Image encryption based on one⁃dimensional quantum walks on a circle

输入: 待加密图片 (OIm) 和P-box2 输出: 整体置换后的图片 (OPIm)
1	[ $h$ $w$ ] ← size (OIm)
2	OIm1 ← reshape(OIm, 1, $h \times w$ ) // 将图片灰度矩阵转化为1 × ( $h \times w$ ) 的像素序列
3 4 5	for $i$ ← 1 to $h \times w$ do OIm2 ( $i$ ) ←OIm1 (P-box2 ( $i$ )) // 进行整体像素置换 endfor
6	OPIm ←OIm2 // 将像素置换后的序列重新转化为灰度矩阵

Table 0. [in Chinese]

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Table 0. [in Chinese]

输入: 一维圆上量子行走所需初始参数 ( $N_{1}$ , $t$ , $α$ , $β$ )
输出: 验证序列S
1	P₁ ← QW ( $N_{1}$ , $t$ , $α$ , $β$ ) // 运行一维圆上量子行走得到概率分布列矩阵
2	SP ← fix (P × 10^8) mod 256 // 得到的N₁维概率分布矩阵放大截取为0~255的整数
3	S ← dec2bin (SP) // 将得到的整数序列转化为二进制序列作为验证序列

Table 0. [in Chinese]
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Table 0. [in Chinese]
输入: 分块加密重组图片 (RIm), xor2
输出: 加密图片 (CIm)
1 CIm ← bitxor (RIm, xor2) //使用xor2 完成整体异或

Table 0. [in Chinese]

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Table 0. [in Chinese]

输入: 待加密图片 (OIm) 和一维圆上量子行走所需初始参数 ( $N$ , $t$ , $α$ , $β$ )
输出: 控制矩阵 (C-box), 替换盒 (P-box1, P-box2), 伪随机数矩阵 (xor1, xor2)
1	P ← QW ( $N$ , $t$ , $α$ , $β$ ) // 运行一维圆上量子行走得到 $N$ 维概率分布列矩阵
2	[ $h$ $w$ ] ← size (OIm) // 获得待加密图片大小 $h \times w$
	// 构建控制矩阵
3	CP ← Resize (P, [1, $h \times w / 256$ ]) // 调整概率分布矩阵尺寸为分块数量大小 $h \times w / 256$
4	C ← fix (CP × 10^8) mod 1 // 缩放后的概率分布矩阵放大截取为0和1的整数
5	C-box ← reshape (C, $h / 16$ , $w / 16$ ) // 将得到的0、1整数序列转化为方块矩阵形式
	// 构建替换盒P-box1
6	BP ← Resize (P, [1, 256]) // 缩放概率分布矩阵尺寸为256得到BP
7	B ← order (BP) // 将BP从小到大排序
8	P-box1 ← index(BP in B) // 对BP中排序前后的数进行对应索引组成替换盒
	// 构建替换盒P-box2
9	BP2 ← Resize (P, [1, $h \times w$ ]) // 缩放概率分布矩阵尺寸为 $h \times w$ 得到BP2
10	B2 ← order (BP2) // 将BP2从小到大排序
11	P-box2 ← index (BP2 in B2) // 对BP2中排序前后的数进行对应索引组成替换盒
	// 构建两个伪随机数矩阵
12	X1 ← fix (BP × 10^8) mod 256//将BP放大截取为0~255的整数组成的序列X1
13	xor1 ← reshape (Key, 16, 16) // 将X1转化为16 × 16方块矩阵xor1
14	X2 ← fix (BP2 × 10^8) mod 256 // 将BP2放大截取为0~255的整数组成的序列X2
15	xor2 ← reshape (Key2, $h$ , $w$ ) // 将X2转化为 $h \times w$ 方块矩阵xor2

Table 0. [in Chinese]

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Table 0. [in Chinese]

输入: 整体置换后的图片(OPIm), C-box, P-box1和X1
输出: 分块加密重组图片 (RIm)
1	[ $h$ $w$ ] ← size (OPIm)
2	Blocks[] ← 将 OPIm 分为16 × 16大小的分块图片
3	for each Block ← Blocks[] do // 对每个分块Blocks[ $i$ ] 标上序号 $i$
4	block ← reshape (Blocks[ $i$ ], 1, 256)
5	if C-box[ $i$ ] = 0 // 根据分块的序号 $i$ 获得控制矩阵中对应的值来决定加密方式
6	BX1 ← bit x or (block, X1)
7	for $j$ ← 1 to 256 do
8	BX2( $j$ ) ←BX1 (P-box1 ( $j$ ))
9	endfor
10	endif // 控制矩阵对应值为0的分块先异或后置换
11	if C-box[ $i$ ] = 1
12	for $j$ ← 1 to 256 do
13	BX1( $j$ ) ←block (P-box1 ( $j$ ))
14	endfor
15	BX2 ← bit xor (BX1, X1)
16	endif // 控制矩阵对应值为1的分块先置换后异或
17	Encblock[ $i$ ] ← reshape (BX2, 16, 16)
18	endfor
19	RIm ← combine Encblock into RIm image // 完成加密的分块根据序号重组

Table 1. Correlation analysis of encrypted image
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Table 1. Correlation analysis of encrypted image
Image Direction
Horizontal Vertical Diagonal
Baboon 0.8653 0.7523 0.7190
Enc-Baboon -0.0018 0.0014 0.0046
Lena 0.9691 0.9841 0.9557
Enc-Lena -0.0013 -0.0012 0.0034
Peppers 0.9755 0.9808 0.9625
Enc-Peppers -0.0005 0.0056 0.0008

Table 2. $R_{N P C}$ value of encrypted image
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Table 2. $R_{N P C}$ value of encrypted image
Image $R_{N P C}$ /%
Baboon 99.600
Lena 99.619
Peppers 99.606

Table 3. Comparison of average values of correlation coefficients and $R_{N P C}$ of the Lena, Baloon, Peppers with other similar schemes based on QWs.

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Table 3. Comparison of average values of correlation coefficients and $R_{N P C}$ of the Lena, Baloon, Peppers with other similar schemes based on QWs.

Algorithm	Correlation (average)			$R_{N P C}$ (average)/%
Algorithm	Horizontal Hertical Diagonal
proposed	-0.0012	0.0017	0.0029	99.608
Ref.[17]	0.0002	0.0014	0.0006	99.614
Ref.[18]	-0.0022	-0.0025	-0.0035	99.619
Ref.[23]	-0.0067	-0.0021	-0.0027	99.58
Ref.[24]	-0.0023	-0.0031	-0.0091	99.598

Table 4. Comparison of hash values of tiny modifications key and true key

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Table 4. Comparison of hash values of tiny modifications key and true key

Key	The corresponding hash values to tiny modifications of the key
Ture key	00FE00D6006200FA003500DA0089003A0055000E0063005A00A700BF004E00490082003000F100CB005100D800A900A0009F001E009B00B3000200BF00CC005500FB009E00E800F8006900C90034004B002B006D008300BB00820028023213499A2547BC8E34B81BFFF64691511BDEA246163222F5323FED71AA50C07CAC4BBACBDDEBD926BA9E76E574D10AFA03FA7BE7BA2F153B0219009D008E0037000A001D0067009E00E600FD00A8006D0035006C007900B400590015000600CA00B10052006000CB00BD0035004400E9002D00E1008A0095006B005F0006000000B000F800DB00AA005500C0009E0090001600CB
Ture key but $N$ =243 (Verification sequence)	5700A5002700910063009E008500C7001900B800F5004A00BA003D00E1002000D700BF007A0075009200CA00570016005200E8002F0031005C0091001E00F700B700A700340054007C0068002D0069005500B3000B007300EF001C0063005800EC00B203461CEEE6564EDE144F857F3C60E1C7E8483EB0DC0F14B2B576B46ABDE55AD2E5140FAA9524B41D8215CE203403830093002400E2006C00F7001C004A0028009B00DC009E003500AB00DC00D70070005B008100AB001700490027007E009C00B80037008600F5009F007D00F2001D0096008D002800CB0034009400A00058009900720064003800D9002E0089009300
Ture key but $t$ =264	6B000B0093007200C0000E00D900FB0027001200C100B9003A00D700CA005900A80001000D000D003C00F6004E003F00170046002A000F0092006C006F00E2000200350091001C00140075000400D4003900E6000D00AB00BC008200E903A51F18FDE5BE4AA3BAE2951D3614E8B3D21006B7EF0F9473C0061BB3DC267A77E6E7623BB0459F7F99F4690331EE4D2D1C6F435D3248304BF0062009E00AD0091000F00DF00CE003A0052008300C4009E006600ED001100AC008500AB003B008A00DB00C10026009F004A000200CD005D00BC007A0019005300CC00A700A100AC0025001A005A002100350026006000E0005E00
Ture key but $β = π / 6$	1D00B200810082049E3EA7E852503552C0F5422AF6CE14B0F80E50AA5A4E21AFB80C513920A0E20E0C2B8FDF5F9D21D574108A918081A4BCFB6E699922A7B80F83B98D9CF2D4B5E1169A8D908BFCD6E05FB35733457D0F977BA420A6704477D212B785F064DAC61871995C2B4183745382C29C375813CCE02B229F8A4F5EE4F022E1516D756C0E12D96B85CA9F8BD196EA4E0EB6CA99B7F8F8551F5AC23F4774A4F0756F16A33D5CAB9BA9A556EBE10960F41E33E5C31B0546D2CBE9910DCCEAF747CE30B3609E7783338FF4CBE000FEF377050E03DF304B5205AE36CDA976F067845EDC4E7F260A3F3312C601A

Table 5. $R_{N P C}$ values of the change key and the true key
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Table 5. $R_{N P C}$ values of the change key and the true key
Key $R_{N P C}$ /%
Ture key 0
Ture key but $N$ = 243 0.9959
Ture key but $t$ = 264 0.9956
Ture key but $β = π / 6$ 0.9956

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Wenjing LIU, Jun LIU, Yuan CAO, Pengyue WANG, Rong ZHANG. Image encryption based on one⁃dimensional quantum walks on a circle[J]. Chinese Journal of Quantum Electronics, 2024, 41(6): 956

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