Federated Learning(FL),a practical solution that leverages distributed data across devices without the need for centralized data storage,which enables multiple participants to jointly train models while preserving dat...Federated Learning(FL),a practical solution that leverages distributed data across devices without the need for centralized data storage,which enables multiple participants to jointly train models while preserving data privacy and avoiding direct data sharing.Despite its privacy-preserving advantages,FL remains vulnerable to backdoor attacks,where malicious participants introduce backdoors into local models that are then propagated to the global model through the aggregation process.While existing differential privacy defenses have demonstrated effectiveness against backdoor attacks in FL,they often incur a significant degradation in the performance of the aggregated models on benign tasks.To address this limitation,we propose a novel backdoor defense mechanism based on differential privacy.Our approach first utilizes the inherent out-of-distribution characteristics of backdoor samples to identify and exclude malicious model updates that significantly deviate from benign models.By filtering out models that are clearly backdoor-infected before applying differential privacy,our method reduces the required noise level for differential privacy,thereby enhancing model robustness while preserving performance.Experimental evaluations on the CIFAR10 and FEMNIST datasets demonstrate that our method effectively limits the backdoor accuracy to below 15%across various backdoor scenarios while maintaining high main task accuracy.展开更多
The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the po...The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.展开更多
The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory ...The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.展开更多
The two-dimensional steady flow of an incompressible second-order viscoelastic fluid between two parallel plates was studied in terms of vorticity, the stream function and temperature equations. The governing equation...The two-dimensional steady flow of an incompressible second-order viscoelastic fluid between two parallel plates was studied in terms of vorticity, the stream function and temperature equations. The governing equations were expanded with respect to a snmll parameter to get the zeroth- and first-order approximate equations. By using the differenl2al quadrature method with only a few grid points, the high-accurate numerical results were obtained.展开更多
Oscillation theorems for a second-order impulsive neutral differential equation are established, which extend the main results developed by Li et alLi et al, Oscillation of second order self-coajugate differential equ...Oscillation theorems for a second-order impulsive neutral differential equation are established, which extend the main results developed by Li et alLi et al, Oscillation of second order self-coajugate differential equation with impuls[es. J Comput Appl Math 197（2006）： 78-88] to the considered equation. Two examples are also inserted to illustrate our main results.展开更多
This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are ...This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.展开更多
In this paper we have established the stability of a generalized nonlinear second-order differential equation in the sense of Hyers and Ulam. We also have proved the Hyers-Ulam stability of Emden-Fowler type equation ...In this paper we have established the stability of a generalized nonlinear second-order differential equation in the sense of Hyers and Ulam. We also have proved the Hyers-Ulam stability of Emden-Fowler type equation with initial conditions.展开更多
This paper discusses a class of forced second-order half-linear differential equations. By using the generalized Riccati technique and the averaging technique, some new interval oscillation criteria are obtained.
In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the ...In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.展开更多
This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder...This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.展开更多
The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),w...The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.展开更多
In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a pol...In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a polynomial. The coefficients of the polynomial are then optimized using simulated annealing technique. Numerical examples with good results show the accuracy of the proposed approach compared with some existing methods.展开更多
In this paper, we consider the following second order retarded differential equations x″(t)+cx′(t)=qx(t-σ)-lx(t-δ) (1) x″(t)+p(t)x(t-τ)=0 (2) We give some sufficient conditions for the oscillation of all solutio...In this paper, we consider the following second order retarded differential equations x″(t)+cx′(t)=qx(t-σ)-lx(t-δ) (1) x″(t)+p(t)x(t-τ)=0 (2) We give some sufficient conditions for the oscillation of all solutions of Eq. (1) in the case where q, ι, σ, δ are positive numbers and c is a real number. And also, we study the asymptotic behavior of the nonoscillatory solutions. If necessary, we give some examples to illustrate our results. At last, we study Eq. (2) with some conditions on p(t).展开更多
After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. ...After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:展开更多
In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the origi...In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.展开更多
In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent opera...In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.展开更多
This paper presents a method for differen- tial collision attack of reduced FOX block cipher based on 4-round distinguishing property. It can be used to attack 5, 6 and 7-round FOX64 and 5-round FOX128. Our attack has...This paper presents a method for differen- tial collision attack of reduced FOX block cipher based on 4-round distinguishing property. It can be used to attack 5, 6 and 7-round FOX64 and 5-round FOX128. Our attack has a precomputation phase, but it can be obtained before attack and computed once for all. This attack on the reduced to 4-round FOX64 requires only 7 chosen plaintexts, and performs 242.8 4-round FOX64 encryptions. It could be extended to 5 （6, 7）-round FOX64 by a key exhaustive search behind the fourth round.展开更多
We investigate the lightweight block cipher KATAN family which consists of three variants with 32, 48 and 64-bit block sizes, called KATAN32, KATAN48 and KATAN64 respectively. However, three variants all have the same...We investigate the lightweight block cipher KATAN family which consists of three variants with 32, 48 and 64-bit block sizes, called KATAN32, KATAN48 and KATAN64 respectively. However, three variants all have the same key length of 80 bits. On the basis of the bit-oriented faulty model and the differential analysis principle, we describe the attack that combines differential fault attack with the meet-in-the-middle (MITM) attack on the KATAN32. More precisely, inducing a fault at a bit, we can recover some linear differential fault equations on the key bits. During solving equations, without the help of computer, we need only algebraic deduction to obtain relations of some key bits. The complexity in this process is neglectable. The secret key of the full cipher can be recovered faster than exhaustive search for all three block sizes in the KATAN family. Our result describes that KATAN32 is vulnerable.展开更多
In this paper we present an attack on 30-round SIMON64, which improves the best results on SIMON64 by 1 round. We use a 23-round differential characteristic which was proposed by Itai et al in 2015 to construct a 30-r...In this paper we present an attack on 30-round SIMON64, which improves the best results on SIMON64 by 1 round. We use a 23-round differential characteristic which was proposed by Itai et al in 2015 to construct a 30-round extended differential characteristized by adding 4 rounds on the top and 3 round on the bottom. Furthermore, we utilize all of the sufficient bit-conditions of the 30-round differential to compute a set of corresponding subkeys. Then we distribute the plaintext pairs over the 286 lists corresponding to the 86-bit subkeys. If a list contains two or more pairs, we regard the subkeys corresponding to the list as candidate subkeys. The time complexity of our attack on 30-round SIMON64/96 （SIMON64/128） is 286.2 （21182） with a success probability of 0.61, while the data complexity and the memory complexity are 263.3 and 290 bytes, respectively.展开更多
文摘Federated Learning(FL),a practical solution that leverages distributed data across devices without the need for centralized data storage,which enables multiple participants to jointly train models while preserving data privacy and avoiding direct data sharing.Despite its privacy-preserving advantages,FL remains vulnerable to backdoor attacks,where malicious participants introduce backdoors into local models that are then propagated to the global model through the aggregation process.While existing differential privacy defenses have demonstrated effectiveness against backdoor attacks in FL,they often incur a significant degradation in the performance of the aggregated models on benign tasks.To address this limitation,we propose a novel backdoor defense mechanism based on differential privacy.Our approach first utilizes the inherent out-of-distribution characteristics of backdoor samples to identify and exclude malicious model updates that significantly deviate from benign models.By filtering out models that are clearly backdoor-infected before applying differential privacy,our method reduces the required noise level for differential privacy,thereby enhancing model robustness while preserving performance.Experimental evaluations on the CIFAR10 and FEMNIST datasets demonstrate that our method effectively limits the backdoor accuracy to below 15%across various backdoor scenarios while maintaining high main task accuracy.
基金supported by the National Board for Higher Mathematics,Mumbai,India under Grant No.2/48(5)/2013/NBHM(R.P.)/RD-II/688 dt 16.01.2014
文摘The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.
基金Supported by the National Natural Science Foundation of China(11101096 )Guangdong Natural Science Foundation (S2012010010376, S201204006711)
文摘The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.
文摘The two-dimensional steady flow of an incompressible second-order viscoelastic fluid between two parallel plates was studied in terms of vorticity, the stream function and temperature equations. The governing equations were expanded with respect to a snmll parameter to get the zeroth- and first-order approximate equations. By using the differenl2al quadrature method with only a few grid points, the high-accurate numerical results were obtained.
基金Supported by the NSF of Guangdong Province(S2011010004447,S2012040006865)
文摘Oscillation theorems for a second-order impulsive neutral differential equation are established, which extend the main results developed by Li et alLi et al, Oscillation of second order self-coajugate differential equation with impuls[es. J Comput Appl Math 197（2006）： 78-88] to the considered equation. Two examples are also inserted to illustrate our main results.
文摘This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.
文摘In this paper we have established the stability of a generalized nonlinear second-order differential equation in the sense of Hyers and Ulam. We also have proved the Hyers-Ulam stability of Emden-Fowler type equation with initial conditions.
文摘This paper discusses a class of forced second-order half-linear differential equations. By using the generalized Riccati technique and the averaging technique, some new interval oscillation criteria are obtained.
文摘In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.
文摘This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.
基金This research is supported by the Shandong Provincial Natural Science Foundation of China(ZR2017MA043).
文摘The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.
文摘In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a polynomial. The coefficients of the polynomial are then optimized using simulated annealing technique. Numerical examples with good results show the accuracy of the proposed approach compared with some existing methods.
文摘In this paper, we consider the following second order retarded differential equations x″(t)+cx′(t)=qx(t-σ)-lx(t-δ) (1) x″(t)+p(t)x(t-τ)=0 (2) We give some sufficient conditions for the oscillation of all solutions of Eq. (1) in the case where q, ι, σ, δ are positive numbers and c is a real number. And also, we study the asymptotic behavior of the nonoscillatory solutions. If necessary, we give some examples to illustrate our results. At last, we study Eq. (2) with some conditions on p(t).
文摘After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:
文摘In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
文摘In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.
文摘In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.
基金This work has been performed in the Project "The Research on the New Analysis in Block Ciphers" supported by the Fundamental Research Funds for the Central Universities of China，the National Natural Science Foundation of China，the 111 Project of China，the Scientific Research Foundation of Education Department of Shaanxi Provincial Government of China
文摘This paper presents a method for differen- tial collision attack of reduced FOX block cipher based on 4-round distinguishing property. It can be used to attack 5, 6 and 7-round FOX64 and 5-round FOX128. Our attack has a precomputation phase, but it can be obtained before attack and computed once for all. This attack on the reduced to 4-round FOX64 requires only 7 chosen plaintexts, and performs 242.8 4-round FOX64 encryptions. It could be extended to 5 （6, 7）-round FOX64 by a key exhaustive search behind the fourth round.
基金the National Natural Science Foundation of China (No. 61272434)the Natural Science Foundation of Shandong Province (Nos. ZR2011FQ032 and ZR2012FM004)+1 种基金the Project of Shandong Province Higher Educational Science and Technology Program(No. J11LG33)the Project of Senior Visiting Scholar of Shandong Province
文摘We investigate the lightweight block cipher KATAN family which consists of three variants with 32, 48 and 64-bit block sizes, called KATAN32, KATAN48 and KATAN64 respectively. However, three variants all have the same key length of 80 bits. On the basis of the bit-oriented faulty model and the differential analysis principle, we describe the attack that combines differential fault attack with the meet-in-the-middle (MITM) attack on the KATAN32. More precisely, inducing a fault at a bit, we can recover some linear differential fault equations on the key bits. During solving equations, without the help of computer, we need only algebraic deduction to obtain relations of some key bits. The complexity in this process is neglectable. The secret key of the full cipher can be recovered faster than exhaustive search for all three block sizes in the KATAN family. Our result describes that KATAN32 is vulnerable.
基金Supported by the National Natural Science Foundation of China(61373142,61572125)Dissertation Innovation Funds(112-06-0019025)
文摘In this paper we present an attack on 30-round SIMON64, which improves the best results on SIMON64 by 1 round. We use a 23-round differential characteristic which was proposed by Itai et al in 2015 to construct a 30-round extended differential characteristized by adding 4 rounds on the top and 3 round on the bottom. Furthermore, we utilize all of the sufficient bit-conditions of the 30-round differential to compute a set of corresponding subkeys. Then we distribute the plaintext pairs over the 286 lists corresponding to the 86-bit subkeys. If a list contains two or more pairs, we regard the subkeys corresponding to the list as candidate subkeys. The time complexity of our attack on 30-round SIMON64/96 （SIMON64/128） is 286.2 （21182） with a success probability of 0.61, while the data complexity and the memory complexity are 263.3 and 290 bytes, respectively.
