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

Table 12

Thirty-one-round DL distinguisher for SIMON128 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	0000000000000000000000000000000000000000100010001000000000000000
	0000000000000000000000000000000000000010000000100000000000000000
10	0000000000000000000000000000000000000000100010001000000000000000
	0000000000000000000000000000000000000000001000000000000000000000
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
	−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
11	−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
	−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	−1.0
12	−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	−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
	−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
	−0.0	−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
13	−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	−0.4688
	−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
	−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	−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
14	−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	−0.0039
	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
	−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
	−0.75	−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	−0.4688
	−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
15	−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	−0.0
	−0.0	−0.00006	−0.0195	−0.0	−0.0	−0.0007	0.1342	−0.0
	−0.0	−0.1868	−0.6426	−0.0	−0.03125	−0.4688	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
	−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	−0.0039
	0.0623	−0.0	−0.0	−0.0039	0.1556	−0.0	−0.0	−0.0469
	0.875	−0.0	−0.0625	−0.46875	−0.875	−0.0	−0.25	−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
16	−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	−0.0390	−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
	0.0003	−0.0	−0.0		−0.0054	−0.0	−0.0	−0.0022
	0.3336	−0.0	−0.00098	−0.1133	0.6426	−0.0	−0.125	−1.0
	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−0.9688	−1.0	−1.0	−1.0	−0.9688	−1.0	−0.77	−0.8573
	−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	−0.017	−0.0	−0.00006	0.0024	−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
	−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
17	−0.4813	−0.5320	−0.2116	−0.1542	−0.2293	−0.4671	−0.1034	−0.1088
	−0.0409	−0.0255	−0.0249	−0.0807	−0.0035	−0.0038	−0.0018	−0.0023
	−0.0004	−0.0029			−0.0	−0.00003
	−0.0			−0.0	−0.0			−0.0
	−0.0		−0.00002	−0.0	−0.0		−0.0126	−0.0
	−0.0	−0.0106	0.2298	−0.0	−0.0020	−0.2637	−1.0	−0.0
	−0.9844	−1.0	−1.0	−1.0	−0.9844	−1.0	−0.8573	−0.9142
	−0.9844	−1.0	−0.8573	−0.9142	−0.5413	−0.5320	−0.7248	−0.9142
	−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	−0.0390	−0.0530
	−0.0143	−0.0117	−0.0086	−0.0424	−0.00025	−0.0008	−0.0	−0.0006
	−0.00002	−0.0007	−0.0		0.00001	−0.0	−0.0
	0.0003	−0.0	−0.0		−0.0054	−0.0	−0.0	−0.0022
	0.3336	−0.0	−0.0010	−0.1133	0.6426	−0.0	−0.125	−1.0
	1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0	−1.0
	−0.9688	−1.0	−1.0	−1.0	−0.9688	−1.0	−0.77	−0.8573
18	−0.0817	−0.0479	−0.0522	−0.1330	−0.0155	−0.0121	−0.0069	−0.0057
	−0.0021	−0.0083	−0.0001	−0.00008	−0.00006	−0.0002		−0.00003
			−0.0				−0.0
		−0.0	−0.0			−0.0	−0.0
		−0.0	−0.0		−0.00001	−0.0	−0.0
	−0.0384	−0.0		−0.0150	0.4028	−0.0	−0.0571	−0.9496
	0.9922	−1.0	−0.9142	−0.9496	−0.6607	−0.6503	−0.8125	−0.9496
	−0.6151	−0.6503	−0.3138	−0.2366	−0.3306	−0.5783	−0.1957	−0.1873
	−0.4813	−0.5320	−0.2116	−0.1542	−0.2293	−0.4671	−0.1034	−0.1088
	−0.0409	−0.0255	−0.0249	−0.0807	−0.0035	−0.0038	−0.0018	−0.0023
	−0.0004	−0.0029			−0.0	−0.00003
	−0.0			−0.0	−0.0			−0.0
	−0.0		−0.00002	−0.0	−0.0		−0.0126	−0.0
	−0.0	−0.0106	0.2298	−0.0	−0.0020	−0.2637	−1.0	−0.0
	−0.9844	−1.0	−1.0	−1.0	−0.9844	−1.0	−0.8573	−0.9142
	−0.9844	−1.0	−0.8573	−0.9142	−0.5413	−0.5320	−0.7248	−0.9142
19	−0.0066	−0.0188	−0.0009	−0.0005	−0.0004	−0.0007	−0.00005	−0.0002
	−0.0				−0.0			−0.0
	−0.0			−0.0	−0.0			−0.0
	−0.0			−0.0	−0.0		−0.0001	−0.0
	−0.0	−0.00007	−0.0450	−0.0	−0.00004	−0.1092	0.8765	−0.0
	−0.7267	−0.7499	−0.4231	−0.3339	−0.4391	−0.6791	−0.3073	−0.2850
	−0.1379	−0.0814	−0.0922	−0.1992	−0.0424	−0.0301	−0.0177	−0.0119
	−0.0817	−0.0479	−0.0522	−0.1330	−0.0155	−0.0121	−0.0069	−0.0057
	−0.0021	−0.0083	−0.0001	−0.00008	−0.00006	−0.0002		−0.00003
				−0.0
		−0.0	−0.0			−0.0	−0.0
		−0.0	−0.0		−0.00001	−0.0	−0.0
	−0.0384	−0.0		−0.0150	0.4028	−0.0	−0.0571	−0.9496
	0.9922	−1.0	−0.9142	−0.9496	−0.6607	−0.6503	−0.8125	−0.9496
	−0.6151	−0.6503	−0.3138	−0.2366	−0.3306	−0.5783	−0.1957	−0.1873
Linear part
	0000000000000000000000000000000000000000000000000000010000000000
	0000000000000000000000000000000000000000000000010001000000000000
20	0000000000000000000000000000000000000000000000010001000000000000
	0000000000000000000000000000000000000000000000000100000000000000
21	0000000000000000000000000000000000000000000000000100000000000000
	0000000000000000000000000000000000000000000000010000000000000000
22	0000000000000000000000000000000000000000000000010000000000000000
	0000000000000000000000000000000000000000000000000000000000000000
23	0000000000000000000000000000000000000000000000000000000000000000
	0000000000000000000000000000000000000000000000010000000000000000
24	0000000000000000000000000000000000000000000000010000000000000000
	0000000000000000000000000000000000000000000000000100000000000000
25	0000000000000000000000000000000000000000000000000100000000000000
	0000000000000000000000000000000000000000000000010001000000000000
26	0000000000000000000000000000000000000000000000010001000000000000
	0000000000000000000000000000000000000000000000000000010000000000
27	0000000000000000000000000000000000000000000000000000010000000000
	0000000000000000000000000000000000000000000000010001000100000000
28	0000000000000000000000000000000000000000000000010001000100000000
	0000000000000000000000000000000000000000000000000100000001000000
29	0000000000000000000000000000000000000000000000000100000001000000
	0000000000000000000000000000000000000000000000010000000100010000
30	0000000000000000000000000000000000000000000000010000000100010000
	0000000000000000000000000000000000000000000000000000000000000100
31	0000000000000000000000000000000000000000000000000000000000000100
	0000010000000000000000000000000000000000000000010000000100010001

Note: The experimental correlation of the first 10 rounds is under sample sizes and 100 random keys, and the experimental correlation of the 9 rounds at the DL part is under sample sizes and 100 random keys. According to piling-up lemma, the experimental correlation is .