Given a k×n integer primitive matrix A(i.e.,a matrix can be extended to an n×n unimodular matrix over the integers)with the maximal absolute value of entries‖A‖bounded by an integer λ from above,the autho...Given a k×n integer primitive matrix A(i.e.,a matrix can be extended to an n×n unimodular matrix over the integers)with the maximal absolute value of entries‖A‖bounded by an integer λ from above,the authors study the probability that the m×n matrix extended from A by appending other m-k row vectors of dimension n with entries chosen randomly and independently from the uniform distribution over{0,1,…,λ-1}is still primitive.The authors present a complete and rigorous proof of a lower bound on the probability,which is at least a constant for fixed m in the range[k+1,n-4].As an application,the authors prove that there exists a fast Las Vegas algorithm that completes a k×n primitive matrix A to an n×n unimodular matrix within expected O(n^(ω)log‖A‖)bit operations,where O is big-O but without log factors,ω is the exponent on the arithmetic operations of matrix multiplication.展开更多
A new meaningful image encryption algorithm based on compressive sensing(CS)and integer wavelet transformation(IWT)is proposed in this study.First of all,the initial values of chaotic system are encrypted by RSA algor...A new meaningful image encryption algorithm based on compressive sensing(CS)and integer wavelet transformation(IWT)is proposed in this study.First of all,the initial values of chaotic system are encrypted by RSA algorithm,and then they are open as public keys.To make the chaotic sequence more random,a mathematical model is constructed to improve the random performance.Then,the plain image is compressed and encrypted to obtain the secret image.Secondly,the secret image is inserted with numbers zero to extend its size same to the plain image.After applying IWT to the carrier image and discrete wavelet transformation(DWT)to the inserted image,the secret image is embedded into the carrier image.Finally,a meaningful carrier image embedded with secret plain image can be obtained by inverse IWT.Here,the measurement matrix is built by both chaotic system and Hadamard matrix,which not only retains the characteristics of Hadamard matrix,but also has the property of control and synchronization of chaotic system.Especially,information entropy of the plain image is employed to produce the initial conditions of chaotic system.As a result,the proposed algorithm can resist known-plaintext attack(KPA)and chosen-plaintext attack(CPA).By the help of asymmetric cipher algorithm RSA,no extra transmission is needed in the communication.Experimental simulations show that the normalized correlation(NC)values between the host image and the cipher image are high.That is to say,the proposed encryption algorithm is imperceptible and has good hiding effect.展开更多
Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that ...Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Consequently,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤n.Finally,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Our results extend and strengthen the theorems of Hong obtained in 2008.展开更多
基金supported by the National Key Research and Development Project under Grant No.2020YFA0712303National Science Foundational of China under Grant No.61903053+1 种基金Youth Innovation Promotion Association of CAS,Guizhou Science and Technology Program under Grant No.[2020]4Y056Chongqing Science and Technology Program under Grant Nos.cstc2021jcyj-msxm X0821,cstc2020yszxjcyj X0005,cstc2021yszx-jcyj X0004,and 2022YSZX-JCX0011CSTB。
文摘Given a k×n integer primitive matrix A(i.e.,a matrix can be extended to an n×n unimodular matrix over the integers)with the maximal absolute value of entries‖A‖bounded by an integer λ from above,the authors study the probability that the m×n matrix extended from A by appending other m-k row vectors of dimension n with entries chosen randomly and independently from the uniform distribution over{0,1,…,λ-1}is still primitive.The authors present a complete and rigorous proof of a lower bound on the probability,which is at least a constant for fixed m in the range[k+1,n-4].As an application,the authors prove that there exists a fast Las Vegas algorithm that completes a k×n primitive matrix A to an n×n unimodular matrix within expected O(n^(ω)log‖A‖)bit operations,where O is big-O but without log factors,ω is the exponent on the arithmetic operations of matrix multiplication.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.61972103,61772371,62172301)the Natural Science Foundation of Guangdong Province of China(2019A1515011361)+2 种基金the Fundamental Research Funds for the Central Universities of China(22120210545)the Key Scientific Research Project of Education Department of Guangdong Province of China(2020ZDZX3064)the Postgraduate Education Innovation Project of Guangdong Ocean University of China(202143).
文摘A new meaningful image encryption algorithm based on compressive sensing(CS)and integer wavelet transformation(IWT)is proposed in this study.First of all,the initial values of chaotic system are encrypted by RSA algorithm,and then they are open as public keys.To make the chaotic sequence more random,a mathematical model is constructed to improve the random performance.Then,the plain image is compressed and encrypted to obtain the secret image.Secondly,the secret image is inserted with numbers zero to extend its size same to the plain image.After applying IWT to the carrier image and discrete wavelet transformation(DWT)to the inserted image,the secret image is embedded into the carrier image.Finally,a meaningful carrier image embedded with secret plain image can be obtained by inverse IWT.Here,the measurement matrix is built by both chaotic system and Hadamard matrix,which not only retains the characteristics of Hadamard matrix,but also has the property of control and synchronization of chaotic system.Especially,information entropy of the plain image is employed to produce the initial conditions of chaotic system.As a result,the proposed algorithm can resist known-plaintext attack(KPA)and chosen-plaintext attack(CPA).By the help of asymmetric cipher algorithm RSA,no extra transmission is needed in the communication.Experimental simulations show that the normalized correlation(NC)values between the host image and the cipher image are high.That is to say,the proposed encryption algorithm is imperceptible and has good hiding effect.
基金supported partially by Doctoral Research Initiation FundProjectof PanzhihuaUniversity(bkqj2019050).
文摘Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Consequently,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤n.Finally,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Our results extend and strengthen the theorems of Hong obtained in 2008.
