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Comprehensive Security Analysis of CRAFT

CRAFT is a light-weight tweakable block cipher, designed with efficient protection against differential fault attacks (DFA) in mind. 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 published in FSE-2020, and this repository contains the codes we have used to find these distinguishers.

craft_round_function

Contents

  1. Prerequisites

  2. Zero-Correlation Cryptanalysis

  3. Differential Cryptanalysis

  4. Experimental Verification

  5. Integral Cryptanalysis

Prerequisites and Installation

In this repository, both MILP, and SMT/SAT based methods are used to analyse CRAFT. For MILP-based method we use Python3 to produce the MILP models, and Gurobi, is used as the solver. Therefore, you need to install Gurobi and link it to Python3. You can find the installation recepies 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 recepies 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.

Zero-Correlation Cryptanalysis

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 a 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 is executed in parallel.

RTK0

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.

ZC-TK0

RKT1

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

RTK2

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

ZC-TK2

RTK3

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

ZC-TK3

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.


Differential Cryptanalysis

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 obtimum 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.

Ein-Even-4r

ein_even 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 

Ein-Even-6r

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

Em-Even-2r

The following 2-round component is used as the middle part in our 12-round differential distinguisher. em_even 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. em_even_2r_matrix

Em-Even-4r

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. em_even_4r_matrix

Eout-Even-6r

The following figure, shows the 6-round component which is used as Eout (last part), in our 10/12/14-round differential distinguishers. eout_even 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 

Ein-Odd-4r

The following 4-round component is used as Ein (first part), in our 9/11/13-rounds differential distinguishers. ein_odd 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

Em-Odd-2r

We use the following 2-round component as the middle part in 11/13-round differential distinguishers. em_odd 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. em_odd_2r_matrix

Eout-Odd-5r

The following 5-round component is used as Eout (last part), in 9/11/13-round differential distinguishers. eout_odd 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 .

Ein-Even-4 Rounds

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])

Ein-Even-6 Rounds

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])

Em-Even-2 Rounds

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, 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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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])

Em-Even-4 Rounds

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, 4.0745362639427185e-09, 2.2351741790771484e-08, 0.0, 1.8277205526828766e-08, 4.0745362639427185e-09, 1.4202669262886047e-08, 0.0, 4.0745362639427185e-09, 4.0745362639427185e-09, 3.1082890927791595e-08, 0.0, 3.3527612686157227e-08, 0.0, 5.005858838558197e-08, 0.0, 4.353933036327362e-08, 4.0745362639427185e-09, 3.3527612686157227e-08, 2.037268131971

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Comprehensive Security Analysis of CRAFT

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