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

Modular Mathematics, Chinese Remainder Theorem Chinese Remainder Theorem 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

Ask A Scientist Mathematics Archive Chinese remainder theorem Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

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

arXiv.org math Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PDF PostScript Other formats Current browse context: math.NT new recent Change to browse by: math NASA ADS 1 blog link what is this? Bookmark what is this? Mathematics > Number Theory Title: Computing Hilbert class polynomials with the Chinese Remainder Theorem 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] Submission history From: Andrew Sutherland [ view email Mon, 16 Mar 2009 15:53:27 GMT (46kb) Thu, 10 Sep 2009 21:34:56 GMT (50kb) Tue, 30 Mar 2010 19:08:24 GMT (47kb) Which authors of this paper are endorsers? Link back to: arXiv form interface contact

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

Rings, Chinese Remainder Theorem showHeader(1, "", "ring", "Rings", ""); Chinese Remainder Theorem 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

Sitemap DI Management Home Cryptography RSA Algorithm 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. Contents Key generation algorithm Encryption Decryption Digital signing ... Comments Key Generation Algorithm 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. n is known as the modulus e is known as the public exponent or encryption exponent or just the exponent d is known as the secret exponent or decryption exponent Encryption 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

Next: Multiplicative Functions Up: Lecture 6: Congruences, Part Previous: Wilson's Theorem The Chinese Remainder 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: The Chinese Remainder Theorem asserts that a solution to Sun's question exists, and the proof gives a method to find a solution. Theorem 2.1 (The Chinese Remainder Theorem) Let and such that . Then there exists such that Proof . The equation has a solution since . Set . We next verify that is a solution to the two equations. Then and Now we can solve Sun's problem: First, we use the theorem to find a solution to the pair of equations Set . Step 1 is to find a solution to . A solution is . Then . Since any with is also a solution to those two equations, we can solve all three equations by finding a solution to the pair of equations Again, we find a solution to

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

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About 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 "Chinese remainder theorem" is owned by bwebste full author list owner history view preamble ... get metadata View style: jsMath HTML HTML with images page images TeX source See Also: Chinese remainder theorem in terms of divisor theory Attachments: proof of Chinese remainder theorem (Proof) by mclase Log in to rate this entry.

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

Cryptology ePrint Archive: Report 2007/107 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

home index units counting ... physics Final Answers , Ph.D. Modular Arithmetic Mathematics is the Queen of sciences, and arithmetic the Queen of mathematics. Carl Friedrich Gauss (1777-1855) Chinese remainder theorem : Remainders define an integer, within limits. Modular arithmetic : The formal algebra of congruences, due to Gauss. Fermat's little theorem : For any prime p not dividing a a p is 1 modulo p. Euler's totient function f (n) counts the integers coprime to n, from 1 to n. Fermat-Euler theorem : If a is coprime to n, a to the f (n) is 1 modulo n. Carmichael's reduced totient function l ) : A special divisor of the totient. 91 is a pseudoprime to half of the bases coprime to itself. Carmichael Numbers : An absolute pseudoprime n divides a n a for any a Chernik's Carmichael numbers : 3 prime factors (6k+1)(12k+1)(18k+1). Large Carmichael numbers may be obtained in various ways. Conjecture : Any odd integer coprime to its totient has Carmichael multiples. Articles previously on this page: Pseudoprimes to base a The above articles have moved...

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

Chinese remainder theorem (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. HTML page formatted Mon Sep 27 10:31:22 2010. Cite this as: Paul E. Black, "Chinese remainder theorem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology . 17 December 2004. (accessed TODAY) Available from: http://xw2k.nist.gov/dads/HTML/chineseRmndr.html

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/

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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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