Aim To study the Lie symmetries and the consered quantities of the holonomic systems with remainder coordinates. Methods Using the invariance of the ordinary differential equations under the infinitesimal transformati...Aim To study the Lie symmetries and the consered quantities of the holonomic systems with remainder coordinates. Methods Using the invariance of the ordinary differential equations under the infinitesimal transformations to establish the determining equations and the restriction equations of the Lie symmetries of the systems. Results and Conclusion the structure equation and the form of conserved quantities were obtained. An example was given to illustrate the application of the result.展开更多
Aerospace relay is one kind of electronic components which is used widely in national defense system and aerospace system. The existence of remainder particles induces the reliability declining, which has become a sev...Aerospace relay is one kind of electronic components which is used widely in national defense system and aerospace system. The existence of remainder particles induces the reliability declining, which has become a severe problem in the development of aerospace relay. Traditional particle impact noise detection （PIND） method for remainder detection is ineffective for small particles, due to its low precision and involvement of subjective factors. An auto-detection method for PIND output signals is proposed in this paper, which is based on direct wavelet de-noising （DWD）, cross-correlation analysis （CCA） and homo-filtering （HF）, the method enhances the affectivity of PIND test about the small particles. In the end, some practical PIND output signals are analysed, and the validity of this new method is proved.展开更多
A novel quantum secret sharing （QSS） scheme is proposed on the basis of Chinese Remainder Theorem （CRT）. In the scheme, the classical messages are mapped to secret sequences according to CRT equations, and distrib...A novel quantum secret sharing （QSS） scheme is proposed on the basis of Chinese Remainder Theorem （CRT）. In the scheme, the classical messages are mapped to secret sequences according to CRT equations, and distributed to different receivers by different dimensional superdense-coding respectively. CRT＇s secret sharing function, together with high-dimensional superdense-eoding, provide convenience, security, and large capability quantum channel for secret distribution and recovering. Analysis shows the security of the scheme.展开更多
This paper focuses on studying the relation between a velocity-dependent symmetry and a generalized Lutzky conserved quantity for a holonomic system with remainder coordinates subjected to unilateral constraints. The ...This paper focuses on studying the relation between a velocity-dependent symmetry and a generalized Lutzky conserved quantity for a holonomic system with remainder coordinates subjected to unilateral constraints. The differential equations of motion of the system are established, and the definition of Lie symmetry for the system is given. The conditions under which a Lie symmetry can directly lead up to a generalized Lutzky conserved quantity and the form of the new conserved quantity are obtained, and an example is given to illustrate the application of the results.展开更多
In this paper, firstly, the p order and pz order of Dirichlet series which converges in the whole plane are studied. Secondly, the equivalence relation between remainder logarithm In En-1 （f, α）, In Rn（f, α） and...In this paper, firstly, the p order and pz order of Dirichlet series which converges in the whole plane are studied. Secondly, the equivalence relation between remainder logarithm In En-1 （f, α）, In Rn（f, α） and coefficients logarithm In ｜an｜ is discussed respectively. Finally, the theory of applying remainder to estimate ρorder and ρβ order can be obtained by using the equivalence relation.展开更多
We propose an unbounded fully homomorphic encryption scheme, i.e. a scheme that allows one to compute on encrypted data for any desired functions without needing to decrypt the data or knowing the decryption keys. Thi...We propose an unbounded fully homomorphic encryption scheme, i.e. a scheme that allows one to compute on encrypted data for any desired functions without needing to decrypt the data or knowing the decryption keys. This is a rational solution to an old problem proposed by Rivest, Adleman, and Dertouzos [1] in 1978, and to some new problems that appeared in Peikert [2] as open questions 10 and open questions 11 a few years ago. Our scheme is completely different from the breakthrough work [3] of Gentry in 2009. Gentry’s bootstrapping technique constructs a fully homomorphic encryption (FHE) scheme from a somewhat homomorphic one that is powerful enough to evaluate its own decryption function. To date, it remains the only known way of obtaining unbounded FHE. Our construction of an unbounded FHE scheme is straightforward and can handle unbounded homomorphic computation on any refreshed ciphertexts without bootstrapping transformation technique.展开更多
In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubi...In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.展开更多
In this paper,some issues concerning the Chinese remaindering representation are discussed.A new converting method is described. An efficient refinement of the division algorithm of Chiu,Davida and Litow is given.
This paper takes further insight into the sparse geometry which offers a larger array aperture than uniform linear array(ULA)with the same number of physical sensors.An efficient method based on closed-form robust Chi...This paper takes further insight into the sparse geometry which offers a larger array aperture than uniform linear array(ULA)with the same number of physical sensors.An efficient method based on closed-form robust Chinese remainder theorem(CFRCRT)is presented to estimate the direction of arrival(DOA)from their wrapped phase with permissible errors.The proposed algorithm has significantly less computational complexity than the searching method while maintaining similar estimation precision.Furthermore,we combine all phase discrete Fourier transfer(APDFT)and the CFRCRT algorithm to achieve a considerably high DOA estimation precision.Both the theoretical analysis and simulation results demonstrate that the proposed algorithm has a higher estimation precision as well as lower computation complexity.展开更多
In the present paper two contents are enclosed .First ,the Fourier analysis approach of the dispersion relation and group velocity effect of finite difference schemes is discussed.the defects of the approach is pointe...In the present paper two contents are enclosed .First ,the Fourier analysis approach of the dispersion relation and group velocity effect of finite difference schemes is discussed.the defects of the approach is pointed out and the correction is made;Second,a new systematic analysis method -remaider -effect analysis (abbr.REAM)is proposed by means of the modified partial differential equations (abbr MPDE)of finite difference schemes.The analysis is based on the synthetical study of the rational dispersion-and dissipation relations of finite difference schemes.And the method clearly possesses constructivity展开更多
Chinese Remainder Codes are constructed by applying weak block designs and Chinese Remainder Theorem of ring theory. The new type of linear codes take the congruence class in the congruence class ring R/I 1∩I 2∩.....Chinese Remainder Codes are constructed by applying weak block designs and Chinese Remainder Theorem of ring theory. The new type of linear codes take the congruence class in the congruence class ring R/I 1∩I 2∩...∩I n for the information bit, embed R/J i into R/I 1∩I 2∩...∩I n, and asssign the cosets of R/J i as the subring of R/I 1∩I 2∩...∩I n and the cosets of R/J i in R/I 1∩I 2∩...∩I n as check lines. There exist many code classes in Chinese Remainder Codes, which have high code rates. Chinese Remainder Codes are the essential generalization of Sun Zi Codes.展开更多
A novel quantum group signature scheme is proposed based on Chinese Remainder Theorem (CRT), in order to improve the security of quantum signature. The generation and verification of the signature can be successfully ...A novel quantum group signature scheme is proposed based on Chinese Remainder Theorem (CRT), in order to improve the security of quantum signature. The generation and verification of the signature can be successfully conducted only if all the participants cooperate with each other and with the message owner's and the arbitrator's help. The quantum parallel algorithm is applied to efficiently compare the restored quantum message to the original quantum message. All the operations in signing and verifying phase can be executed in quantum circuits. It has a wide application to E-payment system, Online contract, Online notarization and etc.展开更多
Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth...Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .展开更多
High-precision vibration frequency acquisition and 3D full-field modal analysis of key rotating components is crucial and has garnered significant attention from the engineering community.However,this requires both a ...High-precision vibration frequency acquisition and 3D full-field modal analysis of key rotating components is crucial and has garnered significant attention from the engineering community.However,this requires both a high sampling frequency to accurately capture vibration frequencies of rotating components and a high spatial resolution to enable full-field modal analysis,making it challenging to achieve both simultaneously when cost control is required.To address these limitations,this study introduces the Chinese remainder theorem(CRT)as the undersampling reconstruction technique and digital image correlation(DIC)to realize high-spatial resolution.Building on this,we propose a framework that integrates 3D DIC,operational modal analysis,modal assurance criterion,and a generalized robust CRT,to enable the decoupling and reconstruction of multi-tone vibration signals,along with full-field modal analysis for rotating structures.The proposed scheme is verified by a comparison experiment with laser Doppler velocimetry and then applied to the full-field undersampling vibration measurement of a rotating disc using low-speed cameras(50 fps).In the experiment,28 harmonic responses and 5 natural responses were identified,demonstrating the scheme's reliability and effectiveness.Furthermore,response frequencies up to 729.22 Hz(error<0.0426 Hz)are identified,and sub-micron amplitude resolution is realized.This method increases the measurement range by 28.32 times and furthermore provides detailed vibration model analysis,providing a cost-effective,high-resolution solution with broad applicability for monitoring rotating vibrations in critical equipment components.展开更多
To avoid Doppler ambiguity,pulse Doppler radar may operate on a high pulse repetition frequency(PRF).The use of a high PRF can,however,lead to range ambiguity in many cases.At present,the major efficient solution to s...To avoid Doppler ambiguity,pulse Doppler radar may operate on a high pulse repetition frequency(PRF).The use of a high PRF can,however,lead to range ambiguity in many cases.At present,the major efficient solution to solve range ambiguity is based on a waveform design scheme.It adds complexity to a radar system.However,the traditional multiple-PRF-based scheme is difficult to be applied in multiple targets because of unknown correspondence between the target range and measured range,especially using the Chinese remainder theorem(CRT)algorithm.We make a study of the CRT algorithm for multiple targets when the residue set contains noise error.In this paper,we present a symmetry polynomial aided CRT algorithm to effectively achieve range estimation of multiple targets when the measured ranges are overlapped with noise error.A closed-form and robust CRT algorithm for single target and the Aitken acceleration algorithm for finding roots of a polynomial equation are used to decrease the computational complexity of the proposed algorithm.展开更多
If an adversary tries to obtain a secret s in a(t,n)threshold secret sharing(SS)scheme,it has to capture no less than t shares instead of the secret s directly.However,if a shareholder keeps a fixed share for a long t...If an adversary tries to obtain a secret s in a(t,n)threshold secret sharing(SS)scheme,it has to capture no less than t shares instead of the secret s directly.However,if a shareholder keeps a fixed share for a long time,an adversary may have chances to filch some shareholders’shares.In a proactive secret sharing(PSS)scheme,shareholders are supposed to refresh shares at fixed period without changing the secret.In this way,an adversary can recover the secret if and only if it captures at least t shares during a period rather than any time,and thus PSS provides enhanced protection to long-lived secrets.The existing PSS schemes are almost based on linear SS but no Chinese Remainder Theorem(CRT)-based PSS scheme was proposed.This paper proposes a PSS scheme based on CRT for integer ring to analyze the reason why traditional CRT-based SS is not suitable to design PSS schemes.Then,an ideal PSS scheme based on CRT for polynomial ring is also proposed.The scheme utilizes isomorphism of CRT to implement efficient share refreshing.展开更多
We give estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group. By considering the variations of associated functionals, we give a stability for two dual i...We give estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group. By considering the variations of associated functionals, we give a stability for two dual inequalities: The fractional Sobolev(FS) and Hardy-Littlewood-Sobolev(HLS) inequalities, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case, the remainder terms of Beckner-Onofri(BO) inequality and its dual logarithmic Hardy-Littlewood-Sobolev(Log-HLS) inequality. Besides, we also list without proof some results for other groups of Iwasawa-type. Our results generalize earlier works on Euclidean spaces of Chen et al.(2013) and Dolbeault and Jankowiak(2014) onto some groups of Heisenberg-type. We worked for "almost"all fractions especially for comparing results, and the stability of HLS is also absolutely new, even for Euclidean case.展开更多
We generalize the Chinese Remainder Theorem. use it to study number theory models, compare and analyse several number theory theorems in non-standard number theory models.
This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree sig...This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree significantly reduces signature size by utilizing polynomial and vector commitments.Compact proofs also accelerate the verification process,reducing computational overhead,which makes Verkle trees particularly useful.The study proposes a new approach based on a non-positional polynomial notation(NPN)employing the Chinese Remainder Theorem(CRT).CRT enables efficient data representation and verification by decomposing data into smaller,indepen-dent components,simplifying computations,reducing overhead,and enhancing scalability.This technique facilitates parallel data processing,which is especially advantageous in cryptographic applications such as commitment and proof construction in Verkle trees,as well as in systems with constrained computational resources.Theoretical foundations of the approach,its advantages,and practical implementation aspects are explored,including resistance to potential attacks,application domains,and a comparative analysis with existing methods based on well-known parameters and characteristics.An analysis of potential attacks and vulnerabilities,including greatest common divisor(GCD)attacks,approximate multiple attacks(LLL lattice-based),brute-force search for irreducible polynomials,and the estimation of their total number,indicates that no vulnerabilities have been identified in the proposed method thus far.Furthermore,the study demonstrates that integrating CRT with Verkle trees ensures high scalability,making this approach promising for blockchain systems and other distributed systems requiring compact and efficient proofs.展开更多
文摘Aim To study the Lie symmetries and the consered quantities of the holonomic systems with remainder coordinates. Methods Using the invariance of the ordinary differential equations under the infinitesimal transformations to establish the determining equations and the restriction equations of the Lie symmetries of the systems. Results and Conclusion the structure equation and the form of conserved quantities were obtained. An example was given to illustrate the application of the result.
基金Chinese Science Technology and Industry Foundation for National Defense(FEBG27100001)
文摘Aerospace relay is one kind of electronic components which is used widely in national defense system and aerospace system. The existence of remainder particles induces the reliability declining, which has become a severe problem in the development of aerospace relay. Traditional particle impact noise detection （PIND） method for remainder detection is ineffective for small particles, due to its low precision and involvement of subjective factors. An auto-detection method for PIND output signals is proposed in this paper, which is based on direct wavelet de-noising （DWD）, cross-correlation analysis （CCA） and homo-filtering （HF）, the method enhances the affectivity of PIND test about the small particles. In the end, some practical PIND output signals are analysed, and the validity of this new method is proved.
基金Supported by the National Natural Science Foundation of China under Grant No.60902044Ph.D.Programs Foundation of Ministry of Education of China under Grant No.20090162120070+2 种基金Postdoctoral Science Foundation of China under Grant No.200801341State Key Laboratory of Advanced Optical Communication Systems and Networks under Grant No.2008SH01in part by the Second stage of Brain Korea 21 programs,Chonbuk National University,Korea
文摘A novel quantum secret sharing （QSS） scheme is proposed on the basis of Chinese Remainder Theorem （CRT）. In the scheme, the classical messages are mapped to secret sequences according to CRT equations, and distributed to different receivers by different dimensional superdense-coding respectively. CRT＇s secret sharing function, together with high-dimensional superdense-eoding, provide convenience, security, and large capability quantum channel for secret distribution and recovering. Analysis shows the security of the scheme.
基金The project supported by National Natural Science Foundation of China under Grant No. 10272021 and the Natural Science Foundation of High Education Department of Jiangsu Province under Grant No. 04KJA130135
文摘This paper focuses on studying the relation between a velocity-dependent symmetry and a generalized Lutzky conserved quantity for a holonomic system with remainder coordinates subjected to unilateral constraints. The differential equations of motion of the system are established, and the definition of Lie symmetry for the system is given. The conditions under which a Lie symmetry can directly lead up to a generalized Lutzky conserved quantity and the form of the new conserved quantity are obtained, and an example is given to illustrate the application of the results.
基金Supported by the National Natural Science Foundation of China(11171119)Supported by the National Science Foundation of Jiangxi Province(20122BAB211005,2010GQS0103)
文摘In this paper, firstly, the p order and pz order of Dirichlet series which converges in the whole plane are studied. Secondly, the equivalence relation between remainder logarithm In En-1 （f, α）, In Rn（f, α） and coefficients logarithm In ｜an｜ is discussed respectively. Finally, the theory of applying remainder to estimate ρorder and ρβ order can be obtained by using the equivalence relation.
文摘We propose an unbounded fully homomorphic encryption scheme, i.e. a scheme that allows one to compute on encrypted data for any desired functions without needing to decrypt the data or knowing the decryption keys. This is a rational solution to an old problem proposed by Rivest, Adleman, and Dertouzos [1] in 1978, and to some new problems that appeared in Peikert [2] as open questions 10 and open questions 11 a few years ago. Our scheme is completely different from the breakthrough work [3] of Gentry in 2009. Gentry’s bootstrapping technique constructs a fully homomorphic encryption (FHE) scheme from a somewhat homomorphic one that is powerful enough to evaluate its own decryption function. To date, it remains the only known way of obtaining unbounded FHE. Our construction of an unbounded FHE scheme is straightforward and can handle unbounded homomorphic computation on any refreshed ciphertexts without bootstrapping transformation technique.
文摘In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.
文摘In this paper,some issues concerning the Chinese remaindering representation are discussed.A new converting method is described. An efficient refinement of the division algorithm of Chiu,Davida and Litow is given.
基金supported by the Fund for Foreign Scholars in University Research and Teaching Programs(the 111 Project)(B18039)
文摘This paper takes further insight into the sparse geometry which offers a larger array aperture than uniform linear array(ULA)with the same number of physical sensors.An efficient method based on closed-form robust Chinese remainder theorem(CFRCRT)is presented to estimate the direction of arrival(DOA)from their wrapped phase with permissible errors.The proposed algorithm has significantly less computational complexity than the searching method while maintaining similar estimation precision.Furthermore,we combine all phase discrete Fourier transfer(APDFT)and the CFRCRT algorithm to achieve a considerably high DOA estimation precision.Both the theoretical analysis and simulation results demonstrate that the proposed algorithm has a higher estimation precision as well as lower computation complexity.
文摘In the present paper two contents are enclosed .First ,the Fourier analysis approach of the dispersion relation and group velocity effect of finite difference schemes is discussed.the defects of the approach is pointed out and the correction is made;Second,a new systematic analysis method -remaider -effect analysis (abbr.REAM)is proposed by means of the modified partial differential equations (abbr MPDE)of finite difference schemes.The analysis is based on the synthetical study of the rational dispersion-and dissipation relations of finite difference schemes.And the method clearly possesses constructivity
文摘Chinese Remainder Codes are constructed by applying weak block designs and Chinese Remainder Theorem of ring theory. The new type of linear codes take the congruence class in the congruence class ring R/I 1∩I 2∩...∩I n for the information bit, embed R/J i into R/I 1∩I 2∩...∩I n, and asssign the cosets of R/J i as the subring of R/I 1∩I 2∩...∩I n and the cosets of R/J i in R/I 1∩I 2∩...∩I n as check lines. There exist many code classes in Chinese Remainder Codes, which have high code rates. Chinese Remainder Codes are the essential generalization of Sun Zi Codes.
文摘A novel quantum group signature scheme is proposed based on Chinese Remainder Theorem (CRT), in order to improve the security of quantum signature. The generation and verification of the signature can be successfully conducted only if all the participants cooperate with each other and with the message owner's and the arbitrator's help. The quantum parallel algorithm is applied to efficiently compare the restored quantum message to the original quantum message. All the operations in signing and verifying phase can be executed in quantum circuits. It has a wide application to E-payment system, Online contract, Online notarization and etc.
文摘Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .
基金supported by the National Science Foundation for Young Scientists of China(Grant No.12402220)the National Natural Science Foundation of China(Grant No.12232017)the National Science and Technology Major Project(Grant No.J2019-V-0006-0100)。
文摘High-precision vibration frequency acquisition and 3D full-field modal analysis of key rotating components is crucial and has garnered significant attention from the engineering community.However,this requires both a high sampling frequency to accurately capture vibration frequencies of rotating components and a high spatial resolution to enable full-field modal analysis,making it challenging to achieve both simultaneously when cost control is required.To address these limitations,this study introduces the Chinese remainder theorem(CRT)as the undersampling reconstruction technique and digital image correlation(DIC)to realize high-spatial resolution.Building on this,we propose a framework that integrates 3D DIC,operational modal analysis,modal assurance criterion,and a generalized robust CRT,to enable the decoupling and reconstruction of multi-tone vibration signals,along with full-field modal analysis for rotating structures.The proposed scheme is verified by a comparison experiment with laser Doppler velocimetry and then applied to the full-field undersampling vibration measurement of a rotating disc using low-speed cameras(50 fps).In the experiment,28 harmonic responses and 5 natural responses were identified,demonstrating the scheme's reliability and effectiveness.Furthermore,response frequencies up to 729.22 Hz(error<0.0426 Hz)are identified,and sub-micron amplitude resolution is realized.This method increases the measurement range by 28.32 times and furthermore provides detailed vibration model analysis,providing a cost-effective,high-resolution solution with broad applicability for monitoring rotating vibrations in critical equipment components.
基金supported by the Fund for Foreign Scholars in University Research and Teaching ProgramsChina(the 111 Project)(No.B18039)。
文摘To avoid Doppler ambiguity,pulse Doppler radar may operate on a high pulse repetition frequency(PRF).The use of a high PRF can,however,lead to range ambiguity in many cases.At present,the major efficient solution to solve range ambiguity is based on a waveform design scheme.It adds complexity to a radar system.However,the traditional multiple-PRF-based scheme is difficult to be applied in multiple targets because of unknown correspondence between the target range and measured range,especially using the Chinese remainder theorem(CRT)algorithm.We make a study of the CRT algorithm for multiple targets when the residue set contains noise error.In this paper,we present a symmetry polynomial aided CRT algorithm to effectively achieve range estimation of multiple targets when the measured ranges are overlapped with noise error.A closed-form and robust CRT algorithm for single target and the Aitken acceleration algorithm for finding roots of a polynomial equation are used to decrease the computational complexity of the proposed algorithm.
基金This work was supported by the National Natural Science Foundation of China(Grant No.61572454)National Key R&D Project(2018YFB2100301,2018YFB0803400)the National Natural Science Foundation of China(Grant Nos.61572453,61520106007).
文摘If an adversary tries to obtain a secret s in a(t,n)threshold secret sharing(SS)scheme,it has to capture no less than t shares instead of the secret s directly.However,if a shareholder keeps a fixed share for a long time,an adversary may have chances to filch some shareholders’shares.In a proactive secret sharing(PSS)scheme,shareholders are supposed to refresh shares at fixed period without changing the secret.In this way,an adversary can recover the secret if and only if it captures at least t shares during a period rather than any time,and thus PSS provides enhanced protection to long-lived secrets.The existing PSS schemes are almost based on linear SS but no Chinese Remainder Theorem(CRT)-based PSS scheme was proposed.This paper proposes a PSS scheme based on CRT for integer ring to analyze the reason why traditional CRT-based SS is not suitable to design PSS schemes.Then,an ideal PSS scheme based on CRT for polynomial ring is also proposed.The scheme utilizes isomorphism of CRT to implement efficient share refreshing.
基金supported by National Natural Science Foundation of China(Grant No.11371036)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.2012000110059)China Scholarship Council(Grant No.201306010009)
文摘We give estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group. By considering the variations of associated functionals, we give a stability for two dual inequalities: The fractional Sobolev(FS) and Hardy-Littlewood-Sobolev(HLS) inequalities, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case, the remainder terms of Beckner-Onofri(BO) inequality and its dual logarithmic Hardy-Littlewood-Sobolev(Log-HLS) inequality. Besides, we also list without proof some results for other groups of Iwasawa-type. Our results generalize earlier works on Euclidean spaces of Chen et al.(2013) and Dolbeault and Jankowiak(2014) onto some groups of Heisenberg-type. We worked for "almost"all fractions especially for comparing results, and the stability of HLS is also absolutely new, even for Euclidean case.
文摘We generalize the Chinese Remainder Theorem. use it to study number theory models, compare and analyse several number theory theorems in non-standard number theory models.
基金funded by the Ministry of Science and Higher Education of Kazakhstan and carried out within the framework of the project AP23488112“Development and study of a quantum-resistant digital signature scheme based on a Verkle tree”at the Institute of Information and Computational Technologies.
文摘This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree significantly reduces signature size by utilizing polynomial and vector commitments.Compact proofs also accelerate the verification process,reducing computational overhead,which makes Verkle trees particularly useful.The study proposes a new approach based on a non-positional polynomial notation(NPN)employing the Chinese Remainder Theorem(CRT).CRT enables efficient data representation and verification by decomposing data into smaller,indepen-dent components,simplifying computations,reducing overhead,and enhancing scalability.This technique facilitates parallel data processing,which is especially advantageous in cryptographic applications such as commitment and proof construction in Verkle trees,as well as in systems with constrained computational resources.Theoretical foundations of the approach,its advantages,and practical implementation aspects are explored,including resistance to potential attacks,application domains,and a comparative analysis with existing methods based on well-known parameters and characteristics.An analysis of potential attacks and vulnerabilities,including greatest common divisor(GCD)attacks,approximate multiple attacks(LLL lattice-based),brute-force search for irreducible polynomials,and the estimation of their total number,indicates that no vulnerabilities have been identified in the proposed method thus far.Furthermore,the study demonstrates that integrating CRT with Verkle trees ensures high scalability,making this approach promising for blockchain systems and other distributed systems requiring compact and efficient proofs.
