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

1. Chinese Remainder Theorem - Wikipedia, The Free Encyclopedia
(2002). Fibonacci s Liber Abaci. SpringerVerlag, 402–403. ISBN 0-387-95419-8. Retrieved from http//en.wikipedia.org/wiki/chinese_remainder_theorem
http://en.wikipedia.org/wiki/Chinese_remainder_theorem
Chinese remainder theorem
From Wikipedia, the free encyclopedia
Jump to: navigation search Chinese remainder theorem refers to a result about congruences in number theory and its generalizations in abstract algebra
Contents
edit Theorem statement
The original form of the theorem, contained in a third-century AD book Sun Zi suanjing(孙子算经 The Mathematical Classic by Sun Zi) by Chinese mathematician Sun Tzu and later republished in a book by Qin Jiushao , Shu shu jiu zhang(数书九章 Mathematical Treatise in Nine Sections) is a statement about simultaneous congruences (see modular arithmetic Suppose n n n k are integers which are pairwise coprime . Then, for any given integers a a a k , there exists an integer x solving the system of simultaneous congruences Furthermore, all solutions x to this system are congruent modulo the product N n n n k Hence for all , if and only if Sometimes, the simultaneous congruences can be solved even if the n i s are not pairwise coprime. A solution

2. Open Dictionary: Top/Engineering/Signal_Processing/Chinese_Remainder_Theorem
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3. Chinese Remainder Theorem LinkedWords™ - Contextually Find And Manage
Added April 6, 2006, 233 am Found in page http//www.linkedwords. com/computers/algorithms/chinese_remainder_theorem.php - Main site
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4. Chinese Remainder Theorem - Wikipedia
Chinese remainder theorem. The Chinese remainder theorem is the name applied to a number of related results in abstract algebra and number theory.
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Chinese remainder theorem
The Chinese remainder theorem is the name applied to a number of related results in abstract algebra and number theory
Simultaneous congruences of integers
The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao[?] published in , is a statement about simultaneous congruences (see modular arithmetic ). Suppose n n k are positive integers which are pairwise coprime (meaning gcd n i n j ) = 1 whenever i j ). Then, for any given integers a a k , there exists an integer x solving the system of simultaneous congruences
x a i mod n i ) for i k
Furthermore, all solutions x to this system are congruent modulo the product n n n k A solution x can be found as follows. For each i , the integers n i and n n i are coprime, and using the extended Euclidean algorithm we can find integers r and s such that r n i s n n i = 1. If we set e i s n n i , then we have
e i mod n i ) and e i mod n j ) for j i
The number x i k a i e i then solves the given system of simultaneous congruences.

5. Chinese Remainder Theorem - AoPSWiki
Retrieved from http//www.mathlinks.ro/Wiki/index.php/chinese_remainder_theorem . Categories Number theory Theorems
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Chinese Remainder Theorem
From AoPSWiki
The Chinese Remainder Theorem is a number theoretic result. it is one of the only theorems named for an oriental person or place, due to the closed development of mathematics in the western world.
Contents
Theorem
Let be relatively prime to . Then each residue class mod is equal to the intersection of a unique residue class mod and a unique residue class mod , and the intersection of each residue class mod with a residue class mod is a residue class mod
Proof
If , then and clearly differ by a multiple of , so and . This is the first part of the theorem. The converse follows because and must differ by a multiple of and , and . This is the second part of the theorem.
Applicability
Much like the Fundamental Theorem of Arithmetic , many people seem to take this theorem for granted before they consciously turn their attention to it. It ubiquity derives from the fact that many results can be easily proven mod (a power of a prime), and can then be generalized to mod using the Chinese Remainder Theorem. For intance

6. Talk:Chinese Remainder Theorem - AoPSWiki
—Boy Soprano II 2000, 4 November 2006 (EST). Retrieved from http//www. artofproblemsolving.com/Wiki/index.php/Talkchinese_remainder_theorem
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It is possible for students to become advanced without becoming knowledgeable about the Chinese Remainder Theorem (much in the manner that students may become good at number theory without studying the Fundamental Theorem of Arithmetic), and it seemed to me that the former article was inappropriate for such students; hence I moved that article to Chinese Remainder Theorem/Introductory and wrote a new advanced Boy Soprano II 20:00, 4 November 2006 (EST)

7. Chinese Remainder Theorem - Indopedia, The Indological Knowledgebase
Retrieved from http//www.indopedia.org/chinese_remainder_theorem.html . This page has been accessed 2099 times. This page was last modified 0607,
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Chinese remainder theorem
ज्ञानकोश: - The Indological Knowledgebase The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory Contents showTocToggle("show","hide") 1 Simultaneous congruences of integers
2 Statement for principal ideal domains

3 Statement for general rings

4 External links
...
edit
Simultaneous congruences of integers
The original form of the theorem, contained in a book by the Chinese mathematician Qin Jiushao published in , is a statement about simultaneous congruences (see modular arithmetic ). Suppose n n k are positive integers which are pairwise coprime (meaning gcd n i n j ) = 1 whenever i j ). Then, for any given integers a a k , there exists an integer x solving the system of simultaneous congruences Pseudocode "subtitle": Furthermore, all solutions x to this system are congruent modulo the product n n n k A solution x can be found as follows. For each

8. Chinese Remainder Theorem -
http//chr.wikigadugi.org/wiki/chinese_remainder_theorem . Modular arithmetic Commutative algebra Mathematical
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9. Chinese Remainder Theorem
The CRT is a method for solving certain systems of congruencies. The CRT reconstructs integers from their residue values modulo a set of relatively prime
http://www.bletchleypark.net/cryptology/Chinese_Remainder_Theorem.html
Chinese Remainder Theorem The CRT is a method for solving certain systems of congruencies. The CRT reconstructs integers from their residue values modulo a set of relatively prime moduli. The CRT is a mechanism for manipulating very large numbers in terms of tuples of smaller integers. The CRT can be calculated in 5 steps. Given a set of congruencies such as: mod m
mod m
mod m
i mod m i Calculate M = m * m * m *, ..., m i Calculate M = M/m , M = M/m and M = M/m , ..., M i = M/m i Calculate your a's (if necessary) Find the multiplicative inverse for each M , M , M , ..., M i , mod m , m , m , ..., m i respectively i * M i * M i ) mod M = Result Example Let's say that you want to find the answer to 3 mod 5005. (Step 1) We know that M = 5005 can be broken down into its prime factors: m = 5, m = 7, m = 11, m (Step 2) In getting the M's we have: M M M M (Step 3) While using Fermat's theorem we can get the following: a a a a (Step 4) Now we have to solve for multiplicative inverses using Euclid's Extended Algorithm for each one: 1001 * M 715 * M 455 * M 385 * M (Step 5) Finally summing everything taken modulo 5005 we have: [(4 * 1001 * 1) + (2 * 715 * 1) + (9 * 435 * 3) + (9 * 385 * 5)] mod 5005 = 35044 mod 5005 Thus, 3

10. Chinese Remainder Theorem
If there are n numbers, a1 to an, that have no factors in common (in other words, are pairwise relatively prime), then any integer greater than or equal to
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Chinese remainder theorem
If there are n numbers, a to a n , that have no factors in common (in other words, are pairwise relatively prime), then any integer greater than or equal to and less than the product of all the numbers n can be uniquely represented by a series consisting of the remainders of division by the numbers n . For example, if a = 3 and a = 5, the Chinese remainder theorem (CRT) says that every integer from to 14 will have a unique set of remainders when divided separately by ( modulo ) 3 and 5. Listing out all the possibilities shows that this is true: has a remainder of modulo 3 and a remainder of modulo 5.
1 has a remainder of 1 modulo 3 and a remainder of 1 modulo 5.
2 has a remainder of 2 modulo 3 and a remainder of 2 modulo 5.
3 has a remainder of modulo 3 and a remainder of 3 modulo 5.
4 has a remainder of 1 modulo 3 and a remainder of 4 modulo 5.
5 has a remainder of 2 modulo 3 and a remainder of modulo 5.
6 has a remainder of modulo 3 and a remainder of 1 modulo 5.

11. A Party Trick Text - Physics Forums Library
http//www.wikipedia.org/wiki/chinese_remainder_theorem http//www.wikipedia.org/wiki/Extended_Euclidean_algorithm but it s not working.
http://www.physicsforums.com/archive/index.php/t-5646.html
Physics Help and Math Help - Physics Forums Mathematics General Math Archives ... PDA View Full Version : A Party Trick Hyperreality I was reading In Code by Saray Flannery,one of the problem in the book bothered me, it is not the problem itself, it is the solution.
A Party Trick
If someone tells me 2, 2 and 3 are the remainders when she divides her age by 3, 5 and 7 respectively then I can work out her age.
Solution:
Let x = 2, y = 2, z = 3 and a = the age of the girl. Then she used this formula:
a = (70x + 21y + 15z)mod n
= (120 + 42 + 45)mod (3 x 5 x 7)
= 17 years old
I have no idea of how this works. Where does this formula come from? There must be a logical way to explain this right? And are there any other ways of solving this kind of problem? selfAdjoint This is an example of a famous result in elementary number theory called the Chinese Remainder Theorem (http://www.cut-the-knot.org/blue/chinese.shtml). The site does a better job of explaining what's involved than I can. HBar I've been playing around with this theorem, but for the life of me i can't get it to work. I've been reading the wikipedia articles on chinese remainder theorem and euclidean algorthim
http://www.wikipedia.org/wiki/Chinese_remainder_theorem

12. Chinese Remainder Theorem
datoteka.com EnglishBosnian/Croatian/Serbian dictionary.
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    13. Essays/Chinese Remainder Theorem - J Wiki
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    Essays/Chinese Remainder Theorem
    Given: integer pairs (m,r) and (n,s) where m and n are bases and r and s are residues with respect to the bases. The Euclidean algorithm computes the GCD of m and n as a linear combination; that is, finds integers a and b such that (m+.n) = (a*m)+(b*n) . The Chinese Remainder Theorem specifies that satisfies and . If m and n are relatively prime, then such an integer c always exists. cr applies to two (base,residue) pairs and produces (LCM,remainder) of the LCM of the bases and the required remainder. If b is a list of bases and r the corresponding residues, c=:cr/b,.r computes the remainder c such that
    Contributed by RogerHui last edited 2007-09-05 21:26:44 by RogerHui
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    14. Wiki Chinese Remainder Theorem
    Wiki Chinese remainder theorem. Contents 1. Theorem statement 2. A constructive algorithm to find the solution 3. Statement for principal ideal domains
    http://wapedia.mobi/en/Chinese_remainder_theorem
    Wiki: Chinese remainder theorem Contents:
    1. Theorem statement

    2. A constructive algorithm to find the solution

    3. Statement for principal ideal domains

    4. Statement for general rings
    ...
    9. External links

    Chinese remainder theorem refers to a result about congruences in number theory and its generalizations in abstract algebra Home Licensing Wapedia: For Wikipedia on mobile phones

    15. Best Way To Calculate 2^a Mod P And 2^-a Mod P? - Dev Shed
    http//en.wikipedia.org/wiki/chinese_remainder_theorem Yes, its all wikipedia, and yes it s just a bunch of links, but one might be helpful.
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    Discuss Best way to calculate 2^a mod P and 2^-a mod P? in the Security and Cryptography forum on Dev Shed.
    Best way to calculate 2^a mod P and 2^-a mod P? Security and Cryptography forum discussing issues related to coding, server applications, network protection, data protection, firewalls, ciphers and the like.
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    Reputation Power: Crypto Algorithm Question - Best way to calculate 2^a mod P and 2^-a mod P? I've decided to implement montgomery modular multiplications as my base modular support. I've found a great algorithm that fits my needs for the actual main process, except it doesn't really go into details of the calculations of the constants... What I'm looking for right now are the best ways to calculate 2 ^ a mod P and 2 ^ -b mod P, where a and b are small number (not greater than 16 bits). I've found great code that is capable of simplifying the inverse with a smaller word size, however in my application I actually need the full inverse of 2 ^ b mod P. Any pointers to any papers/publications/research/etc? I don't want to resort to blatantly subtracting P from 2 ^ a or calculating the euclidean inverse of 2 ^ b mod P, unless there really isnt a better way

    16. Chinese Remainder Theorem | English | Dictionary & Translation By Babylon
    Chinese Remainder Theorem. Dictionary terms for Chinese Remainder Theorem in English, English definition for Chinese Remainder Theorem, Thesaurus and
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    Chinese remainder theorem refers to a result about congruences in number theory and its generalizations in abstract algebra
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    17. Chinese Remainder Theorem : Applications In Computing, Coding, Cryptography By C
    Lowest prices on Chinese Remainder Theorem Applications in Computing, Coding, Cryptography by C. Ding, D. Pei, A. Salomaa.
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    Chinese Remainder Theorem : Applications in Computing, Coding, Cryptography
    Author: C. Ding D. Pei A. Salomaa Format: Hardcover Publish Date: June 1999 ISBN-10: ISBN-13: List Price: For the best price: or just
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    Book Details Summary: The title of this book is Chinese Remainder Theorem : Applications in Computing, Coding, Cryptography and it was written by C. Ding D. Pei A. Salomaa . This edition of Chinese Remainder Theorem : Applications in Computing, Coding, Cryptography is in a Hardcover format. This books publish date is June 1999 and it has a suggested retail price of $46.00. There are pages in the book and it was published by World Scientific Pub Co Inc. The 10 digit ISBN is 9810228279 and the 13 digit ISBN is 9789810228279.
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    18. 2006 Novembre « WSN & VB [Wireless Araştırmalarım Ve Visual Basic Notl
    Chinese Remainder Theorem. Archiviato in linkler — karaltan @ 1230 am. http//bletchleypark.net/cryptology/chinese_remainder_theorem.html
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    19. Chinese Remainder Theorem
    A selection of articles related to Chinese remainder theorem.
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    ARTICLES RELATED TO Chinese remainder theorem
    Chinese remainder theorem: Encyclopedia II - Chinese remainder theorem - Simultaneous congruences of integers The original form of the theorem, contained in a third-century book by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao, is a statement about simultaneous congruences (see modular arithmetic). Suppose n1, ..., nk are positive integers which are pairwise coprime (meaning gcd (ni, nj) = 1 whenever i ≠ j). Then, for any given integers a1, ..., ak, there exists an in ...
    See also:
    Chinese remainder theorem Chinese remainder theorem - Simultaneous congruences of integers Chinese remainder theorem - Statement for principal ideal domains Chinese remainder theorem - Statement for general rings ... Chinese remainder theorem - Applications of the Chinese remainder theorem Read more here: Chinese remainder theorem: Encyclopedia II - Ideal ring theory - Definitions
    See also:

    20. Chinese Remainder Theorem - China-related Topics CE-CH - China-Related Topics
    The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory. Simultaneous congruences of integers The
    http://www.famouschinese.com/virtual/Chinese_remainder_theorem
    March 31, 2008 1 Introduction Simultaneous congruences of integers Statement for principal ideal domains Statement for general rings External links ... China-related Topics CE-CH Chinese remainder theorem
    Wikipedia The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory. [go back to top]
    Simultaneous congruences of integers The original form of the theorem, contained in a book by the Chinese mathematician Qin Jiushao published in 1247, is a statement about simultaneous congruences (see modular arithmetic). Suppose n n k n i n j i j ). Then, for any given integers a a k x solving the system of simultaneous congruences Furthermore, all solutions x to this system are congruent modulo the product n n n k A solution x can be found as follows. For each i ; the integers and n are coprime, and using the extended Euclidean algorithm we can find integers r and s such that s n = 1. If we set s n , then we have for j i The solution to the system of simultaneous congruences is therefore For example, consider the problem of finding an integer

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