CRAFT is a light-weight tweakable block cipher, designed with efficient protection against differential fault attacks (DFA) in mind from design phase. The following figure depicts one round of CRAFT, and you can see it's specification in more details here. We could improve differential, zero-correleation, and integral distinguishers of this cipher during our work which has been accpted to the FSE-2020, and this repository contains the codes we have used to find these distinguishers.
In this repository, both MILP, and SMT/SAT based methods are used to analyse CRAFT. For MILP-based method, Python3 is used to produce the MILP models, and Gurobi, is used as the solver. Therefore, you need to install Gurobi and link it to the Python3. You can find the installation recipes here.
CryptoSMT is used for computing the differential effects. We have improved CryptoSMT's Sbox encoding to make it faster for SPN ciphers. Therefore, If you want to use our SAT/SMT-based tools, you need to do the same installation recipes as CryptoSMT. Note that, if you have already installed CryptoSMT, you merely need to replace the config.py file with your own config.py file in folders SAT-SMT-ST, and SAT-SMT-RT.
While the majority of codes in this repository, have been written with Python3 language, we have used C++, for experimental verification of our results in differential cryptanalysis of CRAFT. We've also used OpenMP to achive a better performance. In order to compile the C++ codes in folder ExperimentalVerification-Diff, you can use g++ compiler with -fopenmp compiler flag.
We use MILP-based method to find zero-correlation distinguishers, and then give a mathematical proof for them. You can find all the codes we've used for zero-correlation attack in the file Zero-Correlation. Since the linear behavior of CRAFT in the related tweak model, depends on the starting round, there are four sub-folders in this file, each one is associated with one out of four cases RTK0, RTK1, RTK2, and RTK3.
In order to find a zero-correlation distinguisher, a MILP model containing all constraints modeling the propagation rules of linear masks through the cipher is extracted at first, and then input/output linear masks are set to be a fixed vector with Hamming weight of one, and finally an MILP solver is called to see whether the obtained MILP problem is feasible or not. If the obtained model is infeasible, we can conclude that the correlation of linear hull with that fixed input/output linear masks must be zero. Since the block-size of CRAFT is 64 bits, and the length of tweak is 64 bits too, there are 262144 possibilities for a fixed input/output masks with Hamming weight of one in the related-tweak model. Therefore 262144 different cases must be probed. In order to check all these cases much faster, we use data-parallel programming, and devide the tasks between 16 threads of one CPU, when each single thread probe (262144/16) 16384 different cases. If you use a CPU equiped with 16 different cores, then all tasks are performed in parallel.
We've found a 14-round zero-correlation distinguisher for the case RTK0. You can see the proof of this case in more details in the paper. If you are interested to know how we could find it, we invite you to take a look at the codes in : /Zero-Correlation/ZeroCorrelation-rev1-tk0/, and then run the following command to reproduce our results:
python3 main.py
If you would like to use this tool for inspecting a specified numer of rounds in case RTK0, you mereley need to change the value of a parameter rounds, in line 34 of main.py file, and use python3 main.py again, to run the modified program.
For this case we have not found a zero-correlation distinguisher covering more than 13 rounds of CRAFT, in the related tweak model, by our MILP-based tool. All the codes we have used for this case, can be found in directory ./Zero-Correlation/ZeroCorrelation-rev1-tk1/. If you want to check our claim for this case, you are invited to take a look at the codes, and run the following command, to reproduce our results:
python3 main.py
For this case we have also found a 14-round distinguisher via our MILP-based tool. The following picture shows a mathematical proof for our distinguiser. The codes we have used for this case, are placed in directory ./Zero-Correlation/ZeroCorrelation-rev1-tk2, and if you want to run the program, and reproduce our results, you need to open your terminal in this directory, and type the following command:
python3 main.py
The following figure, shows a mathematical proof, for the 14-round zero-correlation distinguisher we have found in case RTK3, by our MILP-based tool. The codes associated with this case can be found in directory ./Zero-Correlation/ZeroCorrelation-rev1-tk3, and can be execute by the following command:
python3 main.py
Although we inspected 15 rounds of CRAFT, in cases RTK0, RTK2, and RTK3 by this method, we didn't find a zero-correlation distinguisher, covering more than 14 rounds in these cases.
The first observation that motivated us to begin the cryptanalysis of CRAFT, is related to the influence of clustering effect on the differential cryptanalysis of CRAFT. It dates back to that day, we added CRAFT to the CryptoSMT's cipher suite. When we were checking the corretness of our implementation we obserevd that the existing bounds for the differential effects of CRAFT that were claimed by the designers, can be dramatically improved, but we only could use CryptoSMT, or bit-oriented MILP models, for the small number of rounds, and we couldn't find a bound for large number of rounds, because SAT/SMT or MILP-based methods are not that efficient for large number of rounds.
So we decided to find a way to overcome this difficulty. The first idea, was building a word-oriented MILP model for CRAFT, and using the obtained activity pattern to make the bit-oriented model easier, beacuse in contrast to bit-oriented MILP or SMT/SAT models, a word-oriented one can be solved very fast even for large number of rounds.
In other words, the obtained solutions from the word-roiented MILP problem, showed us, what the activity pattern of an optimum differential trail could be, and then we substituted all passive variables in the bit-oriented models with zeros, which made the problem easier, and we could solve the bit-oriented model faster, to find a real differential trail. A program has been prepared in folder MILP-Diff-Activity-Pattern, to find an optimum activity pattern, and obtain the minimum number of active Sboxes for a given number of rounds in single-tweak, and realted-tweak models. For example if you set the rounds, and related_tweak parameters in the main file as follows:
rounds = 18
related_tweak = 0
and run the following command in directory /MILP-Diff-Activity-Pattern:
python3 main.py
You obtain an optimum activity pattern, covering 18 rounds of CRAFT in single-tweak model like this:
Number of rounds: 18
1100000011001110
0001000000001001
0100010000000010
0001000100011000
0100010001000010
0001000100011000
0100010001000010
0001000100011000
0100010001000010
0001000100011000
0100010001000010
0001000100011000
0100010001000010
0001000100011000
0100010001000010
0001000100011000
0100010001000010
0001000000011000
1100000000000010
Number of active Sboxes: 68
Another interesting fact we found in this stage was that, the activity pattern of an optimum differential trail is unique, when the input/output activity patterns are fixed! However the problem of finding an optimum differential trail for large number of rounds were still time consuming when we used CryptoSMT.
We knew that CryptoSMT uses a naive approach to model differntial behaviour of a given Sbox. Therefore, we improved the Sbox encoding method used in CryptoSMT and then it could find an optimum trial very faster than before. For example if we substitute all passive variables obtained in the above activity pattern for 18 rounds, we can easily find an optimum differential trail covering 18 rounds of CRAFT in the single tweak model, with CryptoSMT
Characteristic for craft - Rounds 18 - Wordsize 64 - Weight 136 - Time 38.15s
Rounds x y z w
---------------------------------------------------------------------------------------
0 0xFF000000FF00FFF0 0xFF000000000000F0 0x000F00000000F00F -6
1 0x000F00000000A00F 0x000F0000000FA000 0x0A000F00000000F0 -6
2 0x0F000F00000000F0 0x0F000F000F0000F0 0x000F000F000FF000 -8
3 0x000F000F000AA000 0x000F000F0005A000 0x0A0005000F0000F0 -8
4 0x0A000A000F0000A0 0x0A000A00050000A0 0x000A0005000AA000 -8
5 0x000A000A000DA000 0x000A000A0007A000 0x0A0007000A0000A0 -8
6 0x050005000F0000A0 0x050005000A0000A0 0x000A000A00055000 -8
7 0x000A000A0007A000 0x000A000A000DA000 0x0A000D000A0000A0 -8
8 0x0A000A00050000F0 0x0A000A000F0000F0 0x000F000F000AA000 -8
9 0x000A000A0005F000 0x000A000A000FF000 0x0F000F000A0000A0 -8
10 0x0F000F000A0000D0 0x0F000F00050000D0 0x000D0005000FF000 -8
11 0x00070007000AF000 0x00070007000DF000 0x0F000D0007000070 -8
12 0x0A000A00050000D0 0x0A000A000F0000D0 0x000D000F000AA000 -8
13 0x000A000A000FF000 0x000A000A0005F000 0x0F0005000A0000A0 -8
14 0x0A000A000F0000A0 0x0A000A00050000A0 0x000A0005000AA000 -8
15 0x000A000A0005D000 0x000A000A000FD000 0x0D000F000A0000A0 -8
16 0x0A000A000A0000A0 0x0A000A00000000A0 0x000A0000000AA000 -6
17 0x000A0000000AA000 0x000A00000000A00A 0xAA000000000000A0 -6
18 0xAA000000000000A0 none none none
Weight: 136
Although we could find an optimum or (non-optimum) tril for large number of rounds very faster than before, but the problem of computing the differential effect for a given input/output differences were still time consuming for large number of rounds. In other words, the number of distinct optimum (and non-optimum) diffrential trails with the same input/output diffrences for CRAFT were so much that we couln't count all of them for large number of rounds. In this stage we used divide and conquer strategy! We divided a long part to some smaller pieces, since we could compute the differential effect for smaller pieces efficiently.
The following pictures, depict those smaller pieces we have used to build differential distinguishers for CRAFT in the single-tweak setting. As you can see in these pictures, all of them are optimimum from the numuber of active Sboxes point of view. We evaluate the probability of each part spereately, and then multiply them together (according to markov assumption) to find the probability of the whole differntial distinguisher.
Since the differences in the middle parts are not fixed and can take 100 different values, we evaluated the probaility for all combination of possible input/output differences for each part and then arranged them into a matrix. In other words we obtain a matrix containing the differential effects corresponding to a fixed input/output differences for each part, and the final bounds for the whole differential effects are obtained by multiplying the matrices corresponding to it's components. We use different componetes to cover even, and odd number of rounds, and you can see the possible input/output differences for each part in these files: even_possible_ios.txt, and odd_possible_ios.txt. All of our distinguishers are consisted of three components, Ein, Em, and Eout, except for 13-round differential distinguisher which is consisted of four parts, Ein, Em, Em, and Eout. In other words istead of using a 4-round middle part we have used a 2-round components twice, to build the middle part of that 13-round distinguisher.
You can reproduce the results related to the above components by running the following command in this direcrtory: /SAT-SMT-ST/:
python3 ein_even_4r.py --inputfile examples/EinEven/ein_even_4r.yaml
We also use an extended version of the above component which covers 6 rounds of CRAFT, and you can reproduce the results related to it, by running the following command, in directory /SAT-SMT-ST/:
python3 ein_even_6r.py --inputfile examples/EinEven/ein_even_6r.yaml
The following 2-round component is used as the middle part in our 12-round differential distinguisher.
You can reproduce the results related to this component, by running the following command in directory /SAT-SMT-ST/:
python3 em_even_2r.py --inputfile examples/EmEven/em_even_2r.yaml
The following figure, is a visualization of the matrix we have produced for the above 2-round middle part.
We use a 4-round component which is obtained by extending the above 2-round middle part by two rounds, as the middle part in our 14-round differential distinguisher for CRAFT. You can reproduce the results related to this component, by running the following command in directory /SAT-SMT-ST/:
python3 em_even_4r.py --inputfile examples/EmEven/em_even_4r.yaml
The following figure, is a visulaization of the matrix we have produced for the 4-round component, which is used as the middle part in our 14-round differential distinguisher.
The following figure, shows the 6-round component which is used as Eout (last part), in our 10/12/14-round differential distinguishers.
You can reproduce the results related to the above component, by running the following command in directory /SAT-SMT-ST/:
python3 eout_even_6r.py --inputfile examples/EoutEven/eout_even_6r.yaml
The following 4-round component is used as Ein (first part), in our 9/11/13-rounds differential distinguishers.
In order to reproduce the results related to the above component, you can run the following comand in directory /SAT-SMT-ST/:
python3 ein_odd_4r.py --inputfile examples/EinOdd/ein_odd_4r.yaml
We use the following 2-round component as the middle part in 11/13-round differential distinguishers.
The results we have obtained for the above component, can be reproduced by running the following command in directory /SAT-SMT-ST/:
python3 em_odd_2r.py --inputfile examples/EmOdd/em_odd_2r.yaml
The followig picture is a visualization of the matrix we have obtained for the above middle part.
The following 5-round component is used as Eout (last part), in 9/11/13-round differential distinguishers.
The results related to the above part, can be reproduced by running the following command in directory /SAT-SMT-ST/:
python3 eout_odd_5r.py --inputfile examples/EoutOdd/eout_odd_5r.yaml
We have recorded all our results related to single-tweak, and related-tweak settings in directories /Results-Diff-ST, and /Results-Diff-RT respectively. We have also prepared two different SageMath worksheets, to calculate a lower bound for differential effects. One worksheet is named ProbabilityComputationEven.ipynb, and located in /Results-Diff-ST/Even, which is used for 10/12/14-rounds differential distinguishers, and another worksheet is named ProbabilityComputationOdd.ipynb, and located in /Results-Diff-ST/Odd, whih is used for 9/11/13-rounds differential distinguishers in the single-tweak setting. Some contents of ProbabilityComputationEven.ipynb which are related to the bounds we have proposed for the differential effect of 10/12/14-rounds of CRAFT, are represented in the following lines .
Optimum weight: 25
p_ein_4r = matrix([2.1010637283325195e-06, 8.544884622097015e-07, 1.0728836059570312e-06, 1.0728836059570312e-06, 1.0728836059570312e-06, 0.0, 1.0952353477478027e-06, 1.098262146115303e-06, 1.1066440492868423e-06, 3.3760443329811096e-08, 1.1289957910776138e-06, 1.0952353477478027e-06, 3.3760443329811096e-08, 8.882489055395126e-07, 3.3760443329811096e-08, 0.0, 2.1348241716623306e-06, 3.3760443329811096e-08, 1.1066440492868423e-06, 1.0728836059570312e-06, 3.688037395477295e-07, 0.0, 0.0, 6.151385605335236e-07, 1.9371509552001953e-07, 1.1641532182693481e-09, 2.3283064365386963e-09, 1.9371509552001953e-07, 2.0721927285194397e-07, 1.3504177331924438e-08, 0.0, 1.9371509552001953e-07, 1.3504177331924438e-08, 1.3504177331924438e-08, 0.0, 2.4633482098579407e-07, 3.8230791687965393e-07, 1.3504177331924438e-08, 0.0, 3.688037395477295e-07, 3.2084062695503235e-07, 0.0, 6.076879799365997e-07, 0.0, 0.0, 3.026798367500305e-09, 1.4901161193847656e-07, 6.05359673500061e-09, 9.778887033462524e-09, 9.778887033462524e-09, 1.4901161193847656e-07, 0.0, 3.3061951398849487e-07, 9.778887033462524e-09, 3.2084062695503235e-07, 0.0, 9.778887033462524e-09, 9.778887033462524e-09, 2.868473529815674e-07, 0.0, 2.980232238769531e-07, 0.0, 8.307397365570068e-07, 0.0, 3.0919909477233887e-07, 4.6566128730773926e-09, 2.980232238769531e-07, 2.3283064365386963e-09, 3.655441105365753e-07, 0.0, 2.980232238769531e-07, 5.634501576423645e-08, 5.634501576423645e-08, 0.0, 2.477318048477173e-07, 5.634501576423645e-08, 3.543682396411896e-07, 0.0, 5.830079317092896e-07, 5.634501576423645e-08, 2.5331974029541016e-07, 1.0128132998943329e-06, 0.0, 0.0, 2.7497299015522003e-07, 2.5331974029541016e-07, 6.05359673500061e-09, 3.026798367500305e-09, 2.7194619178771973e-07, 2.5331974029541016e-07, 0.0, 1.1641532182693481e-08, 0.0, 5.117617547512054e-07, 0.0, 1.1641532182693481e-08, 2.5331974029541016e-07, 5.010515451431274e-07, 0.0, 1.1641532182693481e-08])
num_of_opt_trails_ein_4r = matrix([496, 192, 288, 288, 288, 0, 292, 292, 288, 0, 292, 292, 0, 192, 0, 0, 496, 0, 288, 288, 56, 0, 0, 104, 32, 0, 0, 32, 32, 0, 0, 32, 0, 0, 0, 48, 56, 0, 0, 56, 48, 0, 104, 0, 0, 0, 32, 0, 0, 0, 32, 0, 48, 0, 48, 0, 0, 0, 56, 0, 64, 0, 112, 0, 64, 0, 64, 0, 64, 0, 64, 0, 0, 0, 0, 0, 64, 0, 112, 0, 64, 208, 0, 0, 64, 64, 0, 0, 64, 64, 0, 0, 0, 96, 0, 0, 64, 112, 0, 0])Optimum weight: 44
p_ein_6r = matrix([1.9999717437713116e-09, 2.1114487935847137e-10, 1.669427263095713e-09, 1.669427263095713e-09, 1.669427263095713e-09, 0.0, 1.97400140677928e-09, 1.9759811564767915e-09, 1.6857217843835315e-09, 1.6294521287818498e-11, 1.9902959280670984e-09, 1.97400140677928e-09, 1.6294521287818498e-11, 2.2743940064628987e-10, 1.6294521287818498e-11, 0.0, 2.01626626505913e-09, 1.6294521287818498e-11, 1.6857217843835315e-09, 1.669427263095713e-09, 2.8699531640086207e-10, 0.0, 0.0, 4.952909193889354e-10, 2.3643309532417334e-10, 5.0821569175241166e-12, 1.0164313835048233e-11, 2.3643309532417334e-10, 2.823536959795092e-10, 4.5920600655335875e-11, 0.0, 2.3643309532417334e-10, 4.5920600655335875e-11, 4.5920600655335875e-11, 0.0, 2.0829560298807337e-10, 3.3291591705619794e-10, 4.5920600655335875e-11, 0.0, 2.8699531640086207e-10, 3.8956926573519013e-10, 0.0, 7.948592894990725e-10, 0.0, 0.0, 8.045120125643734e-12, 3.0878410939294554e-10, 1.609024025128747e-11, 3.200462117547431e-11, 3.200462117547431e-11, 3.0878410939294554e-10, 0.0, 4.2157388691066444e-10, 3.200462117547431e-11, 3.8956926573519013e-10, 0.0, 3.200462117547431e-11, 3.200462117547431e-11, 4.0529002376388235e-10, 0.0, 4.0897418784879846e-10, 0.0, 5.759801524618524e-10, 0.0, 4.845368550832063e-10, 1.8326673512092384e-11, 4.0897418784879846e-10, 9.163336756046192e-12, 5.584084306065051e-10, 0.0, 4.0897418784879846e-10, 7.387157552329882e-11, 7.387157552329882e-11, 0.0, 8.874678769643651e-11, 7.387157552329882e-11, 4.828457633720973e-10, 0.0, 4.872333647654159e-10, 7.387157552329882e-11, 4.4802206389249477e-10, 8.822063080060616e-10, 0.0, 0.0, 5.390035084928968e-10, 4.4802206389249477e-10, 2.809485977195436e-11, 1.404742988597718e-11, 5.249560786069196e-10, 4.4802206389249477e-10, 0.0, 4.9338311214341957e-11, 0.0, 3.3141134281322593e-10, 0.0, 4.9338311214341957e-11, 4.4802206389249477e-10, 5.507949651928357e-10, 0.0, 4.9338311214341957e-11])
num_of_opt_trails_ein_6r = matrix([31872, 3168, 26432, 26432, 26432, 0, 31552, 31552, 26432, 0, 31552, 31552, 0, 3168, 0, 0, 31872, 0, 26432, 26432, 3496, 0, 0, 5416, 2920, 0, 0, 2920, 2920, 0, 0, 2920, 0, 0, 0, 1920, 3496, 0, 0, 3496, 1920, 0, 5416, 0, 0, 0, 2920, 0, 0, 0, 2920, 0, 1920, 0, 1920, 0, 0, 0, 3496, 0, 5840, 0, 6992, 0, 6992, 0, 5840, 0, 6992, 0, 5840, 0, 0, 0, 0, 0, 5840, 0, 6992, 0, 5840, 9008, 0, 0, 6992, 5840, 0, 0, 6992, 5840, 0, 0, 0, 2016, 0, 0, 5840, 6992, 0, 0])Optimum weight: 16
p_em_2r = matrix([[0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 0.0, 0.0, 0.0, 0.0, 0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 0.0], [0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 0.0, 0.0, 0.0, 0.0, 0.0001220703125, 0.0, 0.0001220703125, 0.0001220703125, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.52587890625e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 0.0, 3.0517578125e-05, 3.0517578125e-05, 0.0, 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num_of_opt_trails_em_2r = matrix([[8, 0, 8, 8, 8, 0, 8, 8, 8, 0, 8, 8, 0, 0, 0, 0, 8, 0, 8, 8, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0], [8, 0, 8, 8, 8, 0, 8, 8, 8, 0, 8, 8, 0, 0, 0, 0, 8, 0, 8, 8, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [16, 0, 8, 8, 8, 0, 8, 8, 8, 0, 8, 8, 0, 0, 0, 0, 16, 0, 8, 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0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, 2, 0, 4, 4, 2, 0, 4, 4, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, 2, 0, 4, 4, 2, 0, 4, 4, 0, 0, 0, 0, 2, 0, 2, 2, 0, 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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [4, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 4, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])Optimum weight: 32
p_em_4r = matrix([[1.7078127712011337e-07, 1.979060471057892e-08, 1.3597309589385986e-07, 1.3597309589385986e-07, 1.3597309589385986e-07, 0.0, 1.7171259969472885e-07, 1.7398269847035408e-07, 1.3844692148268223e-07, 2.473825588822365e-09, 1.7418642528355122e-07, 1.7171259969472885e-07, 2.473825588822365e-09, 2.2264430299401283e-08, 2.473825588822365e-09, 0.0, 1.7325510270893574e-07, 2.473825588822365e-09, 1.3844692148268223e-07, 1.3597309589385986e-07, 2.5494955480098724e-08, 0.0, 0.0, 3.597233444452286e-08, 1.862645149230957e-08, 1.8044374883174896e-09, 3.6088749766349792e-09, 1.862645149230957e-08, 2.3632310330867767e-08, 5.005858838558197e-09, 0.0, 1.862645149230957e-08, 5.005858838558197e-09, 5.005858838558197e-09, 0.0, 1.0477378964424133e-08, 3.050081431865692e-08, 5.005858838558197e-09, 0.0, 2.5494955480098724e-08, 1.4202669262886047e-08, 0.0, 4.528556019067764e-08, 0.0, 0.0, 4.016328603029251e-09, 2.2351741790771484e-08, 8.032657206058502e-09, 4.0745362639427185e-09, 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num_of_opt_trails_em_4r = matrix([[732, 76, 584, 584, 584, 0, 728, 728, 584, 0, 728, 728, 0, 76, 0, 0, 732, 0, 584, 584, 80, 0, 0, 108, 64, 0, 0, 64, 64, 0, 0, 64, 0, 0, 0, 28, 80, 0, 0, 80, 28, 0, 108, 0, 0, 0, 64, 0, 0, 0, 64, 0, 28, 0, 28, 0, 0, 0, 80, 0, 128, 0, 160, 0, 160, 0, 128, 0, 160, 0, 128, 0, 0, 0, 0, 0, 128, 0, 160, 0, 128, 204, 0, 0, 160, 128, 0, 0, 160, 128, 0, 0, 0, 44, 0, 0, 128, 160, 0, 0], [732, 76, 584, 584, 584, 0, 728, 728, 584, 0, 728, 728, 0, 76, 0, 0, 732, 0, 584, 584, 80, 0, 0, 108, 64, 0, 0, 64, 64, 0, 0, 64, 0, 0, 0, 28, 80, 0, 0, 80, 28, 0, 108, 0, 0, 0, 64, 0, 0, 0, 64, 0, 28, 0, 28, 0, 0, 0, 80, 0, 128, 0, 160, 0, 160, 0, 128, 0, 160, 0, 128, 0, 0, 0, 0, 0, 128, 0, 160, 0, 128, 204, 0, 0, 160, 128, 0, 0, 160, 128, 0, 0, 0, 44, 0, 0, 128, 160, 0, 0], [80, 16, 76, 76, 76, 0, 76, 76, 76, 0, 76, 76, 0, 16, 0, 0, 80, 0, 76, 76, 8, 0, 0, 40, 8, 0, 0, 8, 8, 0, 0, 8, 0, 0, 0, 32, 8, 0, 0, 8, 32, 0, 40, 0, 0, 0, 8, 0, 0, 0, 8, 0, 32, 0, 32, 0, 0, 0, 8, 0, 16, 0, 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0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [192, 16, 160, 160, 160, 0, 192, 192, 160, 0, 192, 192, 0, 16, 0, 0, 192, 0, 160, 160, 24, 0, 0, 28, 20, 0, 0, 20, 20, 0, 0, 20, 0, 0, 0, 4, 24, 0, 0, 24, 4, 0, 28, 0, 0, 0, 20, 0, 0, 0, 20, 0, 4, 0, 4, 0, 0, 0, 24, 0, 40, 0, 48, 0, 48, 0, 40, 0, 48, 0, 40, 0, 0, 0, 0, 0, 40, 0, 48, 0, 40, 56, 0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [240, 32, 204, 204, 204, 0, 236, 236, 204, 0, 236, 236, 0, 32, 0, 0, 240, 0, 204, 204, 28, 0, 0, 64, 24, 0, 0, 24, 24, 0, 0, 24, 0, 0, 0, 36, 28, 0, 0, 28, 36, 0, 64, 0, 0, 0, 24, 0, 0, 0, 24, 0, 36, 0, 36, 0, 0, 0, 28, 0, 48, 0, 56, 0, 56, 0, 48, 0, 56, 0, 48, 0, 0, 0, 0, 0, 48, 0, 56, 0, 48, 80, 0, 0, 56, 48, 0, 0, 56, 48, 0, 0, 0, 24, 0, 0, 48, 56, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [48, 16, 44, 44, 44, 0, 44, 44, 44, 0, 44, 44, 0, 16, 0, 0, 48, 0, 44, 44, 4, 0, 0, 36, 4, 0, 0, 4, 4, 0, 0, 4, 0, 0, 0, 32, 4, 0, 0, 4, 32, 0, 36, 0, 0, 0, 4, 0, 0, 0, 4, 0, 32, 0, 32, 0, 0, 0, 4, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 0, 0, 0, 0, 8, 0, 8, 0, 8, 24, 0, 0, 8, 8, 0, 0, 8, 8, 0, 0, 0, 16, 0, 0, 8, 8, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [192, 16, 160, 160, 160, 0, 192, 192, 160, 0, 192, 192, 0, 16, 0, 0, 192, 0, 160, 160, 24, 0, 0, 28, 20, 0, 0, 20, 20, 0, 0, 20, 0, 0, 0, 4, 24, 0, 0, 24, 4, 0, 28, 0, 0, 0, 20, 0, 0, 0, 20, 0, 4, 0, 4, 0, 0, 0, 24, 0, 40, 0, 48, 0, 48, 0, 40, 0, 48, 0, 40, 0, 0, 0, 0, 0, 40, 0, 48, 0, 40, 56, 0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [192, 16, 160, 160, 160, 0, 192, 192, 160, 0, 192, 192, 0, 16, 0, 0, 192, 0, 160, 160, 24, 0, 0, 28, 20, 0, 0, 20, 20, 0, 0, 20, 0, 0, 0, 4, 24, 0, 0, 24, 4, 0, 28, 0, 0, 0, 20, 0, 0, 0, 20, 0, 4, 0, 4, 0, 0, 0, 24, 0, 40, 0, 48, 0, 48, 0, 40, 0, 48, 0, 40, 0, 0, 0, 0, 0, 40, 0, 48, 0, 40, 56, 0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [192, 16, 160, 160, 160, 0, 192, 192, 160, 0, 192, 192, 0, 16, 0, 0, 192, 0, 160, 160, 24, 0, 0, 28, 20, 0, 0, 20, 20, 0, 0, 20, 0, 0, 0, 4, 24, 0, 0, 24, 4, 0, 28, 0, 0, 0, 20, 0, 0, 0, 20, 0, 4, 0, 4, 0, 0, 0, 24, 0, 40, 0, 48, 0, 48, 0, 40, 0, 48, 0, 40, 0, 0, 0, 0, 0, 40, 0, 48, 0, 40, 56, 0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [192, 16, 160, 160, 160, 0, 192, 192, 160, 0, 192, 192, 0, 16, 0, 0, 192, 0, 160, 160, 24, 0, 0, 28, 20, 0, 0, 20, 20, 0, 0, 20, 0, 0, 0, 4, 24, 0, 0, 24, 4, 0, 28, 0, 0, 0, 20, 0, 0, 0, 20, 0, 4, 0, 4, 0, 0, 0, 24, 0, 40, 0, 48, 0, 48, 0, 40, 0, 48, 0, 40, 0, 0, 0, 0, 0, 40, 0, 48, 0, 40, 56, 0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [48, 16, 44, 44, 44, 0, 44, 44, 44, 0, 44, 44, 0, 16, 0, 0, 48, 0, 44, 44, 4, 0, 0, 36, 4, 0, 0, 4, 4, 0, 0, 4, 0, 0, 0, 32, 4, 0, 0, 4, 32, 0, 36, 0, 0, 0, 4, 0, 0, 0, 4, 0, 32, 0, 32, 0, 0, 0, 4, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 0, 0, 0, 0, 8, 0, 8, 0, 8, 24, 0, 0, 8, 8, 0, 0, 8, 8, 0, 0, 0, 16, 0, 0, 8, 8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [48, 16, 44, 44, 44, 0, 44, 44, 44, 0, 44, 44, 0, 16, 0, 0, 48, 0, 44, 44, 4, 0, 0, 36, 4, 0, 0, 4, 4, 0, 0, 4, 0, 0, 0, 32, 4, 0, 0, 4, 32, 0, 36, 0, 0, 0, 4, 0, 0, 0, 4, 0, 32, 0, 32, 0, 0, 0, 4, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 0, 0, 0, 0, 8, 0, 8, 0, 8, 24, 0, 0, 8, 8, 0, 0, 8, 8, 0, 0, 0, 16, 0, 0, 8, 8, 0, 0], [192, 16, 160, 160, 160, 0, 192, 192, 160, 0, 192, 192, 0, 16, 0, 0, 192, 0, 160, 160, 24, 0, 0, 28, 20, 0, 0, 20, 20, 0, 0, 20, 0, 0, 0, 4, 24, 0, 0, 24, 4, 0, 28, 0, 0, 0, 20, 0, 0, 0, 20, 0, 4, 0, 4, 0, 0, 0, 24, 0, 40, 0, 48, 0, 48, 0, 40, 0, 48, 0, 40, 0, 0, 0, 0, 0, 40, 0, 48, 0, 40, 56, 0, 0, 48, 40, 0, 0, 48, 40, 0, 0, 0, 8, 0, 0, 40, 48, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [160, 16, 128, 128, 128, 0, 160, 160, 128, 0, 160, 160, 0, 16, 0, 0, 160, 0, 128, 128, 20, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 16, 0, 0, 0, 4, 20, 0, 0, 20, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 16, 0, 4, 0, 4, 0, 0, 0, 20, 0, 32, 0, 40, 0, 40, 0, 32, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32, 0, 40, 0, 32, 48, 0, 0, 40, 32, 0, 0, 40, 32, 0, 0, 0, 8, 0, 0, 32, 40, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])Optimum weight: 28
p_eout_4r = matrix([1.0728836059570312e-06, 1.0728836059570312e-06, 8.544884622097015e-07, 2.1010637283325195e-06, 1.098262146115303e-06, 1.0952353477478027e-06, 0.0, 1.0728836059570312e-06, 1.0952353477478027e-06, 1.1289957910776138e-06, 3.3760443329811096e-08, 1.1066440492868423e-06, 0.0, 3.3760443329811096e-08, 8.882489055395126e-07, 3.3760443329811096e-08, 1.0728836059570312e-06, 1.1066440492868423e-06, 3.3760443329811096e-08, 2.1348241716623306e-06, 6.151385605335236e-07, 1.9371509552001953e-07, 2.4633482098579407e-07, 3.688037395477295e-07, 1.9371509552001953e-07, 2.1606683731079102e-07, 0.0, 1.9371509552001953e-07, 1.9371509552001953e-07, 2.2957101464271545e-07, 1.3504177331924438e-08, 2.0721927285194397e-07, 2.4633482098579407e-07, 1.3504177331924438e-08, 2.598389983177185e-07, 1.3504177331924438e-08, 3.688037395477295e-07, 2.0721927285194397e-07, 1.3504177331924438e-08, 3.8230791687965393e-07, 6.076879799365997e-07, 1.4901161193847656e-07, 3.2084062695503235e-07, 2.868473529815674e-07, 1.4901161193847656e-07, 1.7439015209674835e-07, 0.0, 1.4901161193847656e-07, 1.4901161193847656e-07, 1.7136335372924805e-07, 9.778887033462524e-09, 1.587904989719391e-07, 3.2084062695503235e-07, 0.0, 3.3061951398849487e-07, 9.778887033462524e-09, 2.868473529815674e-07, 1.4901161193847656e-07, 9.778887033462524e-09, 2.966262400150299e-07, 8.307397365570068e-07, 2.980232238769531e-07, 2.980232238769531e-07, 5.830079317092896e-07, 2.980232238769531e-07, 3.1152740120887756e-07, 3.0919909477233887e-07, 2.980232238769531e-07, 2.980232238769531e-07, 3.0919909477233887e-07, 3.655441105365753e-07, 3.543682396411896e-07, 2.477318048477173e-07, 0.0, 5.634501576423645e-08, 5.634501576423645e-08, 5.830079317092896e-07, 2.980232238769531e-07, 3.543682396411896e-07, 6.39352947473526e-07, 1.0128132998943329e-06, 2.5331974029541016e-07, 2.5331974029541016e-07, 5.117617547512054e-07, 2.5331974029541016e-07, 2.7497299015522003e-07, 2.7194619178771973e-07, 0.0, 2.5331974029541016e-07, 2.7194619178771973e-07, 2.835877239704132e-07, 1.1641532182693481e-08, 5.117617547512054e-07, 0.0, 1.1641532182693481e-08, 5.234032869338989e-07, 5.010515451431274e-07, 2.5331974029541016e-07, 2.6496127247810364e-07, 1.1641532182693481e-08])
num_of_opt_trails_eout_4r = matrix([288, 288, 192, 496, 292, 292, 0, 288, 292, 292, 0, 288, 0, 0, 192, 0, 288, 288, 0, 496, 104, 32, 48, 56, 32, 32, 0, 32, 32, 32, 0, 32, 48, 0, 48, 0, 56, 32, 0, 56, 104, 32, 48, 56, 32, 32, 0, 32, 32, 32, 0, 32, 48, 0, 48, 0, 56, 32, 0, 56, 112, 64, 64, 112, 64, 64, 64, 64, 64, 64, 64, 64, 0, 0, 0, 0, 112, 64, 64, 112, 208, 64, 64, 96, 64, 64, 64, 0, 64, 64, 64, 0, 96, 0, 0, 96, 112, 64, 64, 0])Optimum weight: 44
p_eout_6r = matrix([1.701625507166682e-09, 1.701625507166682e-09, 6.076348313399649e-10, 2.2644073283117905e-09, 2.28573426852563e-09, 2.2380994835202728e-09, 0.0, 1.701625507166682e-09, 2.2380994835202728e-09, 2.3285906536330003e-09, 9.049117011272756e-11, 1.7921166772794095e-09, 0.0, 9.049117011272756e-11, 6.981260014526924e-10, 9.049117011272756e-11, 1.701625507166682e-09, 1.7921166772794095e-09, 9.049117011272756e-11, 2.354898498424518e-09, 1.0702443375976145e-09, 2.750439875853772e-10, 6.204707858614711e-10, 4.497735517361434e-10, 2.750439875853772e-10, 3.9100456206142553e-10, 0.0, 2.750439875853772e-10, 2.750439875853772e-10, 5.108660161567968e-10, 1.198614540953713e-10, 3.949054416807485e-10, 6.204707858614711e-10, 1.198614540953713e-10, 7.403322399568424e-10, 1.198614540953713e-10, 4.497735517361434e-10, 3.949054416807485e-10, 1.198614540953713e-10, 5.696350058315147e-10, 1.4625456401518022e-09, 3.459561526142352e-10, 8.89322393504699e-10, 5.732232466471032e-10, 3.459561526142352e-10, 5.516938017535722e-10, 0.0, 3.459561526142352e-10, 3.459561526142352e-10, 4.677005449593707e-10, 7.809930480107141e-11, 4.240554574153066e-10, 8.89322393504699e-10, 0.0, 9.674216983057704e-10, 7.809930480107141e-11, 5.732232466471032e-10, 3.459561526142352e-10, 7.809930480107141e-11, 6.513225514481746e-10, 1.081524203527806e-09, 4.675868581216491e-10, 4.675868581216491e-10, 7.208882379927672e-10, 4.675868581216491e-10, 6.779785621802148e-10, 6.370228788910026e-10, 4.675868581216491e-10, 4.675868581216491e-10, 6.370228788910026e-10, 8.173266508038068e-10, 6.478906300344534e-10, 3.6063596553503885e-10, 0.0, 1.8030377191280422e-10, 1.8030377191280422e-10, 7.208882379927672e-10, 4.675868581216491e-10, 6.478906300344534e-10, 9.011920099055715e-10, 1.4573728890354687e-09, 4.788844876202347e-10, 4.788844876202347e-10, 7.813270030965214e-10, 4.788844876202347e-10, 7.742571028757084e-10, 6.693809950775176e-10, 0.0, 4.788844876202347e-10, 6.693809950775176e-10, 7.738663043710403e-10, 1.0448530929352273e-10, 7.813270030965214e-10, 0.0, 1.0448530929352273e-10, 8.858123123900441e-10, 6.760458859389473e-10, 4.788844876202347e-10, 5.833697969137575e-10, 1.0448530929352273e-10])
num_of_opt_trails_eout_6r = matrix([26432, 26432, 3168, 31872, 31552, 31552, 0, 26432, 31552, 31552, 0, 26432, 0, 0, 3168, 0, 26432, 26432, 0, 31872, 5416, 2920, 1920, 3496, 2920, 3496, 0, 2920, 2920, 3496, 0, 2920, 1920, 0, 1920, 0, 3496, 2920, 0, 3496, 5416, 2920, 1920, 3496, 2920, 3496, 0, 2920, 2920, 3496, 0, 2920, 1920, 0, 1920, 0, 3496, 2920, 0, 3496, 6992, 5840, 5840, 6992, 5840, 6992, 6992, 5840, 5840, 6992, 6992, 5840, 0, 0, 0, 0, 6992, 5840, 5840, 6992, 9008, 5840, 5840, 2016, 5840, 6992, 6992, 0, 5840, 6992, 6992, 0, 2016, 0, 0, 2016, 6992, 5840, 5840, 0])Eout-Even-6r + Ein-Even-4r
p_10r = p_ein_4r * p_eout_6r.transpose()
p_10r = p_10r[0, 0]
num_of_optimum_trails_10r = num_of_opt_trails_ein_4r * num_of_opt_trails_eout_6r.transpose()
num_of_optimum_trails_10r = num_of_optimum_trails_10r[0, 0]
print("p_%dr = 2^%0.4f" % (10, p_10r.log2()))
print("number of optimum trails for %d rounds : 2^%0.4f" % (10, num_of_optimum_trails_10r.log(2)))p_10r = 2^-44.8906
number of optimum trails for 10 rounds : 2^26.4958
Eout-Even-6r + Em-Even-2r + Ein-Even-4r
p_12r = p_ein_4r * p_em_2r * p_eout_6r.transpose()
p_12r = p_12r[0, 0]
num_of_optimum_trails_12r = num_of_opt_trails_ein_4r * num_of_opt_trails_em_2r * num_of_opt_trails_eout_6r.transpose()
num_of_optimum_trails_12r = num_of_optimum_trails_12r[0, 0]
print("p_%dr = 2^%0.4f" % (12, p_12r.log2()))
print("number of optimum trails for %d rounds : 2^%0.4f" % (12, num_of_optimum_trails_12r.log(2)))p_12r = 2^-54.4817
number of optimum trails for 12 rounds : 2^32.8375
Eout-Even-6r + Em-Even-4r + Ein-Even-4r
p_14r = p_ein_4r * p_em_4r * p_eout_6r.transpose()
p_14r = p_14r[0, 0]
num_of_optimum_trails_14r = num_of_opt_trails_ein_4r * num_of_opt_trails_em_4r * num_of_opt_trails_eout_6r.transpose()
num_of_optimum_trails_14r = num_of_optimum_trails_14r[0, 0]
print("p_%dr = 2^%0.4f" % (14, p_14r.log2()))
print("number of optimum trails for %d rounds : 2^%0.4f" % (14, num_of_optimum_trails_14r.log(2)))p_14r = 2^-63.8061
number of optimum trails for 14 rounds : 2^39.1814
We use the same strategy for related-twek setting, and you can use the following commands in folder SAT-SMT-RT, to reproduce our results in related-tweak setting.
python3 cryptosmt_ein_tk0_5r_usedfor_15r.py --inputfile examples/craftrtk/rtk0_5_rounds_usedfor_ein_15r.yaml
python3 cryptosmt_eout_tk0_10r_outof_15r.py --inputfile examples/craftrtk/rtk0_10_rounds_usedfor_eout_15r.yaml
python3 cryptosmt_ein_tk1_7r_usedfor_15r.py --inputfile examples/craftrtk/rtk1_7_rounds_usedfor_ein_15r.yaml
python3 cryptosmt_eout_tk1_8r_usedfor_15r.py --inputfile examples/craftrtk/rtk1_8_rounds_usedfor_eout_15r.yaml
python3 cryptosmt_ein_tk1_5r_usedfor_16r.py --inputfile examples/craftrtk/rtk1_5_rounds_usedfor_ein_16r.yaml
python3 cryptosmt_eout_tk1_11r_usedfor_16r.py --inputfile examples/craftrtk/rtk1_11_rounds_usedfor_eout_16r.yaml
python3 cryptosmt_ein_tk1_8r_usedfor_16r.py --inputfile examples/craftrtk/rtk1_8_rounds_usedfor_ein_16r.yaml
python3 cryptosmt_eout_tk1_8r_usedfor_16r.py --inputfile examples/craftrtk/rtk1_8_rounds_usedfor_eout_16r.yaml
python3 cryptosmt_ein_tk1_8r_usedfor_17r.py --inputfile examples/craftrtk/rtk1_8_rounds_usedfor_ein_17r.yaml
python3 cryptosmt_eout_tk1_9r_usedfor_17r.py --inputfile examples/craftrtk/rtk1_9_rounds_usedfor_eout_17r.yaml
python3 cryptosmt_ein_tk2_7r_usedfor_17r.py --inputfile examples/craftrtk/rtk2_7_rounds_usedfor_ein_17r.yaml
python3 cryptosmt_eout_tk2_10r_usedfor_17r.py --inputfile examples/craftrtk/rtk2_10_rounds_usedfor_eout_17r.yaml
python3 cryptosmt_ein_tk3_6r_usedfor_16r.py --inputfile examples/craftrtk/rtk3_6_rounds_usedfor_ein_16r.yaml
python3 cryptosmt_eout_tk3_10r_usedfor_16r.py --inputfile examples/craftrtk/rtk3_10_rounds_usedfor_eout_16r.yaml
In order to verify our results in differential cryptanalysis of CRAFT, we have prepared some codes to simmulate our differential distinguishers. All the codes we have used for these simmulations are located in the folder ExperimentalVerification-Diff. It is needed to use -fopemp comipler flag when you are compiling the codes, since OpenMP is used in these codes. For example craft_ein_odd.cpp has been prepared to compare the empiricial, and the so-called theoretical values we have obtained for the differential effect of the first component Ein-Odd-4r used in our 9/11/13-round differentail distinguishers, and you can compile it by the following command:
g++ craft_ein_odd.cpp -o craft_ein_odd.o -fopenmp
and then run it by the following command:
./craft_ein_odd.o
If you do the above simmulations, you see that the experimental probability is approximately equal to 2^(-18.66), while the so-called theoretical value we have obtained for this component, when the input, and output differences are 0xAA000000AA00AAA0, and 0x0A0005000A0000A0 respectively, is equal to 2^(-18.86). However we invite those, who have more computing power, to do the similar simmultaions for the more number of rounds.
Before we can improve the integral distinguishers of CRAFT, we verfied the designers' claim for this analysis. In order to verify the designers' claim, we have prepared a program to find the integral distinguishers of CRAFT, using division property based on two subsets. The following picture, depicts the variables used in the MILP model which is created by the program to model the propagation of division properties through the cipher.
You can find this program in the folder MILP-Integral, and run it by the following command:
python3 main.py
Copyright (C) 2019 Hosein Hadipour
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
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