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Chinese Remainder Theorem:     more detail

Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography by C. Ding, D. Pei, et all 1999-06 Secret Sharing Using the Chinese Remainder Theorem: Secret Sharing, Chinese Remainder Theorem, Threshold Cryptosystem, Cardinality, Access Structure, Shamir's ... Polynomial Interpolation, George Blakley Remainder: Natural Number, Real Number Modulo Operation, Chinese Remainder Theorem, Division Algorithm, Euclidean Algorithm A hierarchical single-key-lock access control using the Chinese remainder theorem (OSU-CS-TR) by Kim Sin Lee, 1994 Fundamental Number Theory with Applications (Discrete Mathematics and Its Applications) by Richard A. Mollin, 1998-01-31 Fundamental Number Theory with Applications, Second Edition (Discrete Mathematics and Its Applications) by Richard A. Mollin, 2008-02-21

lists with details

41. Chinese Remainder Theorem@Everything2.com
    The following result was known to Chinese mathematics in the first century AD. By way of notation, if a,b are integers and n is a positive integer we write a cgt b (mod n) if n  
    http://www.everything2.com/title/Chinese remainder theorem

42. Chinese Remainder Theorem
    Math reference, the chinese remainder theorem. Modular Mathematics, Chinese Remainder Theorem Chinese Remainder Theorem  
    http://www.mathreference.com/num-mod,chr.html
    
    Extractions: Given a set of values c c n , and a set of mutually coprime moduli m m n , is there an integer x such that x = c i mod m i for each i in 1 through n? Let z be the product of all the moduli. If x is a solution then so is x plus any multiple of z. If w is not a multiple of z, say w is not divisible by m , then x+w will not equal c mod m , and x+w will not be a solution. The solution, if it exists, is well defined mod z. To show that a solution exists, we simply construct one. Let a i be the product of all the moduli other than m i . Verify that a i and m i are coprime. Let b i be the inverse of a i mod m i . Finally, let x be the sum of a i b i c i If the original moduli are not coprime, split each equation up into a set of equations by factoring the composite modulus into prime powers. Then consider all the equations together. Equations sharing a common prime modulus are either redundant or inconsistent. An inconsistent example is x = 4 mod 6 and x = 11 mod 15. This would force x = 1 mod 3 and x = 2 mod 3, which is impossible. The example x = 4 mod 6 and x = 7 mod 15 has the solution x = 22 mod 30. showFooter("num-mod,rsa", "num-mod,note");

43. Theorem 24: The Chinese Remainder Theorem « Theorem Of The Week
    Apr 18, 2010 One special case of the Chinese Remainder Theorem says that is isomorphic to the product . We could prove that by defining a map from to .  
    http://theoremoftheweek.wordpress.com/2010/04/18/theorem-24-the-chinese-remainde

44. Chinese Remainder Theorem
    Chinese remainder theorem Author hasinoff What is the Chinese remainder theorem as it applies to solving equations involving the modulus operator?  
    http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH056.HTM

45. Chinese Remainder Theorem
    In this section we will prove the Chinese Remainder Theorem for rings of integers, deduce several surprising and useful consequences, then learn about  
    http://modular.fas.harvard.edu/129/ant/html/node30.html

46. [0903.2785] Computing Hilbert Class Polynomials With The Chinese Remainder Theor
    by AV Sutherland 2009 - Cited by 13 - Related articles  
    http://arxiv.org/abs/0903.2785
    
    Extractions: new recent math what is this? what is this? Authors: Andrew V. Sutherland (Submitted on 16 Mar 2009 ( ), last revised 30 Mar 2010 (this version, v3)) Abstract: Comments: 37 pages, minor typographical changes, to appear in Mathematics of Computation Subjects: Number Theory (math.NT) MSC classes: 11Y16 (Primary), 11G15, 11G20, 14H52 (Secondary) Cite as: arXiv:0903.2785v3 [math.NT] From: Andrew Sutherland [ view email

47. IEEE Xplore - Chinese Remainder Theorem-Based RSA-Threshold
    by S Sarkar 2009 - Related articles  
    http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5325213

48. Chinese Remainder Theorem
    Math reference, chinese remainder theorem for rings. Rings, Chinese Remainder Theorem Chinese Remainder Theorem The chinese remainder theorem was developed for modular arithmetic  
    http://www.mathreference.com/ring,chr.html
    
    Extractions: showHeader(1, "", "ring", "Rings", ""); The chinese remainder theorem was developed for modular arithmetic , but it generalizes to ideals in a commutative ring R. Let H H n be a set of coprime ideals. By coprime, we mean the sum of any two ideals spans the ring. Let J be the product of all these ideals. We will prove R/J is isomorphic to the direct product of the quotient rings R/H i , as i runs from 1 to n. An element in R/J can be mapped to the i th component in the direct product via R/H i . This is a well defined ring homomorphism, since each H i wholly contains J. We need to show it is 1-1 and onto. Focus on H . We know x + y = 1 for some x in H and y in H . Do the same for each H i in the set. Multiply all these equations together, and something in H + something in the product of the other ideals gives 1. Write this as x + y = 1. Reduce mod H , and y = -1. If y is mapped into any other quotient ring, other than R/H , it maps to 0,as it lives in the product of the other ideals. With y through y n established, let's show the map is onto. Suppose z is an element in the direct product, where z

49. RSA Algorithm
    An alternative method of representing the private key uses the Chinese Remainder Theorem (CRT). See our page The Chinese Remainder Theorem.  
    http://www.di-mgt.com.au/rsa_alg.html
    
    Extractions: Sitemap DI Management Home Cryptography The RSA algorithm is named after Ron Rivest, Adi Shamir and Len Adleman, who invented it in 1977 [ ]. The basic technique was first discovered in 1973 by Clifford Cocks [ ] of CESG (part of the British GCHQ) but this was a secret until 1997. The patent taken out by RSA Labs has expired. The RSA algorithm can be used for both public key encryption and digital signatures. Its security is based on the difficulty of factoring large integers. Generate two large random primes, p and q , of approximately equal size such that their product n = pq is of the required bit length, e.g. 1024 bits. [See note 1 Compute n = pq and Choose an integer e , such that gcd(e, phi) = 1 . [See note 2 Compute the secret exponent d , such that . [See note 3 The public key is (n, e) and the private key is (n, d). Keep all the values d, p, q and phi secret. Sender A does the following:- Obtains the recipient B's public key (n, e).

50. The Chinese Remainder Theorem
    The Chinese Remainder Theorem asserts that a solution to Sun s question exists, Theorem 2.1 (The Chinese Remainder Theorem) Let $ a, b\in\mathbb{Z}$  
    http://modular.math.washington.edu/edu/Fall2001/124/lectures/lecture6/html/node2
    
    Extractions: Next: Multiplicative Functions Up: Lecture 6: Congruences, Part Previous: Wilson's Theorem Sun Tsu Suan-Ching (4th century AD): There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? In modern notation, Sun is asking us to solve the following system of equations:

51. PlanetMath: Chinese Remainder Theorem
    AMS MSC 11N99 (Number theory Multiplicative number theory Miscellaneous) 11A05 (Number theory Elementary number theory Multiplicative structure; Euclidean algorithm  
    http://planetmath.org/encyclopedia/ChineseRemainderTheorem2.html
    
    Extractions: Chinese remainder theorem (Theorem) Let $R$ be a commutative ring with identity . If are ideals of $R$ such that $I_i + I_j = R$ whenever , then let The sum of quotient maps gives an isomorphism This has the slightly weaker consequence that given a system of congruences , there is a solution in $R$ which is unique mod $I$ , as the theorem is usually stated for the integers

52. The Chinese Remainder Theorem
    File Format PDF/Adobe Acrobat Quick View  
    http://www2.edc.org/connect/rmdr_thm.pdf

53. Nrich.maths.org :: Mathematics Enrichment :: The Chinese Remainder Theorem
    In this article we shall consider how to solve problems such as 'Find all integers that leave a remainder of 1 when divided by 2, 3, and 5.'  
    http://nrich.maths.org/public/viewer.php?time=1162928343&obj_id=5466&par

54. Cryptology EPrint Archive: Report 2007/107
    by Y MURAKAMI Cited by 5 - Related articles  
    http://eprint.iacr.org/2007/107
    
    Extractions: Knapsack Public-Key Cryptosystem Using Chinese Remainder Theorem Yasuyuki MURAKAMI, Takeshi NASAKO Abstract: The realization of the quantum computer will enable to break public-key cryptosystems based on factoring problem and discrete logarithm problem. It is considered that even the quantum computer can not solve NP-hard problem in a polynomial time. The subset sum problem is known to be NP-hard. Merkle and Hellman proposed a knapsack cryptosystem using the subset sum problem. However, it was broken by Shamir or Adleman because there exist the linearity of the modular transformation and the specialty in the secret keys. It is also broken with the low-density attack because the density is not sufficiently high. In this paper, we propose a new class of knapsack scheme without modular transformation. The specialty and the linearity can be avoidable by using the Chinese remainder theorem as the trapdoor. The proposed scheme has a high density and a large dimension to be sufficiently secure against a practical low-density attack. Category / Keywords: public-key cryptography / knapsack public-key cryptosystem, subset sum problem, low-density attack, Chinese remainder theorem

55. Constructive Proof Of The Chinese Remainder Theorem — The Endeavour
    The Chinese Remainder Theorem (CRT) is a tool for solving problems involving modular arithmetic. The theorem is called the “Chinese” remainder theorem because the Chinese  
    http://www.johndcook.com/blog/2008/12/11/constructive-proof-chinese-remainder-th

56. Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
    Dec 1, 2003 This result is universally known as the Chinese Remainder Theorem, although it is sometimes butchered and/or generalized beyond recognition.  
    http://www.numericana.com/answer/modular.htm

57. Chinese Remainder Theorem
    Definition of Chinese remainder theorem, possibly with links to more information and implementations.  
    http://xw2k.nist.gov/dads//HTML/chineseRmndr.html
    
    Extractions: (algorithm) Definition: An integer n can be solved uniquely mod LCM(A(i)) Note: For example, knowing the remainder of n when it's divided by 3 and the remainder when it's divided by 5 allows you to determine the remainder of n when it's divided by LCM(3,5) = 15. After LK. Author: PEB Go to the Dictionary of Algorithms and Data Structures home page. If you have suggestions, corrections, or comments, please get in touch with Paul E. Black Entry modified 17 December 2004.

58. The Chinese Remainder Theorem
    The Chinese Remainder Theorem Recall that the Chinese Remainder Theorem from elementary number theory asserts that if are integers that are coprime in pairs, and are integers  
    http://modular.fas.harvard.edu/129/ant/html/node31.html

59. Sensors | Free Full-Text | Broadcast Authentication For Wireless Sensor Networks
    Sep 17, 2010 This protocol uses a nested hash chain of two different hash functions and the Chinese Remainder Theorem (CRT). The two different nested  
    http://www.mdpi.com/1424-8220/10/9/8683/

60. Optics InfoBase: Optics Express - Time Efficient Chinese Remainder Theorem Algor
    Time efficient Chinese remainder theorem algorithm for fullfield fringe phase The Chinese remainder theorem, an algorithm from number theory is used to  
    http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-12-6-1136

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