MILP Based Differential Characteristics Search for Lightweight Block Cipher PIPO-64/128 (https://link.springer.com/chapter/10.1007%2F978-3-030-68890-5_6)
Linear inequalities required for MILP problem are generated using MILES (https://github.com/tarunyadav/MILES).
- MILP_Outer.py
- MILP_Inner.py
- print_diff_characteristic.py
- LinearInequalities_Outer.txt (Generated from MILES)
- LinearInequalities_Inner.txt (Generated from MILES)
Source Code (including README) for inequalities reduction and to search impossible differential & related-key differential characteristics is provided in respective folders.
- MILP_Outer.py is used to minimize number of active S-boxes in PIPO. It uses LinearInequalities_Outer.txt to write linear inequalities of the S-box.
- There are two arugment for MILP_Outer.py. First argument defines number of rounds and second argument define the cutoff bound of active S-box. The second argument is made changable(instead of fix upper/lower bound) because it is obsevered that GUROBI gives quick solution for for some values which are not directly related to the upper/lower bound of active S-box.
- The minimum number of active S-boxes for 7 rounds is searched using following command:
MILP_Outer.py 7 10 - The output (Outer_Result_7.txt) is in the following format:
[[0, 0], [1, 7], [2, 3], [2, 7], [3, 0], [3, 4], [3, 5], [3, 7], [4, 0], [4, 6], [4, 7], [5, 4], [6, 7]]
[a,b] => In the round number a, S-box b is active. - This output is the input for next part (MILP_Inner.py) at line no. 210.
- The output of MILP_Outer.py becomes input for MILP_inner.py at line no. 210. MILP_inner.py uses LinearInequalities_Inner.txt to write linear inequalities of the S-box.
- There is one arugment for MILP_Inner.py. This define the nuumber of rounds.
- High probablitiy differential characteristics for 7-round PIPO-64/128 is searhed using following command:
MILP_Inner.py 7 - The output (Inner_Result_7.txt) is in the following format:
[[0, 0], [1, 7], [2, 3], [2, 7], [3, 0], [3, 4], [3, 5], [3, 7], [4, 0], [4, 6], [4, 7], [5, 4], [6, 7]] 65.0
z0_0_0 z1_7_0 z2_3_3 z2_7_5 z3_0_3 z3_4_0 z3_5_3 z3_7_3 z4_0_0 z4_6_5 z4_7_5 z5_4_6 z6_7_0 x0_8 x0_16 x0_32 x0_40 x0_56 y0_8 x1_15 y1_7 y1_23 x2_19 y2_19 y2_43 y2_51 y2_59 x2_7 y2_7 y2_39 y2_47 y2_55 x3_40 x3_48 y3_8 y3_32 x3_44 x3_52 y3_20 y3_28 y3_60 x3_37 x3_61 y3_29 y3_53 x3_7 x3_23 y3_7 y3_15 y3_55 x4_16 x4_24 x4_48 y4_16 x4_14 x4_38 x4_54 x4_62 y4_38 x4_7 x4_15 x4_31 y4_47 x5_20 x5_36 x5_44 y5_28 x6_31 y6_15 y6_23 y6_31 y6_55 x7_14 x7_19 x7_26 x7_48
- This output can be converted into the differential characteristics using print_diff_characteristic.py
- Differential characteristics can be printed using print_diff_characteristic.py
- There are two argument for print_diff_characteristic.py. First argument is Inner_Result_7.txt and second argument is the characteristic number in Inner_Result_7.txt.
- The differential characteristic for 7-round PIPO-64/128 is printed using following command:
print_diff_characteristic.py Inner_Result_7.txt 1 - The output is in the following format:
Differential Probability for 7 rounds of PIPO-64/128 is 2^{-65.0}
The input difference of plaintext is:
0000 0001 0000 0000 0000 0001 0000 0001 0000 0000 0000 0001 0000 0001 0000 0000 :: Hex => 0100 0101 0001 0100 :: 0x0100010100010100 :: Probability => 2^{-0}
The Output difference of the round 1 is:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0000 0000 :: Hex => 0000 0000 0000 8000 :: 0x0000000000008000 :: Probability => 2^{-4}
The Output difference of the round 2 is:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0000 1000 0000 :: Hex => 0000 0000 0008 0080 :: 0x0000000000080080 :: Probability => 2^{-4}
The Output difference of the round 3 is:
0010 0000 0001 0001 0001 0001 0010 0000 0000 0000 1000 0000 0000 0000 1000 0000 :: Hex => 2011 1120 0080 0080 :: 0x2011112000800080 :: Probability => 2^{-11}
The Output difference of the round 4 is:
0100 0000 0100 0001 0000 0000 0100 0000 1000 0001 0000 0001 1100 0000 1000 0000 :: Hex => 4041 0040 8101 c080 :: 0x404100408101c080 :: Probability => 2^{-19}
The Output difference of the round 5 is:
0000 0000 0000 0000 0001 0000 0001 0000 0000 0000 0001 0000 0000 0000 0000 0000 :: Hex => 0000 1010 0010 0000 :: 0x0000101000100000 :: Probability => 2^{-16}
The Output difference of the round 6 is:
0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0000 0000 0000 0000 0000 0000 :: Hex => 0000 0000 8000 0000 :: 0x0000000080000000 :: Probability => 2^{-7}
The Output difference of the round 7 is:
0000 0000 0000 0001 0000 0000 0000 0000 0000 0100 0000 1000 0100 0000 0000 0000 :: Hex => 0001 0000 0408 4000 :: 0x0001000004084000 :: Probability => 2^{-4}