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November  2019, 13(4): 689-704. doi: 10.3934/amc.2019041

Revisiting design principles of Salsa and ChaCha

Sabyasachi Dey , Tapabrata Roy and  Santanu Sarkar

Department of Mathematics, Indian Institute of Technology Madras, Sardar Patel Road, Chennai 600036, India

Received  October 2018 Published  June 2019

Salsa and ChaCha are well known names in the family of stream ciphers. In this paper, we first revisit the existing attacks on these ciphers. We first perform an accurate computation of the attack complexities of the existing technique instead of the estimation used in previous works. This improves the complexity by some margin. The differential attacks using probabilistic neutral bits against ChaCha and Salsa involve two probability biases: forward probability bias ($ \epsilon_d $) and backward probability bias ($ \epsilon_a $). In the second part of the paper, we suggest a method to increase the backward probability bias, which helps reduce the attack complexity. Finally, we focus on the design principle of ChaCha. We suggest a slight modification in the design of this cipher as a countermeasure of the differential attacks against it. We show that the key recovery attacks proposed against ChaCha will not be effective on this modified version.

Keywords: Stream cipher, ChaCha, cryptanalysis, new design.
Mathematics Subject Classification: Primary: 11Y05; Secondary: 94A60.
Citation: Sabyasachi Dey, Tapabrata Roy, Santanu Sarkar. Revisiting design principles of Salsa and ChaCha. Advances in Mathematics of Communications, 2019, 13 (4) : 689-704. doi: 10.3934/amc.2019041
References:
[1]

J. P. Aumasson, S. Fischer, S. Khazaei, W. Meier and C. Rechberger, New features of latin dances: Analysis of salsa, chacha, and rumba, FSE 2008, LNCS, 5086 (2008), 470-488.  doi: 10.1007/978-3-540-71039-4_30.  Google Scholar

[2]

D. J. Bernstein, Salsa20 specification, eSTREAM Project algorithm description, http://www.ecrypt.eu.org/stream/salsa20pf.html, 2005. Google Scholar

[3]

D. J. Bernstein, ChaCha, a variant of Salsa20, In Workshop Record of SASC, volume 8, 2008. Google Scholar

[4]

A. R. Choudhuri and S. Maitra, Significantly Improved Multi-bit Differentials for Reduced Round Salsa and ChaCha, Accepted in FSE 2017, Available at http://eprint.iacr.org/2016/1034. Google Scholar

[5]

P. Crowley, Truncated differential cryptanalysis of five rounds of Salsa20, IACR 2005. Available at http://eprint.iacr.org/2005/375. Google Scholar

[6]

S. Dey and S. Sarkar, Improved analysis for reduced round Salsa and ChaCha, Discrete Applied Mathematics, 227 (2017), 58-69.  doi: 10.1016/j.dam.2017.04.034.  Google Scholar

[7]

S. Fischer, W. Meier, C. Berbain and J. F. Biasse, Non-randomness in eSTREAM Candidates Salsa20 and TSC-4, Progress in Cryptology–INDOCRYPT 2006, LNCS, 4329 (2006), 2-16.  doi: 10.1007/11941378_2.  Google Scholar

[8]

S. Maitra, Chosen IV cryptanalysis on reduced round chacha and salsa, Discrete Applied Mathematics, 208 (2016), 88-97.  doi: 10.1016/j.dam.2016.02.020.  Google Scholar

[9]

S. Maitra, G. Paul and W. Meier, Salsa20 Cryptanalysis: New Moves and Revisiting Old Styles, WCC 2015. Available at http://eprint.iacr.org/2015/217. Google Scholar

[10]

, Sage: Open Source Mathematics Software, http://www.sagemath.org. Google Scholar

[11]

Z. Shi, B. Zhang, D. Feng and W. Wu, Improved key recovery attacks on reduced-round salsa20 and ChaCha, ICISC 2012, LNCS, 7839 (2012), 337-351.  doi: 10.1007/978-3-642-37682-5_24.  Google Scholar

[12]

Y. Tsunoo, T. Saito, H. Kubo, T. Suzaki and H. Nakashima, Differential Cryptanalysis of Salsa20/8, 2007. Google Scholar

show all references

References:
[1]

J. P. Aumasson, S. Fischer, S. Khazaei, W. Meier and C. Rechberger, New features of latin dances: Analysis of salsa, chacha, and rumba, FSE 2008, LNCS, 5086 (2008), 470-488.  doi: 10.1007/978-3-540-71039-4_30.  Google Scholar

[2]

D. J. Bernstein, Salsa20 specification, eSTREAM Project algorithm description, http://www.ecrypt.eu.org/stream/salsa20pf.html, 2005. Google Scholar

[3]

D. J. Bernstein, ChaCha, a variant of Salsa20, In Workshop Record of SASC, volume 8, 2008. Google Scholar

[4]

A. R. Choudhuri and S. Maitra, Significantly Improved Multi-bit Differentials for Reduced Round Salsa and ChaCha, Accepted in FSE 2017, Available at http://eprint.iacr.org/2016/1034. Google Scholar

[5]

P. Crowley, Truncated differential cryptanalysis of five rounds of Salsa20, IACR 2005. Available at http://eprint.iacr.org/2005/375. Google Scholar

[6]

S. Dey and S. Sarkar, Improved analysis for reduced round Salsa and ChaCha, Discrete Applied Mathematics, 227 (2017), 58-69.  doi: 10.1016/j.dam.2017.04.034.  Google Scholar

[7]

S. Fischer, W. Meier, C. Berbain and J. F. Biasse, Non-randomness in eSTREAM Candidates Salsa20 and TSC-4, Progress in Cryptology–INDOCRYPT 2006, LNCS, 4329 (2006), 2-16.  doi: 10.1007/11941378_2.  Google Scholar

[8]

S. Maitra, Chosen IV cryptanalysis on reduced round chacha and salsa, Discrete Applied Mathematics, 208 (2016), 88-97.  doi: 10.1016/j.dam.2016.02.020.  Google Scholar

[9]

S. Maitra, G. Paul and W. Meier, Salsa20 Cryptanalysis: New Moves and Revisiting Old Styles, WCC 2015. Available at http://eprint.iacr.org/2015/217. Google Scholar

[10]

, Sage: Open Source Mathematics Software, http://www.sagemath.org. Google Scholar

[11]

Z. Shi, B. Zhang, D. Feng and W. Wu, Improved key recovery attacks on reduced-round salsa20 and ChaCha, ICISC 2012, LNCS, 7839 (2012), 337-351.  doi: 10.1007/978-3-642-37682-5_24.  Google Scholar

[12]

Y. Tsunoo, T. Saito, H. Kubo, T. Suzaki and H. Nakashima, Differential Cryptanalysis of Salsa20/8, 2007. Google Scholar

Figure 1.  Two Normal distribution curves
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Figure 2.  Comparison between the probability achieved by random values and our values for ChaCha
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Figure 3.  Comparison between the probability achieved by random values and our values for Salsa
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Table 1.  New attack complexities for 8 rounds Salsa and 7 rounds ChaCha
Cipher $ n $ $ |\epsilon| $ $ \alpha $ $ T $ $ N $ Existing complexity New complexity
Salsa 43 0.000176 16.43 $ 2^{29.66} $ $ 2^{30.66} $ $ 2^{243.93} $ $ 2^{243.74} $
ChaCha 53 0.000100 24.83 $ 2^{31.72} $ $ 2^{32.72} $ $ 2^{235.93} $ $ 2^{235.78} $
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Cipher $ n $ $ |\epsilon| $ $ \alpha $ $ T $ $ N $ Existing complexity New complexity
Salsa 43 0.000176 16.43 $ 2^{29.66} $ $ 2^{30.66} $ $ 2^{243.93} $ $ 2^{243.74} $
ChaCha 53 0.000100 24.83 $ 2^{31.72} $ $ 2^{32.72} $ $ 2^{235.93} $ $ 2^{235.78} $
Table 2.  Attack complexities using chaining distinguisher
Cipher Existing complexity New complexity
Salsa $ 2^{243.67} $ $ 2^{243.23} $
ChaCha $ 2^{235.22} $ $ 2^{234.78} $
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Cipher Existing complexity New complexity
Salsa $ 2^{243.67} $ $ 2^{243.23} $
ChaCha $ 2^{235.22} $ $ 2^{234.78} $
Table 3.  Values for Probabilistic Neutral Bits of ChaCha
3 6 7 15 16 17 18 31 35 38 67 68
1 1 0 1 1 1 0 x 1 0 0 1
71 72 73 91 92 93 94 95 96 97 98 99
1 1 0 x x x x x 1 1 1 1
100 103 104 105 106 107 127 136 137 138 139 156
0 1 1 1 1 0 x 0 0 0 1 x
159 191 223 224 225 226 227 228 248 249 250 251
x x x 1 1 1 1 0 x x x x
252 253 254 255
x x x x
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3 6 7 15 16 17 18 31 35 38 67 68
1 1 0 1 1 1 0 x 1 0 0 1
71 72 73 91 92 93 94 95 96 97 98 99
1 1 0 x x x x x 1 1 1 1
100 103 104 105 106 107 127 136 137 138 139 156
0 1 1 1 1 0 x 0 0 0 1 x
159 191 223 224 225 226 227 228 248 249 250 251
x x x 1 1 1 1 0 x x x x
252 253 254 255
x x x x
Table 4.  Comparison of bias $ \epsilon $ and complexities between existing and our method for ChaCha when $ n = 52 $
Percentage of keys bias (existing) bias (our) existing complexity our complexity
10 0.000200 0.000648 $ 2^{234.97} $ $ 2^{231.56} $
20 0.000178 0.000433 $ 2^{235.28} $ $ 2^{232.68} $
30 0.000165 0.000314 $ 2^{235.50} $ $ 2^{ 233.57} $
40 0.000152 0.000231 $ 2^{235.73} $ $ 2^{234.43} $
50 0.000139 0.000182 $ 2^{235.96} $ $ 2^{235.09} $
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Percentage of keys bias (existing) bias (our) existing complexity our complexity
10 0.000200 0.000648 $ 2^{234.97} $ $ 2^{231.56} $
20 0.000178 0.000433 $ 2^{235.28} $ $ 2^{232.68} $
30 0.000165 0.000314 $ 2^{235.50} $ $ 2^{ 233.57} $
40 0.000152 0.000231 $ 2^{235.73} $ $ 2^{234.43} $
50 0.000139 0.000182 $ 2^{235.96} $ $ 2^{235.09} $
Table 5.  Values for the probabilistic neutral bits of Salsa
25 26 27 28 29 30 31 39 70 71 72 107
x x x x x x x x 1 1 0 1
119 120 121 122 164 165 166 167 168 169 170 171
1 1 1 0 0 1 1 1 1 0 0 0
172 173 174 175 176 209 210 211 212 213 224 225
0 0 0 0 1 1 1 1 1 0 0 1
241 242 243 244 245 246 255
1 1 1 1 1 0 x
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25 26 27 28 29 30 31 39 70 71 72 107
x x x x x x x x 1 1 0 1
119 120 121 122 164 165 166 167 168 169 170 171
1 1 1 0 0 1 1 1 1 0 0 0
172 173 174 175 176 209 210 211 212 213 224 225
0 0 0 0 1 1 1 1 1 0 0 1
241 242 243 244 245 246 255
1 1 1 1 1 0 x
Table 6.  Comparison of bias and complexities between existing and our method for Salsa when $ n = 43 $
Percentage of keys bias (existing) bias (our) existing complexity our complexity
10 -0.000232 -0.000667 $ 2^{243.18} $ $ 2^{240.11} $
20 -0.000207 -0.000397 $ 2^{243.48} $ $ 2^{241.53} $
30 -0.000192 -0.000305 $ 2^{243.69} $ $ 2^{242.24} $
40 -0.000181 -0.000226 $ 2^{ 243.86} $ $ 2^{243.06} $
50 -0.000176 -0.000192 $ 2^{243.93} $ $ 2^{243.51} $
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Percentage of keys bias (existing) bias (our) existing complexity our complexity
10 -0.000232 -0.000667 $ 2^{243.18} $ $ 2^{240.11} $
20 -0.000207 -0.000397 $ 2^{243.48} $ $ 2^{241.53} $
30 -0.000192 -0.000305 $ 2^{243.69} $ $ 2^{242.24} $
40 -0.000181 -0.000226 $ 2^{ 243.86} $ $ 2^{243.06} $
50 -0.000176 -0.000192 $ 2^{243.93} $ $ 2^{243.51} $
Table 7.  Number of $ Random $ $ Value $ $ Positions $ for some size of PNB
Percentage of PNB's Number of $ Random $ $ Value $ $ Positions $
5% 70
10% 127
15% 180
20% 219
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Percentage of PNB's Number of $ Random $ $ Value $ $ Positions $
5% 70
10% 127
15% 180
20% 219
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