MILP/MIQCP-Based Fully Automatic Method of Searching for Differential-Linear Distinguishers for SIMON-Like Ciphers

Table 11

Twenty-six-round DL distinguisher for SIMON96 with theoretical correlation and experimental correlation , where the theoretical probability of the differential part, the theoretical correlation of the DL part, and the theoretical correlation of the linear part are , and , respectively.

Differential part (optimal differential trail)
0	000000010000000100000000000000000000000000000000
	000001000100000001000000000000000000000000000000
11	000000000100010001000000000000000000000000000000
	000000000001000000000000000000000000000000000000
DL part
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	1.0	−1.0	−1.0	−1.0	1.0	−1.0	−1.0
	−1.0	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
12	−1.0	−0.0	−1.0	−1.0	−1.0	−0.0	−1.0	1.0
	−0.0	−0.0	−1.0	−1.0	−0.0	−1.0	−1.0	1.0
	−0.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	1.0	−1.0	−1.0	−1.0	1.0	−1.0	−1.0
	−1.0	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
13	−0.25	−0.5	−1.0	−0.0	−0.25	1.0	−0.0	−0.0
	−0.25	1.0	−0.0	−0.5	−1.0	−1.0	−0.0	−0.5
	−1.0	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−0.5	−1.0	−1.0	−1.0	−0.5	−1.0	−0.0
	−1.0	−0.0	−1.0	−1.0	−1.0	−0.0	−1.0	1.0
	−0.0	−0.0	−1.0	−1.0	−0.0	−1.0	−1.0	1.0
	−0.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
14	−0.4688	−0.0	−0.0625	0.4688	−0.0	−0.0	−0.0625	−0.4688
	−0.0	−0.0	−0.75	−1.0	−0.0	−0.25	−0.75	−1.0
	−0.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−0.75	−1.0	−1.0	−1.0	−0.75	−1.0	−0.25
	−0.4688	−0.75	−1.0	−0.25	−0.4688	−0.0	−0.0625	−0.1563
	−0.25	−0.5	−1.0	−0.0	−0.25	1.0	−0.0	−0.0
	−0.25	1.0	−0.0	−0.5	−1.0	−1.0	−0.0	−0.5
	−1.0	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−0.5	−1.0	−1.0	−1.0	−0.5	−1.0	−0.0
15	−0.0039	0.0623	−0.0	−0.0	−0.0039	0.1556	−0.0	−0.0
	−0.0468	0.875	−0.0	−0.0625	−0.4688	−0.875	−0.0	−0.25
	−1.0	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−0.875	−1.0	−1.0	−1.0	−0.875	−1.0	−0.4688
	−0.6426	−0.875	−1.0	−0.4688	−0.6426	−0.125	−0.1556	−0.3184
	−0.6426	−0.0625	−0.1556	−0.0	−0.0156	−0.0208	−0.1440	−0.0
	−0.4688	−0.0	−0.0625	0.4688	−0.0	−0.0	−0.0625	−0.4688
	−0.0	−0.0	−0.75	−1.0	−0.0	−0.25	−0.75	−1.0
	−0.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−0.75	−1.0	−1.0	−1.0	−0.75	−1.0	−0.25
	−0.4688	−0.75	−1.0	−0.25	−0.4688	−0.0	−0.0625	−0.1563
16	−0.0	−0.0	−0.00006	−0.0195	−0.0	−0.0	−0.0007	0.1342
	−0.0	−0.0	−0.1868	−0.6426	−0.0	−0.0313	−0.4688	1.0
	−0.0	−0.9375	−1.0	−1.0	−1.0	−0.9375	−1.0	−0.6426
	−0.77	−0.9375	−1.0	−0.6426	−0.77	−0.2637	−0.2727	−0.4737
	−0.77	−0.1868	−0.2727	−0.0513	−0.0445	−0.0704	−0.2423	−0.0064
	−0.0195	−0.0	−0.0039	−0.0013	−0.0173	−0.0	−0.00006	0.0024
	−0.0039	0.0623	−0.0	−0.0	−0.0039	0.1556	−0.0	−0.0
	−0.0469	0.875	−0.0	−0.0625	−0.4688	−0.875	−0.0	−0.25
	−1.0	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−1.0	−0.875	−1.0	−1.0	−1.0	−0.875	−1.0	−0.4688
	−0.6426	−0.875	−1.0	−0.4688	−0.6426	−0.125	−0.1556	−0.3184
	−0.6426	−0.0625	−0.1556	−0.0	−0.0156	−0.0208	−0.1440	−0.0
17		0.0003	−0.0	−0.0		−0.0054	−0.0	−0.0
	−0.0022	0.3232	−0.0	−0.00098	−0.1133	0.6225	−0.0	−0.0962
	−0.8573	0.9688	−1.0	−0.77	−0.8573	−0.4060	−0.4024	−0.6114
	−0.8573	−0.3336	−0.4024	−0.1227	−0.09	−0.1412	−0.3524	−0.039
	−0.0530	−0.0143	−0.0117	−0.0086	−0.0424	−0.0002	−0.0008	−0.0
	−0.0006	−0.00002	−0.0007	−0.0		0.00001	−0.0	−0.0
	−0.0	−0.0	−0.00006	−0.0195	−0.0	−0.0	−0.0007	0.1342
	−0.0	−0.0	−0.1868	−0.6426	−0.0	−0.0313	−0.4688	1.0
	−0.0	−0.9375	−1.0	−1.0	−1.0	−0.9375	−1.0	−0.6426
	−0.77	−0.9375	−1.0	−0.6426	−0.77	−0.2637	−0.2727	−0.4737
	−0.77	−0.1868	−0.2727	−0.0513	−0.0445	−0.0704	−0.2423	−0.0064
	−0.0195	−0.0	−0.0039	−0.0013	−0.0173	−0.0	−0.00006	0.0024
Linear Part
	000000000000000010001000100000000000000000000000
	000000000000000000000010000000000000000000000000
18	000000000000000000000010000000000000000000000000
	000000000000000010001000000000000000000000000000
19	000000000000000010001000000000000000000000000000
	000000000000000000100000000000000000000000000000
20	000000000000000000100000000000000000000000000000
	000000000000000010000000000000000000000000000000
21	000000000000000010000000000000000000000000000000
	000000000000000000000000000000000000000000000000
22	000000000000000000000000000000000000000000000000
	000000000000000010000000000000000000000000000000
23	000000000000000010000000000000000000000000000000
	000000000000000000100000000000000000000000000000
24	000000000000000000100000000000000000000000000000
	000000000000000010001000000000000000000000000000
25	000000000000000010001000000000000000000000000000
	000000000000000000000010000000000000000000000000
26	000000000000000000000010000000000000000000000000
	000000000000000010001000100000000000000000000000

Note: The experimental correlation of the first 11 rounds is under sample sizes and 100 random keys, the experimental correlation of the 6 rounds at the DL part is under sample sizes and 100 random keys, the experimental correlation of the 9 rounds at the bottom is under sample sizes and 100 random keys. According to piling-up lemma, the experimental correlation is .