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This work was supported by National Key Research and Development Program of China (Grant No. 2017YFA0303903), National Cryptography Development Fund (Grant Nos. MMJJ20170121, MMJJ20170102), Zhejiang Province Key RD Project (Grant No. 2017C01062), National Natural Science Foundation of China (Grant Nos. 61572293, 61502276, 61692276), Major Scientific and Technological Innovation Projects of Shandong Province (Grant No. 2017CXGC0704), and National Natural Science Foundation of Shandong Province (Grant No. ZR2016FM22).
Appendixes A and B.
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// Step 1: Construct the MILP model. |
Represent the input and output differences for each operation as binary variables. |
Link the binary variables by adding linear inequalities for each target cipher operation. |
// Step 2: Find all the impossible differentials within a given set of input and output differences. |
Determine the sets of input differences $\Delta$ and output differences $\Gamma$. |
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Add all constraints related to the current input and output differences to the MILP model. |
Attempt to solve the model. |
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// The current input and output differences represent a possible differential. |
Break; |
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// The current input and output differences yield an impossible differential. |
Store the current input and output differences. |
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Cipher | Type | Round | Reference |
HIGHT | Impossible differential | 16 | |
Impossible differential | 17 | ||
Impossible differential | 17 | Ours | |
Zero-correlation linear approximation | 16 | ||
Zero-correlation linear approximation | 17 | ||
Zero-correlation linear approximation | 17 | Ours | |
SHACAL-2 | Zero-correlation linear approximation | 12 | |
Impossible differential | 14 | ||
Impossible differential | 15 | Ours | |
LEA | Zero-correlation linear approximation | 7 | |
Zero-correlation linear approximation | 9 | ||
Zero-correlation linear approximation | 10 | Ours | |
LBlock | Related-key impossible differential$^{\rm~a)}$ | 16 | |
Related-key impossible differential | 16 | Ours |