目的 传统误差扩散或恢复函数的多载体密图分存会对嵌密载体视觉质量造成较大影响，同时恢复函数需单独设计，只适用于二值或灰度图像，且通过简单Arnold置乱或异或加密仅能提供有限的安全性。针对此问题，提出结合EMD-c^l嵌入的多载体密图分存方法。方法 采用双哈希MD5和SHA-1值产生多组与密图属性和用户密钥有关的置乱参数，驱动2维双尺度矩形映射来改变载体像素对应关系，然后将置乱后载体同位置像素构成向量，按扩展约瑟夫遍历映射分配基向量，通过EMD-c^l嵌入秘密像素，从而将密图分存到多张载体中。结果 采用EMD-c^l提高了嵌密载体视觉质量且不需额外设计恢复函数，可针对不同分辨率和灰度阶密图分存。所提方法载体像素位置和EMD-c^l基向量都与密图MD5和SHA-1值以及用户密钥紧密绑定，仅有正确用户密钥和密图MD5和SHA-1值才能对密图恢复，并可通过第三方公信方托管的参与者分存信息MD5和SHA-1值使得所述策略具备认证能力。所提方法密钥空间为1.193 6×10¹¹⁸，可抵抗暴力破解。实验结果表明，结合EMD-c^l，所提方法具有较好的嵌密载体视觉质量，NC趋近于1，对于EMD-3^l，嵌密载体PSNR均接近50 dB；对于EMD-5^l和EMD-7^l，PSNR分别达到45 dB和42 dB，而传统方法，PSNR最好仅为42 dB。所提方法可分存不同分辨率和灰度阶密图，可对参与者密钥分存信息的真实性进行检验且对密图哈希和用户密钥极度敏感。结论 所提方法具有较低复杂度，较高安全性和普适性及认证能力，在整体性能上优于传统误差扩散或恢复函数的多载体密图分存方法，适用于对嵌密载体视觉质量要求高和针对不同分辨率和灰度阶密图分存的安全场景中。

Objective Conventional multi-carrier secret image sharing scheme based on error diffusion and recovery function can be used to hide secret image in multiple same size carriers. In these carriers, the error diffusion method leads to low visual quality and the recovery function is only designed for specific gray scale images, such as 1-bit binary images or 8-bit gray scale images. This scheme can only provide limited security by using simple Arnold scrambling or XOR encryption in which Arnold mapping is usually employed for special resolution images, such as square images. However, the security of Arnold mapping is poor due to its dependence on iteration numbers, and its scrambling coefficients are only 1, 1, 1, 2. In addition, Arnold mapping cannot change its mapping type and form by multiple iterations. To address these problems and further improve the visual qualities of carriers and security, a multi-carrier secret image sharing scheme with EMD-c^l embedding is proposed in this study. Method In sharing phase, the double hash function values (MD5 and SHA-1 values) are first used to randomly generate scrambling. Iteration variables of secret images and related user keys are used to change every carrier pixel position by 2D bi-scale rectangular mapping. Vectors composed of same position pixels in scrambled carriers are then randomly allocated to the weights of EMD-c^l basis vector using extended Josephus mapping where the start position and count termination value, count gap, and count direction sequences are added to increase the number of permutations and the kinds of variations in Josephus mapping. Afterward, EMD-c^l embedding strategy is used to embed secret image pixels into multiple carriers to verify the high visual qualities of stego carriers. Finally, both user keys and the double hash function values of secret image are distributed to N participant sharing information by Lagrange (N,N) under module p, where each participant sharing information is kept by one participant and the MD5 and SHA-1 values of each participant sharing information are known to the third trust party to guarantee that every participant sharing information cannot be faked. In recovering phase, all participant sharing information is verified by checking their MD5 and SHA-1 values known to the third trust party. This information is used to recover the related variables, such as the MD5 and SHA-1 values of secret image and the user keys by Lagrange (N,N) under module p. These related variables are used to generate groups of scrambling and iteration variables of a 2D bi-scale rectangular mapping and variables of extended Josephus mapping. Scrambling and iteration variables are used to find the mapping relationships between each secret pixel and N carrier pixels. Extended Josephus mapping variables are used to find the weights of EMD-c^l, and EMD-c^l is adopted to embed the secret image pixels. In recovering phase, if all participant sharing information passes the third trust party checking and the related variables are the same as those of the sharing phase, then the secret image can be recovered correctly. Result Unlike in conventional methods, the visual qualities of all stego carriers are enhanced because EMD-c^l only makes slight modifications, and the most modification for any non-overflow pixel is only 「±c/2」. The proposed scheme need not design any recovery function and can be easily applied in different resolutions or gray scale images. The following are the improved securities:1) The proposed strategy uses different 2D bi-scale rectangular mappings to scramble distinct carrier with varying iteration numbers to avoid the fixed relationship of carrier pixels. 2) A 2D bi-scale rectangular mapping can be used to scramble images in different resolution, and its multiple iterations differ in transform type and form, so this mapping is more complex and secure than Arnold mapping. 3) By introducing the double hash function values, that is, the MD5 and SHA-1 values of the secret image, the proposed strategy easily overcomes the conflict brought by the single hash value (MD5 or SHA-1 value). 4) The proposed strategy uses extended Josephus mapping to scramble weights of EMD-c^l basis vector, guaranteeing high steganography security. 5) All key variables in the proposed strategy, that is, the MD5 and SHA-1 values of the secret image, and the user keys are closely binded where only the correct user keys and MD5 and SHA-1 values can be used to recover the secret image but not vice versa. 6) To avoid tricks, the MD5 and SHA-1 values of key related variable sharing information are known to the third trusted party to provide authentication. 7) The total key space is 1.193 6×10¹¹⁸, which can resist brute-force attack. Experimental results show that, by EMD-c^l embedding, the proposed multi-carrier secret image sharing scheme has enhanced visual qualities in stego carriers. The NC value is close to 1. The PSNR values of all stego carriers are close to 50 dB by EMD-3^l embedding strategy. In EMD-5^l embedding strategy, the PSNR values of all stego carriers are close to 45 dB, and the PSNR values of all stego carriers are close to 42 dB for the EMD-7^l method. However, in conventional multi-carrier secret image sharing schemes, the best results of the PSNR values of all stego carriers are only approximately 42 dB. The proposed strategy can share different resolutions or different gray scale secret images, so it boasts high universality. Everyone can verify the facticity of any key sharing information provided by other participants. Moreover, the strategy is extremely sensitive to any slight modification in the user keys and hash function values:MD5 and SHA-1 or both of them. Therefore, the security of the proposed strategy is higher than that of conventional multi-carrier secret image sharing schemes. Conclusion The proposed method has low complexity and high security and universality, and it provides several authentication capabilities. Thus, the overall performance of the proposed method is superior to that of conventional multi-carrier secret image sharing schemes in terms of error diffusion and recovery function. The proposed method is suitable for multi-carrier secret image sharing schemes that need the high visual qualities of stego carriers and require sharing of secret images in different resolutions or gray scale secret.