Computing by homomorphic images
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Keywords:
Chinese remainder theorem
Homomorphic Encryption
Euclidean algorithm
Modulo operation
Modular arithmetic
Chinese remainder theorem
Homomorphic Encryption
Euclidean algorithm
Modulo operation
Modular arithmetic
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The modular multiplication and exponentiation algorithms based on the Montgomery reduction technique require that the modulus be an odd integer. It is shown that, with the help of the Chinese remainder theorem, the Montgomery reduction algorithm can be used to efficiently perform these modular arithmetic operations with respect to an even modulus.
Chinese remainder theorem
Modular arithmetic
Modular exponentiation
Exponentiation
Modulo operation
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We prove for the (C, 1) summability method several Tauberian remainder theorems using the general control modulo of the oscillatory behavior.
Chinese remainder theorem
Modulo operation
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We present an efficient algorithm based on the robust Chinese remainder theorem (CRT) to perform single frequency determination from multiple undersampled waveforms. The optimal estimate of common remainder in robust CRT, which plays an important role in the final frequency estimation, is first discussed. To avoid the exhausted searching in the optimal estimation, we then provide an improved algorithm with the same performance but less computation. Besides, the sufficient and necessary condition of the robust estimation was proposed. Numerical examples are also provided to verify the effectiveness of the proposed algorithm and related conclusions.
Chinese remainder theorem
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Modular multiplication plays an important role to several public-key cryptosystems such as the RSA cryptosystem.This paper proposes an efficient modulo p multiplication algorithm with moderate factors of p + 1 and p -1.In order to improve the RSA decryption performance, users can utilize our proposed algorithm and the strong prime criterion.It will prove that the decryption method based on our proposed algorithm can run at a speed almost 6.5 times faster than that of the traditional method, or almost 2 times faster than that of the method based on the Chinese Remainder Theorem.Furthermore, the proposed algorithm can greatly enhance the performance of RSA encryption.
Modular arithmetic
Chinese remainder theorem
Multiplication algorithm
Modular exponentiation
Modulo operation
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The residue-to-binary conversion is the crucial step for residue arithmetic. The traditional methods are the Chinese remainder theorem (CRT) and the mixed radix conversion. This paper presents new Chinese remainder theorems I, II, and Ill for the residue-to-binary conversion, with the following detailed results. (1) The big weights in the original CRT are reduced to a matrix of numbers less than the moduli P/sub i/. (2) The new Chinese remainder theorem I is a parallel algorithm in mixed radix format. The delay is reduced from O(n) to O(logn). (3) The new Chinese remainder theorem II reduces the modulo operation from the size M to a size less than /spl radic/M. (4) The new Chinese remainder theorem II can be easily extended to the new Chinese remainder theorem III for non-prime moduli sets. (5) A summary of a long list of references on residue-to-binary conversion is also presented.
Chinese remainder theorem
Residue number system
Modulo operation
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We prove the equidistribution of subsets of $(\Rr/\Zz)^n$ defined by fractional parts of subsets of~$(\Zz/q\Zz)^n$ that are constructed using the Chinese Remainder Theorem.
Chinese remainder theorem
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Chinese remainder theorem
Greatest common divisor
Modulo operation
Robustness
Nyquist–Shannon sampling theorem
Signal reconstruction
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Chinese remainder theorem
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Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra.
Chinese remainder theorem
Calculator
Multiple
Arithmetic progression
Modulo operation
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