Abstract
Recently, a faster variant of SNOW-V, called SNOW-Vi, was proposed for fast enough performance not only in cloud settings but also on low grade CPUs, in response to the requirements of confidentiality and integrity protection in 5G with wider applicability. SNOW-Vi differs in the way how the LFSR is updated and a new location of tap T2 from SNOW-V, but otherwise employs the same 896-bit internal state and provides 256-bit security level. In this paper, we present guess-and-determine attacks on SNOW-Vi which are affected by the changes. Firstly, we analyze the upper bound on the complexity of guess-and-determine attacks against SNOW-Vi, which is still with a complexity of \(2^{512}\). Secondly, we verify that the new improved guess-and-determine attacks on SNOW-V are no more valid to SNOW-Vi, and point out that it mainly benefits from the new location of tap T2. Thirdly, we present a byte-based guess-and-determine attack on SNOW-Vi with a complexity of \(2^{408}\) using 7 consecutive 128-bit keystream output words. The attack initially rewrites the description of SNOW-Vi into byte-oriented mode, then launches the heuristic algorithm to get an intermediate guessing path with simplified connection equations, finally corrects the path by the carry rules. Meanwhile, we first present a detailed description of the heuristic guessing-path auto-searching algorithm with initial guessing set, and propose the guess-and-determine path correcting framework and its automatic algorithm. This is the first complete guess-and-determine attack on SNOW-Vi. These attacks do not threaten SNOW-Vi, but provide more in-depth details for understanding its security and give new ideas for cryptanalysis of other ciphers.
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Notes
The cipher has been submitted to SAGE (Security Algorithms Group of Experts) within 3GPP for consideration for possible use in 5G and is under evaluation [14].
The update of LFSRs we talk about later in this paper are all for 8 times clocked
Actually, the Round keys are both added and defined 0 in the implementation of the AES encryption round functions.
We have illustrated that the recursive enumeration technique from [YJM21] cannot be used directly to speed up the attack, since the first step in the attack that the form of Eq. (1) no more remains.
Usually, we plus 1 to tell apart from none operation.
In the process of designing such guess-and-determine attacks of many stream ciphers, we find that the final complexity of the attacks is related to the number of consecutive time instants chosen for building the attack. The experimental results show that: When the number of time instants is less than some bound, the attack is the same as exhaustive search attack, while when the number of time instants increases and arrives at the bound, it begins to output real guess-and-determine attacks. The complexity of the guess-and-determine attacks decreases with the increasing number of time instants until it reaches the inflection point. Once the number of time instants comes to some point (before there may be some complexity fluctuation), the complexity will not change with the number of time instants any more.
If we do not need to derive all the guessing path with the same scale of guessing basis, this step of judging guessing basis can be done after the third-out loop for one objective node.
For the carries, it is not easy to handle during the search process because its size is not consistent with the search variable, so the final complexity needs to be given through the correction algorithm.
We have also tried to correct the path derived by heuristic auto-searching algorithm when define the initial guessing set with 8 bytes variables as Set (29) and \(T=5\), while it calls for more modified guess bits.
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The authors have no relevant financial or non-financial interests to disclose. The authors have no conflicts of interest to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors have no financial or proprietary interests in any material discussed in this article.
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The datasets generated during and/or analysed during the current study are available in the GitLab repository, https://gitlab.com/jiaolin2019/guess-and-determine-attack-on-snow-vi.git.
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Communicated by M. Naya-Plasencia.
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This work was supported by National Natural Science Foundation of China (Grant Nos. 61902030, 61772517, 62002024, 62022018.
Appendices
Appendix 1: Guess-and-determine path of SNOW-Vi
step | derived variable | used equation | add guess | reduce guess | guess bits |
|---|---|---|---|---|---|
1 | \(R1_{12}^{2}\) | Guess | 8 | ||
2 | \(R1_{1}^{2}\) | Guess | 16 | ||
3 | \(R1_{6}^{2}\) | Guess | 24 | ||
4 | \(R1_{11}^{2}\) | Guess | 32 | ||
5 | \(R1_{8}^{2}\) | Guess | 40 | ||
6 | \(R1_{13}^{2}\) | Guess | 48 | ||
7 | \(R1_{2}^{2}\) | Guess | 56 | ||
8 | \(R1_{7}^{2}\) | Guess | 64 | ||
9 | \(b_{28}^{2}\) | Guess | 72 | ||
10 | \(b_{17}^{2}\) | Guess | 80 | ||
11 | \(b_{22}^{2}\) | Guess | 88 | ||
12 | \(b_{27}^{2}\) | Guess | 96 | ||
13 | \(R2_{12}^{3}\) | Equation 14, \(t=2\) | 96 | ||
14 | \(R2_{13}^{3}\) | Equation 14, \(t=2\) | 96 | ||
15 | \(R2_{14}^{3}\) | Equation 14, \(t=2\) | 96 | ||
16 | \(R2_{15}^{3}\) | Equation 14, \(t=2\) | 96 | ||
17 | \(R2_{8}^{3}\) | Equation 15, \(t=2\) | 96 | ||
18 | \(R2_{9}^{3}\) | Equation 15, \(t=2\) | 96 | ||
19 | \(R2_{10}^{3}\) | Equation 15, \(t=2\) | 96 | ||
20 | \(R2_{11}^{3}\) | Equation 15, \(t=2\) | 96 | ||
21 | \(R2_{12}^{2}\) | Equation 10, \(i=12\), \(t=2\) | 96 | ||
22 | \(b_{12}^{3}\) | Equation 7, \(i=12\), \(t=2\) | 96 | ||
23 | \(R2_{1}^{2}\) | Equation 10, \(i=1\), \(t=2\) | \(cz_{1}^{2}\) | 97 | |
24 | \(R2_{6}^{2}\) | Equation 10, \(i=6\), \(t=2\) | \(cz_{6}^{2}\) | 98 | |
25 | \(b_{6}^{3}\) | Equation 7, \(i=6\), \(t=2\) | 98 | ||
26 | \(R2_{11}^{2}\) | Equation 10, \(i=11\), \(t=2\) | \(cz_{11}^{2}\) | 99 | |
27 | \(b_{11}^{3}\) | Equation 7, \(i=11\), \(t=2\) | 99 | ||
28 | \(R3_{12}^{3}\) | Equation 18, \(t=2\) | 99 | ||
29 | \(R3_{13}^{3}\) | Equation 18, \(t=2\) | 99 | ||
30 | \(R3_{14}^{3}\) | Equation 18, \(t=2\) | 99 | ||
31 | \(R3_{15}^{3}\) | Equation 18, \(t=2\) | 99 | ||
32 | \(R1_{8}^{1}\) | Guess | 107 | ||
33 | \(b_{24}^{1}\) | Guess | 115 | ||
34 | \(R2_{8}^{1}\) | Equation 10, \(i=8\), \(t=1\) | 115 | ||
35 | \(b_{8}^{2}\) | Equation 7, \(i=8\), \(t=1\) | 115 | ||
36 | \(a_{24}^{1}\) | Guess | 123 | ||
37 | \(R3_{8}^{1}\) | Equation 12, \(i=8\), \(t=1\) | 123 | ||
38 | \(a_{8}^{2}\) | Equation 4, \(i=8\), \(t=1\) | 123 | ||
39 | \(b_{24}^{2}\) | Guess | 131 | ||
40 | \(R2_{8}^{2}\) | Equation 10, \(i=8\), \(t=2\) | 131 | ||
41 | \(b_{24}^{3}\) | Equation 6, \(i=8\), \(t=2\) | \(b_{9,7}^{2}\) | 132 | |
42 | \(b_{8}^{3}\) | Equation 7, \(i=8\), \(t=2\) | 132 | ||
43 | \(R1_{8}^{3}\) | Equation 10, \(i=8\), \(t=3\) | 132 | ||
44 | \(R1_{2}^{1}\) | Guess | 140 | ||
45 | \(b_{8}^{4}\) | Equation 7, \(i=8\), \(t=3\) | 140 | ||
46 | \(R2_{9}^{2}\) | Equation 15, \(t=1\) | 140 | ||
47 | \(R2_{10}^{2}\) | Equation 15, \(t=1\) | 140 | ||
48 | \(R1_{13}^{1}\) | Equation 15, \(t=1\) | 140 | ||
49 | \(R1_{7}^{1}\) | Equation 15, \(t=1\) | 140 | ||
50 | \(b_{18}^{1}\) | Guess | 148 | ||
51 | \(R2_{2}^{1}\) | Equation 10, \(i=2\), \(t=1\) | \(cz_{2}^{1}\) | 149 | |
52 | \(b_{2}^{2}\) | Equation 7, \(i=2\), \(t=1\) | 149 | ||
53 | \(b_{29}^{1}\) | Guess | 157 | ||
54 | \(R2_{13}^{1}\) | Equation 10, \(i=13\), \(t=1\) | \(cz_{13}^{1}\) | 158 | |
55 | \(b_{13}^{2}\) | Equation 7, \(i=13\), \(t=1\) | 158 | ||
56 | \(b_{23}^{1}\) | Guess | 166 | ||
57 | \(R2_{7}^{1}\) | Equation 10, \(i=7\), \(t=1\) | \(cz_{7}^{1}\) | 167 | |
58 | \(b_{7}^{2}\) | Equation 7, \(i=7\), \(t=1\) | 167 | ||
59 | \(R3_{8}^{2}\) | Equation 19, \(t=1\) | 167 | ||
60 | \(R3_{9}^{2}\) | Equation 19, \(t=1\) | 167 | ||
61 | \(R3_{10}^{2}\) | Equation 19, \(t=1\) | 167 | ||
62 | \(R3_{11}^{2}\) | Equation 19, \(t=1\) | 167 | ||
63 | \(a_{24}^{0}\) | Guess | 175 | ||
64 | \(a_{8}^{1}\) | Equation 4, \(i=8\), \(t=0\) | 175 | ||
65 | \(b_{8}^{1}\) | Equation 6, \(i=8\), \(t=1\) | \(b_{8,7}^{1}\) | 176 | |
66 | \(b_{24}^{0}\) | Equation 7, \(i=8\), \(t=0\) | 176 | ||
67 | \(R1_{6}^{3}\) | Guess | 184 | ||
68 | \(a_{25}^{2}\) | Equation 12, \(i=9\), \(t=2\) | \(cR1_{9}^{2}\) | 185 | |
69 | \(a_{9}^{3}\) | Equation 4, \(i=9\), \(t=2\) | 185 | ||
70 | \(R2_{6}^{3}\) | Guess | 193 | ||
71 | \(b_{22}^{3}\) | Equation 10, \(i=6\), \(t=3\) | \(cz_{6}^{3}\) | 194 | |
72 | \(b_{6}^{4}\) | Equation 7, \(i=6\), \(t=3\) | 194 | ||
73 | \(b_{22}^{1}\) | Guess | 202 | ||
74 | \(b_{6}^{2}\) | equation 7, \(i=6\), \(t=1\) | 202 | ||
75 | \(a_{6}^{2}\) | Equation 6, \(i=6\), \(t=2\) | 202 | ||
76 | \(a_{22}^{1}\) | Equation 4, \(i=6\), \(t=1\) | 202 | ||
77 | \(a_{24}^{2}\) | Equation 3, \(i=8\), \(t=1\) | \(a_{9,7}^{1}\) | 203 | |
78 | \(R1_{2}^{3}\) | Equation 12, \(i=8\), \(t=2\) | \(cR1_{9}^{2}\) | 202 | |
79 | \(a_{8}^{3}\) | Equation 4, \(i=8\), \(t=2\) | 202 | ||
80 | \(b_{24}^{4}\) | Equation 6, \(i=8\), \(t=3\) | \(b_{9,7}^{3}\) | 203 | |
81 | \(a_{8}^{0}\) | Guess | 211 | ||
82 | \(b_{8}^{0}\) | Equation 6, \(i=8\), \(t=0\) | \(b_{8,7}^{0}\) | 212 | |
83 | \(a_{22}^{0}\) | Equation 3, \(i=8\), \(t=0\) | \(a_{9,7}^{0}\) | 213 | |
84 | \(a_{6}^{1}\) | Equation 4, \(i=6\), \(t=0\) | 213 | ||
85 | \(b_{6}^{1}\) | Equation 6, \(i=6\), \(t=1\) | \(b_{6,7}^{1}\) | 214 | |
86 | \(b_{22}^{0}\) | Equation 7, \(i=6\), \(t=0\) | 214 | ||
87 | \(R3_{7}^{1}\) | Guess | 222 | ||
88 | \(a_{23}^{1}\) | Equation 12, \(i=7\), \(t=1\) | \(cR1_{7}^{1}\) | 223 | |
89 | \(a_{7}^{2}\) | Equation 4, \(i=7\), \(t=1\) | 223 | ||
90 | \(a_{25}^{0}\) | Guess | 231 | ||
91 | \(a_{9}^{1}\) | Equation 4, \(i=9\), \(t=0\) | \(a_{9,7}^{1}\) | 230 | |
92 | \(b_{9}^{1}\) | Equation 2, \(i=9\), \(t=1\) | \(b_{9,7}^{1}\) | 229 | |
93 | \(b_{25}^{0}\) | Equation 7, \(i=9\), \(t=0\) | 229 | ||
94 | \(a_{6}^{0}\) | Guess | 237 | ||
95 | \(b_{6}^{0}\) | Equation 6, \(i=6\), \(t=0\) | \(b_{6,7}^{0}\) | 238 | |
96 | \(a_{20}^{0}\) | Equation 3, \(i=6\), \(t=0\) | \(a_{7,7}^{0}\) | 239 | |
97 | \(a_{4}^{1}\) | Equation 4, \(i=4\), \(t=0\) | 239 | ||
98 | \(R1_{8}^{0}\) | Guess | 247 | ||
99 | \(R2_{8}^{0}\) | Equation 10, \(i=8\), \(t=0\) | 247 | ||
100 | \(R1_{9}^{2}\) | Guess | 255 | ||
101 | \(b_{25}^{2}\) | Equation 10, \(i=9\), \(t=2\) | 255 | ||
102 | \(b_{25}^{1}\) | Equation 5, \(i=9\), \(t=1\) | 255 | ||
103 | \(b_{9}^{3}\) | Equation 7, \(i=9\), \(t=2\) | \(b_{9,7}^{3}\) | 254 | |
104 | \(b_{9}^{2}\) | Equation 7, \(i=9\), \(t=1\) | \(b_{9,7}^{2}\) | 253 | |
105 | \(R3_{6}^{1}\) | Guess | 261 | ||
106 | \(R2_{6}^{1}\) | Equation 12, \(i=6\), \(t=1\) | \(cR1_{6}^{1}\) | \(cR1_{7}^{1}\) | 261 |
107 | \(R1_{6}^{1}\) | Equation 10, \(i=6\), \(t=1\) | \(cz_{6}^{1}\) | \(cz_{7}^{1}\) | 261 |
108 | \(b_{25}^{4}\) | Guess | 269 | ||
109 | \(b_{25}^{3}\) | Equation 5, \(i=9\), \(t=3\) | 269 | ||
110 | \(R1_{9}^{3}\) | Equation 10, \(i=9\), \(t=3\) | 269 | ||
111 | \(a_{9}^{2}\) | Equation 5, \(i=9\), \(t=2\) | 269 | ||
112 | \(b_{9}^{4}\) | Equation 7, \(i=9\), \(t=3\) | 269 | ||
113 | \(a_{25}^{1}\) | Equation 4, \(i=9\), \(t=1\) | 269 | ||
114 | \(a_{20}^{1}\) | Guess | 277 | ||
115 | \(a_{22}^{2}\) | Equation 3, \(i=6\), \(t=1\) | \(a_{7,7}^{1}\) | 278 | |
116 | \(a_{4}^{2}\) | Equation 4, \(i=4\), \(t=1\) | 278 | ||
117 | \(R3_{6}^{2}\) | Equation 12, \(i=6\), \(t=2\) | \(cR1_{6}^{2}\) | 279 | |
118 | \(a_{24}^{3}\) | Equation 3, \(i=8\), \(t=2\) | 279 | ||
119 | \(a_{6}^{3}\) | Equation 4, \(i=6\), \(t=2\) | 279 | ||
120 | \(b_{22}^{4}\) | Equation 6, \(i=6\), \(t=3\) | \(b_{7,7}^{3}\) | 280 | |
121 | \(R2_{9}^{0}\) | Guess | 288 | ||
122 | \(R1_{9}^{0}\) | Equation 10, \(i=9\), \(t=0\) | 288 | ||
123 | \(R1_{3}^{0}\) | Guess | 296 | ||
124 | \(R2_{4}^{1}\) | Equation 16, \(t=0\) | 296 | ||
125 | \(R2_{5}^{1}\) | Equation 16, \(t=0\) | 296 | ||
126 | \(R1_{4}^{0}\) | Equation 16, \(t=0\) | 296 | ||
127 | \(R1_{14}^{0}\) | Equation 16, \(t=0\) | 296 | ||
128 | \(R3_{4}^{1}\) | Equation 12, \(i=4\), \(t=1\) | 296 | ||
129 | \(R3_{5}^{1}\) | Equation 20, \(t=0\) | 296 | ||
130 | \(R2_{4}^{0}\) | Equation 20, \(t=0\) | 296 | ||
131 | \(R2_{14}^{0}\) | Equation 20, \(t=0\) | 296 | ||
132 | \(R2_{3}^{0}\) | Equation 20, \(t=0\) | 296 | ||
133 | \(b_{20}^{0}\) | Equation 10, \(i=4\), \(t=0\) | 296 | ||
134 | \(b_{30}^{0}\) | Equation 10, \(i=14\), \(t=0\) | \(cz_{14}^{0}\) | 297 | |
135 | \(b_{19}^{0}\) | Equation 10, \(i=3\), \(t=0\) | \(cz_{3}^{0}\) | 298 | |
136 | \(b_{4}^{1}\) | Equation 7, \(i=4\), \(t=0\) | 298 | ||
137 | \(b_{14}^{1}\) | Equation 7, \(i=14\), \(t=0\) | 298 | ||
138 | \(b_{3}^{1}\) | Equation 7, \(i=3\), \(t=0\) | 298 | ||
139 | \(R3_{4}^{0}\) | Guess | 306 | ||
140 | \(R1_{1}^{1}\) | Equation 12, \(i=4\), \(t=0\) | 306 | ||
141 | \(R1_{11}^{1}\) | Guess | 314 | ||
142 | \(R2_{13}^{2}\) | Equation 14, \(t=1\) | 314 | ||
143 | \(R2_{14}^{2}\) | Equation 14, \(t=1\) | 314 | ||
144 | \(R2_{15}^{2}\) | Equation 14, \(t=1\) | 314 | ||
145 | \(R1_{12}^{1}\) | Equation 14, \(t=1\) | 314 | ||
146 | \(b_{29}^{2}\) | Equation 10, \(i=13\), \(t=2\) | 314 | ||
147 | \(b_{13}^{3}\) | Equation 7, \(i=13\), \(t=2\) | 314 | ||
148 | \(R2_{11}^{1}\) | Guess | 322 | ||
149 | \(b_{27}^{1}\) | Equation 10, \(i=11\), \(t=1\) | \(cz_{11}^{1}\) | 323 | |
150 | \(b_{11}^{2}\) | Equation 7, \(i=11\), \(t=1\) | 323 | ||
151 | \(a_{27}^{1}\) | Guess | 331 | ||
152 | \(a_{11}^{2}\) | Equation 4, \(i=11\), \(t=1\) | 331 | ||
153 | \(a_{27}^{3}\) | Equation 2, \(i=11\), \(t=2\) | \(a_{10,7}^{2}\) | 332 | |
154 | \(b_{27}^{3}\) | Equation 5, \(i=11\), \(t=2\) | \(b_{10,7}^{2}\) | 333 | |
155 | \(R1_{11}^{3}\) | Equation 10, \(i=11\), \(t=3\) | \(cz_{11}^{3}\) | 334 | |
156 | \(R1_{2}^{0}\) | Guess | 342 | ||
157 | \(R2_{9}^{1}\) | Equation 15, \(t=0\) | 342 | ||
158 | \(R2_{10}^{1}\) | Equation 15, \(t=0\) | 342 | ||
159 | \(R1_{13}^{0}\) | Equation 15, \(t=0\) | 342 | ||
160 | \(R1_{7}^{0}\) | Equation 15, \(t=0\) | 342 | ||
161 | \(R1_{9}^{1}\) | Equation 10, \(i=9\), \(t=1\) | 342 | ||
162 | \(R3_{9}^{1}\) | Equation 12, \(i=9\), \(t=1\) | 342 | ||
163 | \(a_{11}^{0}\) | Guess | 350 | ||
164 | \(b_{11}^{0}\) | Equation 2, \(i=11\), \(t=0\) | \(a_{10,7}^{0}\) | 351 | |
165 | \(b_{27}^{0}\) | Equation 5, \(i=11\), \(t=0\) | \(b_{10,7}^{0}\) | 352 | |
166 | \(b_{11}^{1}\) | Equation 7, \(i=11\), \(t=0\) | 352 | ||
167 | \(a_{11}^{1}\) | Equation 5, \(i=11\), \(t=1\) | \(b_{10,7}^{1}\) | 353 | |
168 | \(a_{27}^{2}\) | Equation 2, \(i=11\), \(t=1\) | \(a_{10,7}^{1}\) | 354 | |
169 | \(R1_{14}^{3}\) | Equation 12, \(i=11\), \(t=2\) | \(cR1_{11}^{2}\) | 355 | |
170 | \(a_{11}^{3}\) | Equation 4, \(i=11\), \(t=2\) | 355 | ||
171 | \(b_{30}^{3}\) | Equation 10, \(i=14\), \(t=3\) | \(cz_{14}^{3}\) | 356 | |
172 | \(b_{27}^{4}\) | Equation 5, \(i=11\), \(t=3\) | \(b_{10,7}^{3}\) | 357 | |
173 | \(b_{14}^{4}\) | Equation 7, \(i=14\), \(t=3\) | 357 | ||
174 | \(R1_{4}^{1}\) | Guess | 365 | ||
175 | \(b_{20}^{1}\) | Equation 10, \(i=4\), \(t=1\) | 365 | ||
176 | \(b_{20}^{2}\) | Equation 6, \(i=4\), \(t=1\) | \(b_{5,7}^{1}\) | 366 | |
177 | \(b_{4}^{2}\) | Equation 7, \(i=4\), \(t=1\) | 366 | ||
178 | \(b_{4}^{3}\) | Equation 7, \(i=4\), \(t=2\) | 366 | ||
179 | \(b_{20}^{3}\) | Equation 6, \(i=4\), \(t=2\) | \(b_{5,7}^{2}\) | 367 | |
180 | \(R2_{4}^{2}\) | Guess | 375 | ||
181 | \(R1_{4}^{2}\) | Equation 10, \(i=4\), \(t=2\) | 375 | ||
182 | \(R2_{5}^{2}\) | Equation 16, \(t=1\) | 375 | ||
183 | \(R2_{7}^{2}\) | Equation 16, \(t=1\) | 375 | ||
184 | \(R1_{14}^{1}\) | Equation 16, \(t=1\) | 375 | ||
185 | \(R1_{3}^{1}\) | Equation 16, \(t=1\) | 375 | ||
186 | \(b_{23}^{2}\) | Equation 10, \(i=7\), \(t=2\) | 375 | ||
187 | \(b_{23}^{3}\) | Equation 5, \(i=7\), \(t=2\) | 375 | ||
188 | \(b_{7}^{3}\) | Equation 7, \(i=7\), \(t=2\) | \(b_{7,7}^{3}\) | 374 | |
189 | \(b_{7}^{4}\) | Equation 7, \(i=7\), \(t=3\) | 374 | ||
190 | \(a_{23}^{0}\) | Guess | 382 | ||
191 | \(b_{9}^{0}\) | Equation 8, \(i=9\), \(t=0\) | \(b_{9,7}^{0}\) | 381 | |
192 | \(a_{9}^{0}\) | Equation 8, \(i=9\), \(t=0\) | \(a_{9,7}^{0}\) | 380 | |
193 | \(a_{7}^{1}\) | Equation 4, \(i=7\), \(t=0\) | \(a_{7,7}^{1}\) | 379 | |
194 | \(b_{7}^{1}\) | Equation 5, \(i=7\), \(t=1\) | \(b_{6,7}^{1}\) | 378 | |
195 | \(b_{23}^{0}\) | Equation 7, \(i=7\), \(t=0\) | 378 | ||
196 | \(R2_{7}^{0}\) | Equation 10, \(i=7\), \(t=0\) | \(cz_{7}^{0}\) | 379 | |
197 | \(R3_{10}^{1}\) | Equation 19, \(t=0\) | 379 | ||
198 | \(R3_{11}^{1}\) | Equation 19, \(t=0\) | 379 | ||
199 | \(R2_{13}^{0}\) | Equation 19, \(t=0\) | 379 | ||
200 | \(R2_{2}^{0}\) | Equation 19, \(t=0\) | 379 | ||
201 | \(R1_{14}^{2}\) | Equation 12, \(i=11\), \(t=1\) | \(cR1_{11}^{1}\) | 380 | |
202 | \(b_{29}^{0}\) | Equation 10, \(i=13\), \(t=0\) | \(cz_{13}^{0}\) | \(cz_{14}^{0}\) | 380 |
203 | \(b_{18}^{0}\) | Equation 10, \(i=2\), \(t=0\) | \(cz_{2}^{0}\) | \(cz_{3}^{0}\) | 380 |
204 | \(b_{30}^{2}\) | Equation 10, \(i=14\), \(t=2\) | 380 | ||
205 | \(R2_{4}^{3}\) | Equation 16, \(t=2\) | 380 | ||
206 | \(R2_{5}^{3}\) | Equation 16, \(t=2\) | 380 | ||
207 | \(R2_{7}^{3}\) | Equation 16, \(t=2\) | 380 | ||
208 | \(R1_{3}^{2}\) | Equation 16, \(t=2\) | 380 | ||
209 | \(b_{13}^{1}\) | Equation 7, \(i=13\), \(t=0\) | 380 | ||
210 | \(b_{2}^{1}\) | Equation 7, \(i=2\), \(t=0\) | 380 | ||
211 | \(b_{14}^{3}\) | Equation 7, \(i=14\), \(t=2\) | 380 | ||
212 | \(R1_{4}^{3}\) | Equation 10, \(i=4\), \(t=3\) | 380 | ||
213 | \(R1_{7}^{3}\) | Equation 10, \(i=7\), \(t=3\) | 380 | ||
214 | \(a_{13}^{1}\) | Equation 5, \(i=13\), \(t=1\) | \(b_{12,7}^{1}\) | 381 | |
215 | \(a_{29}^{2}\) | Equation 2, \(i=13\), \(t=1\) | \(a_{12,7}^{1}\) | 382 | |
216 | \(R3_{13}^{2}\) | Equation 12, \(i=13\), \(t=2\) | \(cR1_{13}^{2}\) | 383 | |
217 | \(a_{13}^{3}\) | Equation 4, \(i=13\), \(t=2\) | 383 | ||
218 | \(a_{21}^{1}\) | Guess | 391 | ||
219 | \(R1_{5}^{2}\) | Equation 12, \(i=5\), \(t=1\) | \(cR1_{6}^{1}\) | 390 | |
220 | \(a_{23}^{2}\) | Equation 2, \(i=7\), \(t=1\) | 390 | ||
221 | \(a_{5}^{2}\) | Equation 4, \(i=5\), \(t=1\) | 390 | ||
222 | \(b_{21}^{2}\) | Equation 10, \(i=5\), \(t=2\) | \(cz_{6}^{2}\) | 389 | |
223 | \(a_{25}^{3}\) | Equation 2, \(i=9\), \(t=2\) | 389 | ||
224 | \(a_{7}^{3}\) | Equation 4, \(i=7\), \(t=2\) | 389 | ||
225 | \(b_{5}^{3}\) | Equation 7, \(i=5\), \(t=2\) | 389 | ||
226 | \(a_{27}^{4}\) | Equation 2, \(i=11\), \(t=3\) | \(a_{10,7}^{3}\) | 390 | |
227 | \(a_{9}^{4}\) | Equation 4, \(i=9\), \(t=3\) | 390 | ||
228 | \(b_{23}^{4}\) | Equation 5, \(i=7\), \(t=3\) | 390 | ||
229 | \(b_{25}^{5}\) | Equation 5, \(i=9\), \(t=4\) | 390 | ||
230 | \(a_{4}^{0}\) | Guess | 398 | ||
231 | \(b_{4}^{0}\) | Equation 6, \(i=4\), \(t=0\) | \(b_{4,7}^{0}\) | 399 | |
232 | \(a_{18}^{0}\) | Equation 3, \(i=4\), \(t=0\) | \(a_{5,7}^{0}\) | 400 | |
233 | \(a_{2}^{1}\) | Equation 4, \(i=2\), \(t=0\) | 400 | ||
234 | \(b_{18}^{2}\) | Equation 6, \(i=2\), \(t=1\) | 400 | ||
235 | \(R2_{2}^{2}\) | Equation 10, \(i=2\), \(t=2\) | 400 | ||
236 | \(b_{2}^{3}\) | Equation 7, \(i=2\), \(t=2\) | 400 | ||
237 | \(R3_{8}^{3}\) | Equation 19, \(t=2\) | 400 | ||
238 | \(R3_{9}^{3}\) | Equation 19, \(t=2\) | 400 | ||
239 | \(R3_{10}^{3}\) | Equation 19, \(t=2\) | 400 | ||
240 | \(R3_{11}^{3}\) | Equation 19, \(t=2\) | 400 | ||
241 | \(R1_{2}^{4}\) | Equation 12, \(i=8\), \(t=3\) | 400 | ||
242 | \(R1_{6}^{4}\) | Equation 12, \(i=9\), \(t=3\) | 400 | ||
243 | \(R1_{14}^{4}\) | Equation 12, \(i=11\), \(t=3\) | \(cR1_{11}^{3}\) | 401 | |
244 | \(R2_{6}^{4}\) | Equation 10, \(i=6\), \(t=4\) | \(cz_{6}^{4}\) | 402 | |
245 | \(R2_{4}^{4}\) | Equation 16, \(t=3\) | 402 | ||
246 | \(R2_{5}^{4}\) | Equation 16, \(t=3\) | 402 | ||
247 | \(R2_{7}^{4}\) | Equation 16, \(t=3\) | 402 | ||
248 | \(R1_{3}^{3}\) | Equation 16, \(t=3\) | 402 | ||
249 | \(R1_{7}^{4}\) | Equation 10, \(i=7\), \(t=4\) | 402 | ||
250 | \(a_{29}^{3}\) | Equation 12, \(i=13\), \(t=3\) | \(cR1_{13}^{3}\) | 403 | |
251 | \(a_{13}^{2}\) | Equation 2, \(i=13\), \(t=2\) | \(a_{13,7}^{2}\) | 404 | |
252 | \(a_{13}^{4}\) | Equation 4, \(i=13\), \(t=3\) | 404 | ||
253 | \(b_{29}^{3}\) | Equation 5, \(i=13\), \(t=2\) | \(b_{12,7}^{2}\) | 405 | |
254 | \(R1_{13}^{3}\) | Equation 10, \(i=13\), \(t=3\) | \(cz_{13}^{3}\) | \(cz_{14}^{3}\) | 405 |
255 | \(b_{29}^{4}\) | Equation 5, \(i=13\), \(t=3\) | 405 | ||
256 | \(b_{13}^{4}\) | Equation 7, \(i=13\), \(t=3\) | 405 | ||
257 | \(R3_{7}^{2}\) | Equation 12, \(i=7\), \(t=2\) | 405 | ||
258 | \(R2_{8}^{4}\) | Equation 15, \(t=3\) | 405 | ||
259 | \(R2_{9}^{4}\) | Equation 15, \(t=3\) | 405 | ||
260 | \(R2_{10}^{4}\) | Equation 15, \(t=3\) | 405 | ||
261 | \(R2_{11}^{4}\) | Equation 15, \(t=3\) | 405 | ||
262 | \(R3_{4}^{2}\) | Equation 20, \(t=1\) | 405 | ||
263 | \(R3_{5}^{2}\) | Equation 20, \(t=1\) | 405 | ||
264 | \(R2_{14}^{1}\) | Equation 20, \(t=1\) | 405 | ||
265 | \(R2_{3}^{1}\) | Equation 20, \(t=1\) | 405 | ||
266 | \(b_{30}^{1}\) | Equation 10, \(i=14\), \(t=1\) | 405 | ||
267 | \(b_{14}^{2}\) | Equation 7, \(i=14\), \(t=1\) | 405 | ||
268 | \(R1_{11}^{4}\) | Equation 10, \(i=11\), \(t=4\) | \(cz_{11}^{4}\) | 406 | |
269 | \(a_{30}^{3}\) | Equation 12, \(i=14\), \(t=3\) | 406 | ||
270 | \(a_{14}^{2}\) | Equation 6, \(i=14\), \(t=2\) | \(b_{15,7}^{2}\) | 407 | |
271 | \(a_{28}^{2}\) | Equation 3, \(i=14\), \(t=2\) | \(a_{15,7}^{2}\) | 408 | |
272 | \(a_{12}^{3}\) | Equation 4, \(i=12\), \(t=2\) | 408 | ||
273 | \(R3_{12}^{2}\) | Equation 12, \(i=12\), \(t=2\) | \(cR1_{13}^{2}\) | 407 | |
274 | \(R3_{14}^{2}\) | Equation 18, \(t=1\) | 407 | ||
275 | \(R3_{15}^{2}\) | Equation 18, \(t=1\) | 407 | ||
276 | \(R2_{12}^{1}\) | Equation 18, \(t=1\) | 407 | ||
277 | \(R2_{1}^{1}\) | Equation 18, \(t=1\) | 407 | ||
278 | \(b_{28}^{1}\) | Equation 10, \(i=12\), \(t=1\) | \(cz_{13}^{1}\) | 406 | |
279 | \(R1_{8}^{4}\) | Equation 10, \(i=8\), \(t=4\) | 406 | ||
280 | \(R1_{9}^{4}\) | Equation 10, \(i=9\), \(t=4\) | 406 | ||
281 | \(b_{19}^{1}\) | Equation 10, \(i=3\), \(t=1\) | 406 | ||
282 | \(a_{14}^{1}\) | Equation 6, \(i=14\), \(t=1\) | \(b_{15,7}^{1}\) | 407 | |
283 | \(b_{3}^{2}\) | Equation 7, \(i=3\), \(t=1\) | 407 | ||
284 | \(a_{14}^{4}\) | Equation 4, \(i=14\), \(t=3\) | 407 | ||
285 | \(a_{30}^{2}\) | Equation 12, \(i=14\), \(t=2\) | 407 | ||
286 | \(a_{28}^{1}\) | Equation 3, \(i=14\), \(t=1\) | \(a_{15,7}^{1}\) | 408 | |
287 | \(a_{12}^{2}\) | Equation 4, \(i=12\), \(t=1\) | \(a_{12,7}^{2}\) | 407 | |
288 | \(a_{14}^{3}\) | Equation 4, \(i=14\), \(t=2\) | 407 | ||
289 | \(b_{12}^{2}\) | Equation 7, \(i=12\), \(t=1\) | \(b_{12,7}^{2}\) | 406 | |
290 | \(b_{17}^{1}\) | Equation 10, \(i=1\), \(t=1\) | \(cz_{1}^{1}\) | \(cz_{2}^{1}\) | 406 |
291 | \(b_{1}^{2}\) | Equation 7, \(i=1\), \(t=1\) | 406 | ||
292 | \(b_{30}^{4}\) | Equation 6, \(i=14\), \(t=3\) | \(b_{15,7}^{3}\) | 407 | |
293 | \(b_{28}^{3}\) | Equation 6, \(i=12\), \(t=2\) | 407 | ||
294 | \(b_{12}^{4}\) | Equation 7, \(i=12\), \(t=3\) | 407 | ||
295 | \(R1_{12}^{3}\) | Equation 10, \(i=12\), \(t=3\) | \(cz_{13}^{3}\) | 406 | |
296 | \(R2_{14}^{4}\) | Equation 10, \(i=14\), \(t=4\) | \(cz_{14}^{4}\) | 407 | |
297 | \(b_{28}^{4}\) | Equation 6, \(i=12\), \(t=3\) | 407 | ||
298 | \(R2_{12}^{4}\) | Equation 14, \(t=3\) | 407 | ||
299 | \(R2_{13}^{4}\) | Equation 14, \(t=3\) | 407 | ||
300 | \(R2_{15}^{4}\) | Equation 14, \(t=3\) | 407 | ||
301 | \(R1_{1}^{3}\) | Equation 14, \(t=3\) | 407 | ||
302 | \(R1_{12}^{4}\) | Equation 10, \(i=12\), \(t=4\) | 407 | ||
303 | \(R1_{13}^{4}\) | Equation 10, \(i=13\), \(t=4\) | \(cz_{14}^{4}\) | 406 | |
304 | \(a_{20}^{2}\) | Equation 12, \(i=4\), \(t=2\) | 406 | ||
305 | \(a_{4}^{3}\) | Equation 4, \(i=4\), \(t=2\) | 406 | ||
306 | \(b_{20}^{4}\) | Equation 6, \(i=4\), \(t=3\) | 406 | ||
307 | \(R1_{4}^{4}\) | Equation 10, \(i=4\), \(t=4\) | 406 | ||
308 | \(R2_{8}^{5}\) | Equation 15, \(t=4\) | 406 | ||
309 | \(R2_{9}^{5}\) | Equation 15, \(t=4\) | 406 | ||
310 | \(R2_{10}^{5}\) | Equation 15, \(t=4\) | 406 | ||
311 | \(R2_{11}^{5}\) | Equation 15, \(t=4\) | 406 | ||
312 | \(a_{22}^{3}\) | Equation 3, \(i=6\), \(t=2\) | 406 | ||
313 | \(R3_{6}^{3}\) | Equation 12, \(i=6\), \(t=3\) | \(cR1_{6}^{3}\) | 407 | |
314 | \(R3_{4}^{3}\) | Equation 20, \(t=2\) | 407 | ||
315 | \(R3_{5}^{3}\) | Equation 20, \(t=2\) | 407 | ||
316 | \(R3_{7}^{3}\) | Equation 20, \(t=2\) | 407 | ||
317 | \(R2_{3}^{2}\) | Equation 20, \(t=2\) | 407 | ||
318 | \(a_{23}^{3}\) | Equation 12, \(i=7\), \(t=3\) | 407 | ||
319 | \(a_{21}^{2}\) | Equation 2, \(i=7\), \(t=2\) | 407 | ||
320 | \(R1_{5}^{3}\) | Equation 12, \(i=5\), \(t=2\) | \(cR1_{6}^{2}\) | 406 | |
321 | \(b_{21}^{3}\) | Equation 10, \(i=5\), \(t=3\) | \(cz_{6}^{3}\) | 405 | |
322 | \(a_{5}^{3}\) | Equation 4, \(i=5\), \(t=2\) | 405 | ||
323 | \(b_{21}^{4}\) | Equation 5, \(i=5\), \(t=3\) | 405 | ||
324 | \(R1_{5}^{4}\) | Equation 10, \(i=5\), \(t=4\) | \(cz_{6}^{4}\) | 404 | |
325 | \(a_{21}^{3}\) | Equation 12, \(i=5\), \(t=3\) | \(cR1_{5}^{3}\) | \(cR1_{6}^{3}\) | 404 |
326 | \(a_{18}^{1}\) | Equation 3, \(i=4\), \(t=1\) | \(a_{5,7}^{1}\) | 405 | |
327 | \(a_{2}^{2}\) | Equation 4, \(i=2\), \(t=1\) | 405 | ||
328 | \(b_{18}^{3}\) | Equation 6, \(i=2\), \(t=2\) | 405 | ||
329 | \(b_{5}^{2}\) | Equation 5, \(i=5\), \(t=2\) | \(b_{4,7}^{2}\) | 404 | |
330 | \(b_{21}^{1}\) | Equation 7, \(i=5\), \(t=1\) | 404 | ||
331 | \(R1_{5}^{1}\) | Equation 10, \(i=5\), \(t=1\) | \(cz_{6}^{1}\) | 403 | |
332 | \(R2_{2}^{3}\) | Equation 10, \(i=2\), \(t=3\) | \(cz_{2}^{3}\) | 404 | |
333 | \(R3_{8}^{4}\) | Equation 19, \(t=3\) | 404 | ||
334 | \(R3_{9}^{4}\) | Equation 19, \(t=3\) | 404 | ||
335 | \(R3_{10}^{4}\) | Equation 19, \(t=3\) | 404 | ||
336 | \(R3_{11}^{4}\) | Equation 19, \(t=3\) | 404 | ||
337 | \(R1_{14}^{5}\) | Equation 12, \(i=11\), \(t=4\) | \(cR1_{11}^{4}\) | 405 | |
338 | \(b_{30}^{5}\) | Equation 6, \(i=14\), \(t=4\) | \(b_{15,7}^{4}\) | 406 | |
339 | \(R2_{14}^{5}\) | Equation 10, \(i=14\), \(t=5\) | \(cz_{14}^{5}\) | 407 | |
340 | \(R2_{12}^{5}\) | Equation 14, \(t=4\) | 407 | ||
341 | \(R2_{13}^{5}\) | Equation 14, \(t=4\) | 407 | ||
342 | \(R2_{15}^{5}\) | Equation 14, \(t=4\) | 407 | ||
343 | \(R1_{1}^{4}\) | Equation 14, \(t=4\) | 407 | ||
344 | \(b_{29}^{5}\) | Equation 5, \(i=13\), \(t=4\) | 407 | ||
345 | \(R1_{13}^{5}\) | Equation 10, \(i=13\), \(t=5\) | \(cz_{13}^{5}\) | \(cz_{14}^{5}\) | 407 |
346 | \(a_{20}^{3}\) | Equation 12, \(i=4\), \(t=3\) | \(cR1_{5}^{3}\) | 406 | |
347 | \(a_{22}^{4}\) | Equation 3, \(i=6\), \(t=3\) | 406 | ||
348 | \(a_{23}^{4}\) | Equation 2, \(i=7\), \(t=3\) | 406 | ||
349 | \(R1_{9}^{5}\) | Equation 10, \(i=9\), \(t=5\) | \(cz_{9}^{5}\) | 407 | |
350 | \(R3_{7}^{4}\) | Equation 12, \(i=7\), \(t=4\) | \(cR1_{7}^{4}\) | 408 | |
351 | \(R3_{4}^{4}\) | Equation 20, \(t=3\) | 408 | ||
352 | \(R3_{5}^{4}\) | Equation 20, \(t=3\) | 408 | ||
353 | \(R2_{3}^{3}\) | Equation 20, \(t=3\) | 408 | ||
354 | \(R3_{6}^{4}\) | Equation 20, \(t=3\) | \(cR1_{5}^{4}\) | Equation 12, \(i=6\), \(t=4\) | 401 |
355 | \(a_{6}^{4}\) | Equation 4, \(i=6\), \(t=3\) | 401 | ||
356 | \(b_{22}^{5}\) | Equation 6, \(i=6\), \(t=4\) | 401 | ||
357 | \(b_{19}^{2}\) | Equation 10, \(i=3\), \(t=2\) | 401 | ||
358 | \(a_{25}^{4}\) | Equation 2, \(i=9\), \(t=3\) | 401 | ||
359 | \(a_{7}^{4}\) | Equation 4, \(i=7\), \(t=3\) | 401 | ||
360 | \(a_{3}^{1}\) | Equation 5, \(i=3\), \(t=1\) | 401 | ||
361 | \(b_{3}^{3}\) | Equation 7, \(i=3\), \(t=2\) | 401 | ||
362 | \(b_{23}^{5}\) | Equation 5, \(i=7\), \(t=4\) | 401 | ||
363 | \(R1_{6}^{5}\) | Equation 12, \(i=9\), \(t=4\) | \(cR1_{9}^{4}\) | 402 | |
364 | \(R2_{6}^{5}\) | Equation 10, \(i=6\), \(t=5\) | \(cz_{6}^{5}\) | 403 | |
365 | \(R2_{4}^{5}\) | Equation 16, \(t=4\) | 403 | ||
366 | \(R2_{5}^{5}\) | Equation 16, \(t=4\) | 403 | ||
367 | \(R2_{7}^{5}\) | Equation 16, \(t=4\) | 403 | ||
368 | \(R1_{3}^{4}\) | Equation 16, \(t=4\) | 403 | ||
369 | \(a_{28}^{3}\) | Equation 12, \(i=12\), \(t=3\) | \(cR1_{13}^{3}\) | 402 | |
370 | \(R1_{7}^{5}\) | Equation 10, \(i=7\), \(t=5\) | 402 | ||
371 | \(a_{29}^{4}\) | Equation 2, \(i=13\), \(t=3\) | 402 | ||
372 | \(R3_{13}^{4}\) | Equation 12, \(i=13\), \(t=4\) | \(cR1_{13}^{4}\) | 403 | |
373 | \(a_{26}^{2}\) | Equation 3, \(i=12\), \(t=2\) | 403 | ||
374 | \(a_{18}^{2}\) | Equation 3, \(i=4\), \(t=2\) | 403 | ||
375 | \(R2_{0}^{2}\) | Equation 17, \(t=1\) | 403 | ||
376 | \(R1_{0}^{1}\) | Equation 17, \(t=1\) | 403 | ||
377 | \(R1_{10}^{1}\) | Equation 17, \(t=1\) | 403 | ||
378 | \(R1_{15}^{1}\) | Equation 17, \(t=1\) | 403 | ||
379 | \(a_{19}^{2}\) | Equation 2, \(i=5\), \(t=2\) | 403 | ||
380 | \(R3_{12}^{4}\) | Equation 18, \(t=3\) | 403 | ||
381 | \(R3_{14}^{4}\) | Equation 18, \(t=3\) | 403 | ||
382 | \(R3_{15}^{4}\) | Equation 18, \(t=3\) | 403 | ||
383 | \(R2_{1}^{3}\) | Equation 18, \(t=3\) | 403 | ||
384 | \(R1_{10}^{3}\) | Equation 12, \(i=10\), \(t=2\) | \(cR1_{11}^{2}\) | 402 | |
385 | \(R3_{2}^{2}\) | Equation 12, \(i=2\), \(t=2\) | \(cR1_{2}^{2}\) | 403 | |
386 | \(a_{16}^{1}\) | Equation 3, \(i=2\), \(t=1\) | 403 | ||
387 | \(a_{2}^{3}\) | Equation 4, \(i=2\), \(t=2\) | 403 | ||
388 | \(R3_{0}^{3}\) | Equation 21, \(t=2\) | 403 | ||
389 | \(R3_{1}^{3}\) | Equation 21, \(t=2\) | 403 | ||
390 | \(R3_{2}^{3}\) | Equation 21, \(t=2\) | 403 | ||
391 | \(R3_{3}^{3}\) | Equation 21, \(t=2\) | 403 | ||
392 | \(b_{26}^{1}\) | Equation 10, \(i=10\), \(t=1\) | \(cz_{11}^{1}\) | 402 | |
393 | \(R3_{3}^{2}\) | Equation 12, \(i=3\), \(t=2\) | 402 | ||
394 | \(a_{17}^{1}\) | Equation 2, \(i=3\), \(t=1\) | 402 | ||
395 | \(a_{3}^{3}\) | Equation 4, \(i=3\), \(t=2\) | 402 | ||
396 | \(b_{26}^{3}\) | Equation 10, \(i=10\), \(t=3\) | \(cz_{11}^{3}\) | 401 | |
397 | \(a_{0}^{2}\) | Equation 4, \(i=0\), \(t=1\) | 401 | ||
398 | \(b_{18}^{4}\) | Equation 6, \(i=2\), \(t=3\) | 401 | ||
399 | \(a_{17}^{3}\) | Equation 12, \(i=1\), \(t=3\) | \(cR1_{1}^{3}\) | 402 | |
400 | \(a_{18}^{3}\) | Equation 12, \(i=2\), \(t=3\) | 402 | ||
401 | \(b_{10}^{2}\) | Equation 7, \(i=10\), \(t=1\) | \(b_{10,7}^{2}\) | 401 | |
402 | \(R3_{0}^{2}\) | Equation 21, \(t=1\) | 401 | ||
403 | \(R3_{1}^{2}\) | Equation 21, \(t=1\) | 401 | ||
404 | \(R2_{0}^{1}\) | Equation 21, \(t=1\) | 401 | ||
405 | \(R2_{15}^{1}\) | Equation 21, \(t=1\) | 401 | ||
406 | \(a_{1}^{2}\) | Equation 4, \(i=1\), \(t=1\) | 401 | ||
407 | \(b_{19}^{3}\) | Equation 10, \(i=3\), \(t=3\) | 401 | ||
408 | \(R2_{0}^{3}\) | Equation 17, \(t=2\) | 401 | ||
409 | \(R1_{0}^{2}\) | Equation 17, \(t=2\) | 401 | ||
410 | \(R1_{10}^{2}\) | Equation 17, \(t=2\) | 401 | ||
411 | \(R1_{15}^{2}\) | Equation 17, \(t=2\) | 401 | ||
412 | \(R2_{2}^{4}\) | Equation 10, \(i=2\), \(t=4\) | \(cz_{2}^{4}\) | 402 | |
413 | \(a_{16}^{2}\) | Equation 3, \(i=2\), \(t=2\) | \(a_{3,7}^{2}\) | 403 | |
414 | \(a_{17}^{2}\) | Equation 12, \(i=1\), \(t=2\) | \(cR1_{1}^{2}\) | \(cR1_{2}^{2}\) | 403 |
415 | \(b_{16}^{1}\) | Equation 10, \(i=0\), \(t=1\) | \(cz_{1}^{1}\) | 402 | |
416 | \(b_{31}^{1}\) | Equation 10, \(i=15\), \(t=1\) | 402 | ||
417 | \(a_{15}^{2}\) | Equation 2, \(i=1\), \(t=2\) | \(a_{15,7}^{2}\) | 401 | |
418 | \(a_{3}^{2}\) | Equation 5, \(i=3\), \(t=2\) | \(a_{3,7}^{2}\) | 400 | |
419 | \(b_{19}^{4}\) | Equation 5, \(i=3\), \(t=3\) | 400 | ||
420 | \(b_{16}^{2}\) | Equation 10, \(i=0\), \(t=2\) | \(cz_{1}^{2}\) | 399 | |
421 | \(b_{26}^{2}\) | Equation 10, \(i=10\), \(t=2\) | \(cz_{11}^{2}\) | 398 | |
422 | \(b_{31}^{2}\) | Equation 10, \(i=15\), \(t=2\) | 398 | ||
423 | \(R1_{0}^{3}\) | Equation 12, \(i=0\), \(t=2\) | \(cR1_{1}^{2}\) | 397 | |
424 | \(b_{0}^{2}\) | Equation 7, \(i=0\), \(t=1\) | 397 | ||
425 | \(b_{15}^{2}\) | Equation 7, \(i=15\), \(t=1\) | \(b_{15,7}^{2}\) | 396 | |
426 | \(R2_{3}^{4}\) | Equation 10, \(i=3\), \(t=4\) | 396 | ||
427 | \(a_{10}^{2}\) | Equation 6, \(i=10\), \(t=2\) | \(a_{10,7}^{2}\) | 395 | |
428 | \(R2_{0}^{4}\) | Equation 17, \(t=3\) | 395 | ||
429 | \(R2_{1}^{4}\) | Equation 17, \(t=3\) | 395 | ||
430 | \(R1_{15}^{3}\) | Equation 17, \(t=3\) | Equation 17, \(t=3\) | 387 | |
431 | \(a_{31}^{2}\) | Equation 12, \(i=15\), \(t=2\) | 387 |
Appendix 2: source code
The source code of the heuristic guess-and-determine attack on SNOW-Vi is attached at https://gitlab.com/jiaolin2019/guess-and-determine-attack-on-snow-vi.git
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Jiao, L., Hao, Y. & Li, Y. Guess-and-determine attacks on SNOW-Vi stream cipher. Des. Codes Cryptogr. 91, 2021–2055 (2023). https://doi.org/10.1007/s10623-022-01150-z
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DOI: https://doi.org/10.1007/s10623-022-01150-z