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.

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. .

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

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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.

Fast Arithmetics Using Chinese Remaindering

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.

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.

Range estimation based on symmetry polynomial aided Chinese remainder theorem for multiple targets in a pulse Doppler radar

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.

The Generalization of the Chinese Remainder Theorem

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.

A proactive secret sharing scheme based on Chinese remainder theorem

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.

全平面上Dirichlet级数的级与余项估计（英文）

In this paper, firstly, the ρ order and ρβorder of Dirichlet series which converges in the whole plane are studied. Secondly, the equivalence relation between remainder logarithm ln En-1(f, α), ln Rn(f, α) and coefficients logarithm ln |an| is discussed respectively. Finally,the theory of applying remainder to estimate ρ order and ρβorder can be obtained by using the equivalence relation.

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.

Informix数据库分片策略的大规模应用研究及相关问题的探讨

采用数据库日期分区策略，成功解决了大规模应用中的批量数据清理的问题，对于行宽超过32K，记录数超过200万行的大数据量库表数据的清理操作，处理时间由过去的（4～6）h，缩短到2s钟以内，消除了批量处理过程对生产系统的影响，保证系统实现24h不间断运行。该方案应用透明，方便应用系统的方案改造升级，具有极高的推广价值。

A FUNCTION MINIMIZATION CALCULATION OF CYCLONE 10 MAGNETIC FIELDS

By app(?)ing function minimization calculation method, two function expressions are used to simulate the magnetic field measured for cyclotron Cyclone 10 in azimuth and radius The numerical fitting curves are consistent with magnetic field measured. In most pl(?)es, the accuracies are several thousandth, except those errors to be pointed out in paper.

PUMTD:Privacy-Preserving User-Profile Matching Protocol in Social Networks

User profile matching can establish social relationships between different users in the social network.If the user profile is matched in plaintext,the user's privacy might face a security challenge.Although there exist some schemes realizing privacypreserving user profile matching,the resource-limited users or social service providers in these schemes need to take higher computational complexity to ensure the privacy or matching of the data.To overcome the problems,a novel privacy-preserving user profile matching protocol in social networks is proposed by using t-out-of n servers and the bloom filter technique,in which the computational complexity of a user is reduced by applying the Chinese Remainder Theorem,the matching users can be found with the help of any t matching servers,and the privacy of the user profile is not compromised.Furthermore,if at most t-1 servers are allowed to collude,our scheme can still fulfill user profile privacy and user query privacy.Finally,the performance of the proposed scheme is compared with the other two schemes,and the results show that our scheme is superior to them.

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.

关于数值微分公式的两个注记(英文)

Some new conclusions on asymptotic properties and inverse problems of numerical differentiation formulae have been drawn in this paper.In the first place,several asymptotic properties of intermediate points of numerical differentiation formulae are presented by using Taylor's formula.And then,based on the ideas of algebraic accuracy,several inverse problems of numerical differentiation formulae are given.

New Reverse Conversion for Four-Moduli Set and Five-Moduli Set

Most reverse conversions in Residue Number Systems (RNS) are based on the Chinese Remainder Theorem (CRT) and the Mixed Radix Conversion (MRC). The complexity of the circuitry of the CRT is high due to the large modulo-M operation. The MRC has a simple circuitry but it’s a sequential process in nature. The purpose of this research is to obtain an efficient reverse conversion method to reduce the computational overhead found in the conventional reverse conversion algorithms. In this paper, new algorithms for reverse conversion in RNS for four-moduli set and five-moduli set have been proposed and their correctness evaluated. Numerical evaluations to ascertain the correctness and simplicity of the algorithm have been presented. These algorithms have fewer multiplicative index operations than those in the conventional CRT and MRC. The large modulo-M operation has been eliminated which reduces the computational overhead.

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.

Parallel Algorithms for Residue Scaling and Error Correction in Residue Arithmetic

In this paper, we present two new algorithms in residue number systems for scaling and error correction. The first algorithm is the Cyclic Property of Residue-Digit Difference (CPRDD). It is used to speed up the residue multiple error correction due to its parallel processes. The second is called the Target Race Distance (TRD). It is used to speed up residue scaling. Both of these two algorithms are used without the need for Mixed Radix Conversion (MRC) or Chinese Residue Theorem (CRT) techniques, which are time consuming and require hardware complexity. Furthermore, the residue scaling can be performed in parallel for any combination of moduli set members without using lookup tables.