar

[26]

M. J. MihaljevićS. GangopadhyayG. Paul and H. Imai, Generic cryptographic weakness of $k$-normal Boolean functions in certain stream ciphers and cryptanalysis of Grain-128, Periodica Mathematica Hungarica, 65 (2012), 205-227.  doi: 10.1007/s10998-012-4631-8.  Google Scholar

[27]

V. MikhalevF. Armknecht and C. Müller, On ciphers that continuously access the non-volatile key, IACR Transactions on Symmetric Cryptology, 2016 (2016), 52-79.  doi: 10.46586/tosc.v2016.i2.52-79.  Google Scholar

[28]

R. Posteuca, Related-key differential slide attack against Fountain V1, Proceedings of the Romanian Academy, Series A, 21 (2020), 61–68.  Google Scholar

[29]

S. SarkarS. Maitra and A. Baksi, Observing biases in the state: Case studies with Trivium and Trivia-sc, Designs, Codes and Cryptography, 82 (2017), 351-375.  doi: 10.1007/s10623-016-0211-x.  Google Scholar

[30]

Q. Wang, Y. Hao, Y. Todo, C. Li, T. Isobe and W. Meier, Improved division property based cube attacks exploiting algebraic properties of superpoly, International Cryptology Conference (Crypto), LNCS, Springer, 10991 (2018), 275–305. doi: 10.1007/978-3-319-96884-1_10.  Google Scholar

[31] D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1st Edition, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511813658.  Google Scholar
[32]

B. Zhang, Fountain: A lightweight authenticated cipher (v1), NIST Lightweight Cryptography Competition, (2019), 1, https://csrc.nist.gov/CSRC/media/Projects/Lightweight-Cryptography/documents/round-1/spec-doc/fountain-spec.pdf. Google Scholar

[33]

B. Zhang and X. Gong, Another tradeoff attack on Sprout-like stream ciphers, International Conference on the Theory and Application of Cryptology and Information Security (Asiacrypt), LNCS, Springer, 9453 (2015), 561–585. doi: 10.1007/978-3-662-48800-3_23.  Google Scholar

[34]

B. ZhangX. Gong and W. Meier, Fast correlation attacks on Grain-like small state stream ciphers, IACR Transactions on Symmetric Cryptology, 2017 (2017), 58-81.   Google Scholar

Figure 1.  Design Specification of Fountain v1.
Table 1.  S-box for key-IV initialization phase
$ x $ : $ 0 $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ $ A $ $ B $ $ C $ $ D $ $ E $ $ F $
$ S(x) $ : $ 1 $ $ A $ $ 4 $ $ C $ $ 6 $ $ F $ $ 3 $ $ 9 $ $ 2 $ $ D $ $ B $ $ 7 $ $ 5 $ $ 0 $ $ 8 $ $ E $
$ x $ : $ 0 $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ $ A $ $ B $ $ C $ $ D $ $ E $ $ F $
$ S(x) $ : $ 1 $ $ A $ $ 4 $ $ C $ $ 6 $ $ F $ $ 3 $ $ 9 $ $ 2 $ $ D $ $ B $ $ 7 $ $ 5 $ $ 0 $ $ 8 $ $ E $
Table 2.  Integrated S-box for key-IV initialization phase
$ x $ : $ 0 $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ $ A $ $ B $ $ C $ $ D $ $ E $ $ F $
$ S(x) $ : $ 9 $ $ 5 $ $ 6 $ $ D $ $ 8 $ $ A $ $ 7 $ $ 2 $ $ E $ $ 4 $ $ C $ $ 1 $ $ F $ $ 0 $ $ B $ $ 3 $
$ x $ : $ 0 $ $ 1 $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ $ A $ $ B $ $ C $ $ D $ $ E $ $ F $
$ S(x) $ : $ 9 $ $ 5 $ $ 6 $ $ D $ $ 8 $ $ A $ $ 7 $ $ 2 $ $ E $ $ 4 $ $ C $ $ 1 $ $ F $ $ 0 $ $ B $ $ 3 $
Table 3.  Distinguishing attack on Fountain v1
Cube Cube variable indices Probability of superpoly = 0
Size Rounds
$ 187 $ $ 188 $ $ 189 $ $ 190 $ $ 191 $ $ 192 $ $ 193 $ $ 194 $ $ 195 $
$ |I_4|=31 $ $ I_4 $ $ 1.00 $ $ 1.00 $ $ 0.93 $ $ 0.50 $ $ 0.51 $ $ 0.48 $ $ 0.48 $ $ 0.52 $ $ 0.49 $
Cube Cube variable indices Probability of superpoly = 0
Size Rounds
$ 187 $ $ 188 $ $ 189 $ $ 190 $ $ 191 $ $ 192 $ $ 193 $ $ 194 $ $ 195 $
$ |I_4|=31 $ $ I_4 $ $ 1.00 $ $ 1.00 $ $ 0.93 $ $ 0.50 $ $ 0.51 $ $ 0.48 $ $ 0.48 $ $ 0.52 $ $ 0.49 $
Table 4.  TMDTO parameters for Fountain v1
Time Memory Data Pre-processing
$ 110 $ $ 110 $ $ 110 $ $ 146 $
Time Memory Data Pre-processing
$ 110 $ $ 110 $ $ 110 $ $ 146 $
Table 5.  State bits recovery
Keystream bit Equation Guessed bits Feedback Recovered
bits bit
$ z_{34} $ $ s_{45}^{(1)} = z_{24} + s_{37}^{(1)} + s_{54}^{(2)} +s_{39}^{(3)} + s_{42}^{(3)} $ $ s_{1}^{(1)}, s_{1}^{(2)} $, $ s_{1}^{(3)} $ $ s_{45}^{(1)} $
$ + s_{42}^{(4)}+s_{63}^{(4)}+s_{36}^{(4)}s_{31}^{(1)}+s_{28}^{(2)}s_{29}^{(3)} $ $ s_{1}^{(4)}, s_{0}^{(4)} $
$ + s_{37}^{(4)} s_{61}^{(3)} + s_{36}^{(4)} s_{57}^{(4)} s_{64}^{(4)} $ $ s_{61}^{(3)}, s_{64}^{(4)} $ $ s_{64}^{(4)} $
$ z_{35} $ $ s_{46}^{(1)} = z_{25} + s_{38}^{(1)} + s_{55}^{(2)} +s_{40}^{(3)} + s_{43}^{(3)} $ $ s_{2}^{(1)}, s_{2}^{(2)} $, $ s_{2}^{(3)} $ $ s_{46}^{(1)} $
$ + s_{43}^{(4)}+s_{64}^{(4)}+s_{37}^{(4)}s_{32}^{(1)}+s_{28}^{(2)}s_{29}^{(3)} $
$ + s_{38}^{(4)} s_{62}^{(3)} + s_{37}^{(4)} s_{58}^{(4)} s_{65}^{(4)} $ $ s_{62}^{(3)}, s_{65}^{(4)} $ $ s_{65}^{(4)} $
Keystream bit Equation Guessed bits Feedback Recovered
bits bit
$ z_{34} $ $ s_{45}^{(1)} = z_{24} + s_{37}^{(1)} + s_{54}^{(2)} +s_{39}^{(3)} + s_{42}^{(3)} $ $ s_{1}^{(1)}, s_{1}^{(2)} $, $ s_{1}^{(3)} $ $ s_{45}^{(1)} $
$ + s_{42}^{(4)}+s_{63}^{(4)}+s_{36}^{(4)}s_{31}^{(1)}+s_{28}^{(2)}s_{29}^{(3)} $ $ s_{1}^{(4)}, s_{0}^{(4)} $
$ + s_{37}^{(4)} s_{61}^{(3)} + s_{36}^{(4)} s_{57}^{(4)} s_{64}^{(4)} $ $ s_{61}^{(3)}, s_{64}^{(4)} $ $ s_{64}^{(4)} $
$ z_{35} $ $ s_{46}^{(1)} = z_{25} + s_{38}^{(1)} + s_{55}^{(2)} +s_{40}^{(3)} + s_{43}^{(3)} $ $ s_{2}^{(1)}, s_{2}^{(2)} $, $ s_{2}^{(3)} $ $ s_{46}^{(1)} $
$ + s_{43}^{(4)}+s_{64}^{(4)}+s_{37}^{(4)}s_{32}^{(1)}+s_{28}^{(2)}s_{29}^{(3)} $
$ + s_{38}^{(4)} s_{62}^{(3)} + s_{37}^{(4)} s_{58}^{(4)} s_{65}^{(4)} $ $ s_{62}^{(3)}, s_{65}^{(4)} $ $ s_{65}^{(4)} $
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