In this paper, we use CryptoMiniSat 5.0 [Soo16a], which won several parts in SAT 2016 competition [SAT16]. CryptoMiniSat can read an extended Conjunctive
Nor-19
doi:10.6342/NTU201602059 mal Form that contains XOR-clauses. This feature is very good to our analysis since
there are many clauses in the constructed equations. One advantage of XOR-clause is that no cutting is needed, which avoids the numerous dummy variables.
And the other is that the solver may reduce the variables by XORing two clauses.
For example, in the Linear Transformation phase
z1,31= y1,3+ y1,118
z1,48= y1,3+ y1,14+ y1,25+ y1,100+ y1,104+ y1,118
are generated. If we XOR two clause,
0 = z1,31+ z1,48+ y1,14+ y1,25+ y1,100+ y1,104
is obtained. This is simpler and if all the variables are assigned temporary values, SAT solver can determine whether the assignment is valid immediately. The author of CryptoMiniSat claims that preserving XOR-clauses may achieve 2 times speed up [Soo16b]. In Chapter 4, we will show the dierence of solving time between using CNF and using extended CNF.
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doi:10.6342/NTU201602059
Chapter 4 Experiments
4.1 Setup
The experiment is run on Intel Xeon CPU with two 2.40GHz cores and the memory is 36GB. And the SAT solver used in this experiment is CryptoMinisat 5.0 [Soo16a].
The transforming process, from ANF to CNF, as discussed in Chapter 4 is performed with the help of SageMath [Dev16], an open-source mathematics software based on Python.
As the analysis in Chapter 3, the intermediate bytes used are the 128 bits after key mixing, 128 bits after substitution boxs and 128 bits after the Linear Transfor-mation, for each round. Nevertheless, in the following experiment, only one round information is included to extract the rst round key and the function of the rst S-Box. Because once the rst round information is totally encoded, the input of the second round is also known. Then the situation is just like the rst round. So we only perform the experiment in the rst round and the following rounds are expected to be successful too.
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doi:10.6342/NTU201602059
4.2 Results
In the rst experiment the Template is assumed totally correct for all bytes, i.e.
the Hamming Weight of all bytes are known. The result is shown in Table 4.1 (without XOR-clause) and 4.2 (with XOR-clause). In fact, the unknown part of the experiment is only S-Box but round key at the beginning. However, the CNFs can be solved for both S-Box and round key unknown. The least number of traces needed is 3. But as the table shows, there is a trade-o between the solving time and the number of traces. The more traces, the more information contained such that the variables leading to con ict can be removed sooner. Note that the solving time is median here since median is more representative due to the large variance of solving time.
Besides, the solving time with XOR-clause is faster than without XOR-clause.
The number of literal and clause are also fewer. We have discussed the reason in Chapter 3. This result shows that CryptoMiniSat is really good at dealing with XOR-clauses as the author states.
Table 4.1: Result: Without Error Tolerance (No XOR-clause)
#Trace #Literal #Clause Solving Time (Median)
3 17,331 235,211 545 sec
4 24,203 333,036 140 sec
5 30,366 416,701 34 sec
Table 4.2: Result: Without Error Tolerance (XOR-clause)
#Trace #Literal #Clause Solving Time (Median)
3 7,104 37,784 423 sec
4 9,408 50,437 21 sec
5 11,712 63,007 13 sec
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doi:10.6342/NTU201602059 To make the algebraic SCA successful with the existence of error, some
modi -cations are indispensable. The details are described in Section 3.4. It is undoubted that more traces are needed to make the solution of SAT problem unique. And the size of CNFs generated by the encoder increases certainly. Thus, it takes much more time to solve the equations. Table 4.3 shows the results with 1 − HW error, i.e.
the system allows the Hamming Weight given by the template is away from the real Hamming Weight by at most 1.
Table 4.3: Result: With Error Tolerance (No XOR-clause)
#Trace #Literal #Clause Solving Time (Median) Success Rate
17 165,719 2,402,210 513 sec 92%
18 177,340 2,568,688 568 sec 98%
19 186,512 2,701,551 616 sec 98%
20 196,341 2,851,993 600 sec 98%
21 203,488 2,948,686 578 sec 100%
22 214,972 3,113,149 555 sec 100%
Table 4.4: Result: With Error Tolerance (XOR-clause)
#Trace #Literal #Clause Solving Time (Median) Success Rate
17 39,360 192,490 175 sec 96%
18 41,664 203,769 169 sec 90%
19 43,968 215,116 153 sec 98%
20 46,272 226,445 156 sec 98%
21 48,576 237,777 167 sec 100%
22 50,880 249,125 112 sec 100%
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doi:10.6342/NTU201602059
Chapter 5 Conclusion
In this work, an algebraic analysis combined with side-channel information is applied to solve he unknown S-Box problem. Based on Template Attack and Algebraic Side-Channel Analysis, many equations corresponding to operations of the cryptographic algorithm are constructed. And by an ANF-to-CNF converter, the equations can then be solved by a SAT solver.
The target block cipher is Serpent, which is a nalist of AES competition. The experimental results show that this method is successful once a good template is given. Moreover, this method can also tolerate error up to 1-HW error. That is, a template that gives a wrong Hamming Weight that is ±1 away from the correct Hamming Weight is still work.
5.1 Future Works
A sound template is the base of our analysis. Thus, an important thing is to build a template in reality. The technique of building template is not di cult. For a device that does not have any countermeasure against SCA, high success rate templates are expected to be able to achieve. But some experiences are required to perform trace alignment, points choosing and numerical problem.
This type of Side-Channel Analysis is suitable for those countermeasures that 24
doi:10.6342/NTU201602059 aim at changing round functions of block ciphers. In this work, we only deal with
unknown S-Box situation. However, a practical countermeasure may even replace other non-linear parts. So nding out an e cient algebraic representation to write down the equations of other parts, for example, including key schedule, is another work. Finally, this method can be also applied to other block cipher like AES. This is another challenge since the S-Box in AES is 8 × 8.
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