An Unbounded Fully Homomorphic Encryption Scheme Based on Ideal Lattices and Chinese Remainder Theorem

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.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

THE REMAINDER－EFFECT ANALYSIS OF FINITE DIFFERENCE SCHEMES AND THE APPLICATIONS

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

Research on Auto-detection for Remainder Particles of Aerospace Relay Based on Wavelet Analysis

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.

The Numbers of Thousand Place of Mersenne Primes

Mersenne primes are a special kind of primes, which are an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. Searching for Mersenne primes is very challenging in scientific researches. In this paper, the numbers of thousand place of Mersenne primes are studied, and the conclusion is presented by using the Chinese remainder theorem.

Quantum Secret Sharing Based on Chinese Remainder Theorem

分享的一个新奇的量秘密(QSS ) 计划根据中国剩余物定理(CRT ) 被建议。在计划，古典消息根据 CRT 方程被印射到秘密序列，并且分别地由不同维的编码 superdense 散布了到不同接收装置。CRT 的秘密分享功能和高度维的编码 superdense，为秘密分发提供便利，安全，和大能力量隧道并且恢复。分析显示出计划的安全。

Generalized Lutzky Conserved Quantities of Holonomic Systems with Remainder Coordinates Subjected to Unilateral Constraints

<正> This paper focuses on studying the relation between a velocity-dependent symmetry and a generalizedLutzky conserved quantity for a holonomic system with remainder coordinates subjected to unilateral constraints.Thedifferential 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 theform of the new conserved quantity are obtained,and an example is given to illustrate the application of the results.

Robustness Proof on A United Watermarking Based on CRT Theorem

A new method of embedding and detecting a joint watermarking is proposed. Itapplies the asmuth-bloom secret sharing scheme, which is based on CRT (Chinese remainder theorem)theorem, to the digital watermarking technology. On the base of describing the watermarkingembedding proceeding and analyzing the watermarking detection proceeding, a series of experiments isdone. The experiments emphasize on the method's robust proving and security analysis. And theexperiments show that the method can resistthe attacks of JPEG compress, geometry, noise and grayadjusting. The results of the experiments show that the method has a nice recognition of copyrightfor joint ownership.

ASIAN CARP EXPANSION IN THE MISSISSIPPI RIVER:FOCUSING ON THE LEADING EDGE OF THE INVASION FRONT

Asian carp are expanding their range throughout the Mississippi River; however, abundance is thought to be highest in reaches within close proximity to the Illinois River. In the Mississippi River, Lock and Dam 19(L&D 19) at Keokuk, Iowa is the primary barrier to slow the expansion upstream. As Asian carp abundance increases below L&D 19, it is important to investigate potential means of control(i.e., reduction through harvest and barriers) that will prevent complete invasion of the Mississippi River above L&D 19. Silver Carp and Bighead Carp were collected below L&D 19, a subsample were implanted with ultrasonic transmitters to evaluate passage through the lock chamber and the remainder were used to determine population dynamics at the leading edge of invasion. Although the dam portion of the structure poses a complete barrier to upstream expansion, we documented lock chamber passage demonstrating the lock chamber has the capability to provide passage upstream. Based on the results of the population assessment, in order to induce recruitment overfishing at this leading edge of invasion, Asian carp will need to be intensively harvested at 300 mm and larger. The combination of commercial fishing efforts and research investigating ways to prevent passage upstream must be employed.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

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.

A method based on Chinese remainder theorem with all phase DFT for DOA estimation in sparse array

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.

Quantum Group Signature Scheme Based on Chinese Remainder Theorem

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.

What the Volume of Matrices Says?

We know matrices and their transposes and we also know flip matrices. In my previous paper Matrices-One Review, I introduced transprocal matrix. Flip matrices are transpose of transprocal matrices. Now I would like to introduce water image of four matrices said above and properties of such matrices. Also we know, determinant of sum of matrices is not equal to sum of determinant of matrices. Why can’t we get equal value on addition side and additive side of determinant of matrix addition and subtraction? This question triggered me to find the reason. The basic algebra of mensuration gave ideas to retreat determinant of matrix addition and subtraction. I extent that ideas for matrices sum. Further, in 1812, French mathematician Jacques Philippe Marie Binet described how to multiply matrices. Matrices are defined on addition, subtraction and multiplication but not in division. By the inspiration of Binet, I would like to describe how to do divisions on matrices. This idea is derived from division of fractions. In division of fraction, reciprocal of divisor fraction multiplies with dividend fraction. I do the same in division on matrices with some modifications. By this way, we could find quotient matrix and remainder matrix which satisfy division algorithm. So we could say, determinant of division of dividend matrix and divisor matrix is equal to division of determinant of dividend matrix and determinant of divisor matrix.

Design of an Efficient Binary Vedic Multiplier for High Speed Applications Using Vedic Mathematics with Bit Reduction Technique

Vedic mathematics is the system of mathematics followed in ancient Indian and it is applied in various mathematical branches. The word “Vedic” represents the storehouse of all knowledge. Because using Vedic Mathematics, the arithmetical problems are solved easily. The mathematical algorithms are formed from 16 sutras and 13 up-sutras. But there are some limitations in each sutra. Here, two sutras Nikhilam sutra and Karatsuba algorithm are considered. In this research paper, a novel algorithm for binary multiplication based on Vedic mathematics is designed using bit reduction technique. Though Nikhilam sutra is used for multiplication, it is not used in all applications. Because it is special in multiplication. The remainder is derived from this sutra by reducing the remainder bit size to N-2 bit. Here, the number of bits of the remainder is constantly maintained as N-2 bits. By using Karatsuba algorithm, the overall structure of the multiplier is designed. Unlike the conventional Karatsuba algorithm, the proposed algorithm requires only one multiplier with N-2 bits only. The speed of the proposed algorithm is improved with balancing the area and the power. Even though there is a deviation in lower order bits, this method shows larger difference in higher bit lengths.

Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations

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. .

密封电子元器件与装置多余物检测发展综述

密封电子元器件与装置作为航天系统中的重要组成部分,具有高精密性、高可靠性以及高复杂性的典型特点,而多余物问题是严重影响其高可靠性的主要因素之一。随着航天技术的快速发展,多余物检测技术也在不断改进与深入。该文以认识多余物、控制多余物产生以及检测多余物为主线,围绕多余物的防控方法、检测方法、检测标准进行综合论述。着重分析了检测方法中的颗粒碰撞噪声检测(PIND)方法,分别从小型元器件多余物检测、中大型装置多余物检测两个角度,针对现有研究进展逐一进行了详细的介绍。在此基础上,结合现有国内多余物检测研究的现状,针对多余物检测的难点与未来发展趋势进行归纳,并提出了期望与目标。

定积分在两点展开的渐近公式

对于具有m+n阶连续导数的被积函数的定积分,通过多次分部积分给出了在积分上限和积分下限两点处同时展开的定积分的渐近展开式,其余项类似于Taylor公式的积分型余项,这种展开式可看作是Taylor公式的一种推广.被积函数在积分上下限处的值有不同情形,展开式也会随之变化而具有多种形式.通过分析与举例也发现这种展开式的近似计算优于Taylor公式的近似计算,而且在某些积分不等式的证明中也体现了其快捷方便的优点.

基于中国剩余定理的NFC安全认证算法

针对近场通信技术在应用中出现的安全隐患问题,给出一种基于中国剩余定理的算法。算法利用中国剩余定理实现对传送信息进行加密,中国剩余定理基于数学中大素数分解难题,使得攻击者无法进行破解;所有信息加密过程中混入随机数,用于保证消息的新鲜性;算法在进行信息更新时采用伪随机函数计算,因伪随机函数具备的单向性,使得攻击者无法分析出有用隐私信息。将不同算法对比安全分析,表明该算法能够抵抗重放攻击、异步攻击等多种攻击。通过性能角度及仿真实验对多个算法进行分析,结果表明该算法计算时间复杂度低于其他算法。

基于卷积—反残差和组合注意力机制的航天器多余物检测

航天器密闭电子设备内腔多余物给航天器飞行安全带来了巨大隐患。由于多余物体积小、与设备内常规组件形态结构相似且易被其他组件遮挡,采用现有的方法对其进行检测时误检、漏检频发。为解决上述问题,提出一种基于卷积—反残差和组合注意力机制的航天器密闭电子设备多余物检测网络RPDN。首先,网络通过构建卷积—反残差模块,保证了多余物细粒度特征的完整性;其次,设计组合注意力机制,增强了多余物特征的表征能力;最后,结合多尺度特征融合模块与目标检测层从多维度进行目标预测。实验结果表明RPDN在各项评价指标上均取得了良好的效果,mAP达到92.16%,检测效率达到了13FPS,实现了航天器密闭电子设备内腔多余物高效、精准检测。

分布式智能车载网联系统的匿名认证与密钥协商协议

智能车载网联系统作为智慧城市建设的重要组成部分,近年来受到学术界与工业界越来越多的关注。智能车载网联系统中提升了智能车辆的行驶安全性与出行效率,但在开放的环境下数据传输容易被截取,造成敏感信息泄漏。因此需要实现匿名认证并且协商正确的会话密钥,来确保智能车载网联系统敏感信息的安全。该文提出面向分布式智能车载网联系统架构的匿名认证与密钥协商协议。该协议基于中国剩余定理秘密分享技术来保护认证标识符,智能车辆能够以线性的计算开销在不同的区域恢复出对应的标识符,该标识符能够长期安全使用且智能车辆能够在不使用防篡改设备的情况下完成安全认证,路侧通信基站能够检测信息的匿名性和完整性,并与智能车辆协商到后续安全通信的会话密钥,同时实现双向认证。此外,协议能够在复杂的分布式智能车载网联系统中拓展批量匿名认证、域密钥更新、车对车的匿名认证、匿名身份可追踪等实用性功能。安全性与性能分析表明该协议能够安全高效地部署在分布式智能车载网联环境。