An Image Encryption Algorithm Based on Hyperchaotic System and Variable-Step Josephus Problem

Table 12

Cipher randomness test.

Test items	Encrypted cameraman image	Encrypted Lena image	Encrypted white image
The frequency (monobit) test	0.0145	0.3445	0.0145
Frequency test within a block	0.5579	0.2092	0.5579
The runs test	0.9938	0.4367	0.9938
Tests for the longest-run-of-ones in a block	0.2798	0.7677	0.2798
The binary matrix rank test	0.0223	0.1924	0.0225
The discrete Fourier transform (spectral) test	0.9600	0.3087	0.9600
The nonoverlapping template matching test	0.0775	0.0478	0.0775
The overlapping template matching test	0.3450	0.5307	0.3450
Maurer’s “universal statistical” test	0.1878	0.0182	0.1878
The serial test	0.0161	0.4132	0.0161
	0.8952	0.6253	0.8952
The approximate entropy test	0.8776	0.9952	0.8776
The cumulative sums (CUSUMs) test	0.4752	0.8580	0.4752
	0.9431	0.6384	0.9431
The random excursions test	0.3075	0.0178	0.3075
	0.4309	0.0364	0.4309
	0.9371	0.0595	0.9371
	0.8469	0.0869	0.8469
	0.8645	0.2959	0.8645
	0.1806	0.6863	0.1806
	0.2538	0.7197	0.2538
	0.4315	0.4917	0.4315
The random excursions variant test	0.6441	0.2875	0.6441
	0.5550	0.3591	0.5550
	0.6597	0.3490	0.6597
	0.8183	0.3757	0.8183
	0.7190	0.3885	0.7190
	0.8103	0.1848	0.8103
	0.8203	0.0682	0.8203
	0.9415	0.0334	0.9415
	0.6567	0.0097	0.6567
	0.4849	0.2926	0.4849
	0.2126	0.9798	0.2126
	0.0649	0.7537	0.0649
	0.0187	0.7909	0.0187
	0.0049	0.9650	0.0049
	0.0005	0.8222	0.0005
	0.0009	0.7611	0.0009
	0.0091	0.6836	0.0091
	0.0349	0.5373	0.0349