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 Theorem statement A constructive algorithm to find the solution Statement for principal ideal domains Statement for general rings ... 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 Competing Definitions. Modify Term Chinese Remainder Theorem hits=1 votes=1 rating=5.0 Modify Term Chinese Remainder Theorem hits=1 votes=1 rating=5.0 http://w3k.org/pages/Top/Engineering/Signal_Processing/Chinese_Remainder_Theorem

Home Help Login Logout ... Random Open Dictionary Dedicated to the pursuit of knowledge Advanced Top Engineering Signal Processing ... Chinese Remainder Theorem Competing Definitions Chinese Remainder Theorem [hits=1 votes=1 rating=5.0] Chinese Remainder Theorem [hits=1 votes=1 rating=5.0] : Chinese Remainder Theorem from MathWorld Chinese Remainder Theorem [hits=1 votes=1 rating=5.0] Questions? Bug report? Send email to the Open Dictionary Administrator The Open Dictionary is a World Wide Web of Knowledge (W3K) project.

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 http://www.linkedwords.com/computers/algorithms/chinese_remainder_theorem.php

March 31, 2008 chinese remainder theorem Web Users Web Publishers About Help Path: Home computers algorithms chinese remainder theorem Page Tools Clustered InsideWeb Language Contextual Path (URL): http://www.linkedwords.com/computers/algorithms/chinese_remainder_theorem.php Keyword LookUp: all keyword pages categories linked keyword pages only Choose Category Arts Brands Business Computers Games Health Home Local News / Media Recreation Reference Science Shopping Society Sports More Search Options: Search within this category only Search the web... Main Categories Arts Brands Business Computers ... Sports Page Tools: (RSS, Email, Print, etc..) Generate RSS Email to Friend / Colleague Printer Friendly Format Add to Favorites ... Make Home Page Change Language Default French German Spanish ... Help Us Translate Archives: Home July August September ... February chinese remainder theorem This page contains listings contextually, grammatically and meaningfully related to the keyword/phrase '... chinese remainder theorem ...'. If you have text in any page containing the keyword/phrase '...chinese remainder theorem...'

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. http://facetroughgemstones.com/wikipedia/ch/Chinese_remainder_theorem.html

Office Supplies Buy Posters A-Z Products Website Advertising ... 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 http://www.mathlinks.ro/Wiki/index.php/Chinese_Remainder_Theorem

Home Art of Problem Solving WOOT! Books ... Log in 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 Proof Applicability Extended version of the theorem ... Discussion 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 http://www.artofproblemsolving.com/Wiki/index.php/Talk:Chinese_Remainder_Theorem

AoPSWiki Looking for a challenging algebra text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Algebra by Richard Rusczyk. Page Discussion View source History Personal tools Log in / create account Navigation Main Page Community portal Current events Recent changes ... Donations Search Toolbox What links here Related changes Upload file Special pages ... Permanent link Talk:Chinese Remainder Theorem From AoPSWiki 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, http://www.indopedia.org/Chinese_remainder_theorem.html

Indopedia Main Page FORUM Help ... Log in The Indology CMS In other languages: Deutsch Categories Modular arithmetic Commutative algebra ... Wikipedia Article 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 http://chr.wikigadugi.org/wiki/Chinese_remainder_theorem

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 http://www.daviddarling.info/encyclopedia/C/Chinese_remainder_theorem.html

NUMBER THEORY A B C ... CONTACT entire Web this site 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. http://www.datoteka.com/rjecnik/en-bhs/c/chinese_remainder_theorem.html

datoteka.com/rjecnik englesko-bosanski/hrvatski/srpski rjeènik English-Bosnian/Croatian/Serbian Dictionary Chinese remainder theorem kineski teorem ostatka (u kripto grafiji) datoteka.com home dictionary search

13. Essays/Chinese Remainder Theorem - J Wiki Show Parent; Immutable Page; Show Changes; Get Info. More Actions. Show Raw Text, Show Print View, Delete Cache, , Attachments, Check Spelling http://www.jsoftware.com/jwiki/Essays/Chinese_Remainder_Theorem

J Wiki Search: Login Home FrontPage RecentChanges ... Show Parent Immutable Page Show Changes Get Info More Actions: Show Raw Text Show Print View Delete Cache Attachments Check Spelling Show Like Pages Show Local Site Map Rename Page Delete Page Literate 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 Show Parent Immutable Page Show Changes Get Info More Actions: Show Raw Text Show Print View Delete Cache Attachments Check Spelling Show Like Pages Show Local Site Map Rename Page Delete Page Literate

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. http://forums.devshed.com/security-and-cryptography-17/best-way-to-calculate-2-a

Security and Cryptography Forums Register User CP ... Sitemap Support Dev Shed Site Dev Shed Forums Search Network Advanced Forums User Name: Password: Remember me Dev Shed Forums System Administration Security and Cryptography Crypto Algorithm Question - Best way to calculate 2^a mod P and 2^-a mod P? 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. Add This Thread To: Del.icio.us Digg Google Spurl Blink Furl Simpy Y! MyWeb Previous Thread Next Thread Thread Tools Search this Thread ... Display Modes Dev Shed Forums Sponsor: Free Webcast! March 25th Register Today! November 1st, 2007, 02:37 PM FearHQ vbmenu_register("postmenu_1924801", true); Registered User Join Date: Nov 2007 Posts: 4 Time spent in forums: 49 m 47 sec 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 http://www.babylon.com/definition/Chinese_Remainder_Theorem/English

Chinese Remainder Theorem Define Chinese Remainder Theorem Translate Chinese Remainder Theorem Chinese Remainder Theorem in Spanish Chinese Remainder Theorem in Croatian Chinese Remainder Theorem in Serbian Wikipedia English The Free Encyclopedia ... Download this dictionary Chinese remainder theorem Chinese remainder theorem refers to a result about congruences in number theory and its generalizations in abstract algebra See more at Wikipedia.org... This article uses material from Wikipedia and is licensed under the GNU Free Documentation License Translate: English Chinese (S) Chinese (T) Croatian Dutch French German Hebrew Italian Japanese Korean Portuguese Russian Serbian Spanish Swedish Turkish Additional Use Babylon to translate to various languages

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. http://www.allbookstores.com/book/9789810228279/A_Salomaa/Chinese_Remainder_Theo

Search By: Keyword Title Author ISBN Browse: Choose a subject: Art Children's Books Christian Books History Gardening Horror Mystery Oprah Book Club Photography Reference Romance Science Science Fiction Sports Technology Travel Art Children's Books Christian Books History ... Store Directory 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 Publisher: World Scientific Pub Co Inc Add this book to your wish list View your wish list Link to this book Click on this books subject categories to see related titles: Subjects MATHEMATICS Set Theory Subjects ... Add your own review! 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. Search By: Keyword Title Author ISBN Browse: Choose a subject: Art Children's Books Christian Books History Gardening Horror Mystery Oprah Book Club Photography Reference Romance Science Science Fiction Sports Technology Travel This site works with Firefox plugins.

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 http://karaltan.wordpress.com/2006/11/

@import url( http://s.wordpress.com/wp-content/themes/pub/silver-black/style.css?m=1203105036 ); var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www."); document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E")); Novembre 30, 2006 Introduction to Cryptography Archiviato in: yazılar cryptography has been growing outside of military and highly diplomatic circles, and into the public domain. E-commerce, personal and corporate privacy have continued the pervasive need for secure communications. Additionally, cryptographic techniques are not limited to the purpose of hiding messages, for instance, archeologists can use cryptographic techniques to interpret ancient or lost languages. Cryptography requires problems that are computationally hard so that scrambled messages are hard to break without the correct key. Checking whether or not a key is correct is easily verifiable, which might suggest that cryptographic problems are class NP problems. However, cryptographic strength does not relate to NP-complete problems because NP-complete problems deal with difficulty in the worst case, which means that the problem may be easily solvable in all other cases. Complexity for cryptography requires analyzing best case solutions, such that a candidate for a cipher system would be difficult in the best case to solve, thus not being concerned with worst case complexity. Hence, a cipher system requires that average case complexity needs to be difficult, to achieve a cipher system that is practically unbreakable.

19. Chinese Remainder Theorem A selection of articles related to Chinese remainder theorem. http://www.experiencefestival.com/chinese_remainder_theorem

Articles Archives Start page News Contact Community General Newsletter Contact information Site map Most recommended Search the site Archive Photo Archive Video Archive Articles Archive More ... Wisdom Archive Body Mind and Soul Faith and Belief God and Religion ... Yoga Positions Site map 2 Site map Chinese remainder theorem A Wisdom Archive on Chinese remainder theorem Chinese remainder theorem A selection of articles related to Chinese remainder theorem More material related to Chinese Remainder Theorem can be found here: Index of Articles related to Chinese Remainder Theorem Chinese remainder theorem Page 2 Page 3 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

Page 1     1-20 of 76    1  | 2  | 3  | 4  | Next 20