Focused on the problems of no protection for the number of zero elements in original matrix and no verification for the result returned by cloud in outsourcing algorithm of matrix full rank decomposition, a verifiable and secure outsourcing scheme of matrix full rank decomposition was proposed. Firstly, in the phase of encryption, a dense invertible matrix was constructed by using the Sherman-Morrison formula for encryption. Secondly, in the phase of cloud computing, the cloud computing of the full rank decomposition for the encryption matrix was required. And when the results of full rank decomposition for encryption matrix (a column full rank matrix and a row full rank matrix) were obtained, the cloud computing of the left inverse of the column full rank matrix and the right inverse of the row full rank matrix was required respectively. Thirdly, in the phase of verification, the client not only needed to verify whether these two matrices returned by cloud are row-full-rank or column-full-rank respectively, but also needed to verify whether the multiplication of these two matrices is equal to the encryption matrix. Finally, if the verification was passed, the client was able to use the private key to perform the decryption. In the protocol analysis, the proposed scheme is proved to satisfy correctness, security, efficiency, and verifiability. At the same time, when the dimension of the selected original matrix is 512×512, with different densities of non-zero elements in the matrix, the entropy of the encryption matrix calculated by this scheme is identically equal to 18, indicating that the scheme can protect the number of zero elements effectively. Experimental results show the effectiveness of the proposed scheme.

The arithmetic correlation function is a new method for studying the cryptographic properties of Boolean functions. Based on the basic definitions of addition and multiplication of multi-2-adic integer, the study constructed a new algebraic ring and realized the arithmetic or “with-carry” analogs of classic correlation functions. In this paper the definition of arithmetic autocorrelation function was introduced. The arithmetic correlation value of symmetric Boolean functions was studied. The results show that the arithmetic autocorrelation function of symmetric Boolean functions is a real symmetric function with at most n1 values.