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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. Recommended Cryptography Books: Prerequisites For Chinese Remainder Theorem
    chinese remainder theorem Applications in Computing, Coding, Cryptography Ding,C. / Pei, D. / Salomaa, A.. 1996. 213 pages. Categories Mathematics.  
    http://www.youdzone.com/cryptobooks_9810228279_prereqs.html
    
    Extractions: Ding, C. / Pei, D. / Salomaa, A.. 1996. 213 pages. Categories: Mathematics The book covers some fun stuff, like the Chinese using the CRT to compute the numbers of different size blocks required for building the Great Wall of China, and of Chinese generals that had their soldiers line up in group of which the general merely counted the remainder to keep their true numbers secret. But on the whole, this book is not for someone with a casual interest in the subject, and definitely requires background in number theory, information theory, and abstract algebra. Not light reading.

42. Chinese Remainder Theorem
    Divide your number by 3 and enter the remainder in the first box below on the left. Divideyour number by 5 and enter the remainder in the box in the center.  
    http://www.doylestown.com/chinese.html

43. Chinese Remainder Theorem Can Be Seen
    See here for the ps.gz version. Andrzej Solecki's cecinestpasunepeep Show. proudlypresents 5 as m, 4 as n and x as x in the great. chinese remainder theorem. !  
    http://www.ilhadamagia.com.br/~andsol/english/mat/china.html

44. Solving Congruences: The Chinese Remainder Theorem
    Solving Congruences The chinese remainder theorem. This is done by the Chinese RemainderTheorem, socalled because it appeared in ancient Chinese manuscripts.  
    http://www.math.okstate.edu/~wrightd/crypt/lecnotes/node21.html
    
    Extractions: Next: Challenges! Up: Cryptology Class Notes Previous: Square roots In considering the problem of finding modular square roots, we found that the problem for a general modulus m could be reduced to that for a prime power modulus. The next problem would be how to piece the solutions for prime powers together to solve the original congruence. This is done by the Chinese Remainder Theorem, so-called because it appeared in ancient Chinese manuscripts. A typical problem is to find integers x that simultaneously solve have a unique solution x modulo mn Our example problem would have a unique solution modulo It's better than this; there is a relatively simple algorithm to find the solution. Since all number theory algorithms ultimately come down to Euclid's algorithm, you can be sure it happens here as well. First let's consider an even simpler example. Suppose we want all numbers

45. Numlib::ichrem -- Chinese Remainder Theorem For Integers
    numlibichrem chinese remainder theorem for integers. Introduction.numlibichrem (a,m) returns the least nonnegative integer  
    http://www.mupad.com/doc/de/numlib/ichrem.shtml
    
    Extractions: numlib::ichrem (a,m) returns the least nonnegative integer x such that (x-a[i]) mod m[i]=0 for i=1,..,m if such a number exists; otherwise numlib::ichrem (a,m) returns FAIL numlib::ichrem(a, m) a a list of integers m a list of natural numbers of the same length as a either a nonnegative integer or FAIL numlib::lincongruence Here the moduli are pairwise coprime. In this case, a solution always exists: numlib::ichrem([2,3,2],[3,5,7]) Here the moduli are not pairwise coprime, and a solution does not exist: numlib::ichrem([5,6,8],[20,21,22]) FAIL Also here the moduli are not pairwise coprime, but a solution nevertheless exists: numlib::ichrem([5,6,7],[20,21,22]) Do you have questions or comments?

46. Chinese Remainder Theorem
    chinese remainder theorem. Solve the followinglinear congruences x a i (mod m i ) x=.  
    http://people.ucsc.edu/~erowland/crt.html

47. Untitled
    and is the origin of the chinese remainder theorem. The song appeared firstin Sun ZI Suan Jing of the 4th Century chinese remainder theorem.  
    http://www.weizmann.ac.il/~feshtrik/sunzig.html
    
    Extractions: Is a Chinese folk song hiding the answer to: What is the number that gives when devided by 3 a remainder of 2 when devided by 5 a remainder of 3 when devided by 7 a remainder of 2 and is the origin of the Chinese Remainder Theorem The song appeared first in "Sun ZI Suan Jing" of the 4th Century The Song of Master Sun 3 men walk together for 70 miles 5 plum trees blossom 21 branches 7 persons reunion at June 15 Circulating at a period of 105. Back to Shmuel Shtrikman Home Page

48. Chinese Remainder Theorem
    AA Jagers.  
    http://wwwhome.math.utwente.nl/~jagersaa/152063/CRS/Main.html

49. MathGroup Archive: November 1998 Re: The Chinese Remainder Theorem
    Re the chinese remainder theorem. Prev by thread Re the Chinese RemainderTheorem; Next by thread Re the chinese remainder theorem.  
    http://forums.wolfram.com/mathgroup/archive/1998/Nov/msg00034.html

50. MathGroup Archive: November 1998 Re: The Chinese Remainder Theorem
    Re the chinese remainder theorem. To mathgroup@smc.vnet.net; Subjectmg14630 Re mg14597 the chinese remainder theorem; From  
    http://forums.wolfram.com/mathgroup/archive/1998/Nov/msg00055.html

51. Bookmarks2 For Patrick Reany
    generalizations Rings. chinese remainder theorem. Reany's Heuristicsof chinese remainder theorem Math 5410 chinese remainder theorem  
    http://www.ajnpx.com/html/Bookmarks2.html

52. An English-Persian Dictionary Of Algorithms And Data Structures
    chinese remainder theorem, ÞÖíåí ÈÇÞíãÇäÏåí íäí.algorithm. íԝäåÇÏ ÊÑÌãåí ÌÏíÏ. Definition  
    http://ce.sharif.edu/~dic-ads/d.php?r=Chinese remainder theorem.1.8

53. Chinese Remainder Theorem
    next up previous Next Up Previous chinese remainder theorem. Find integersx that leave remainders 2, 3, 2 when divided by 3, 5, 7, respectively.  
    http://ranger.uta.edu/~cook/aa/lectures/l23/node22.html

54. Chinese Remainder Theorem (33.27 Cont.)
    next up previous Next Up Previous chinese remainder theorem (33.27cont.). If and. Then (a+b) mod n ((a 1 + b 1 ) mod n 1 , , (a  
    http://ranger.uta.edu/~cook/aa/lectures/l23/node23.html

55. Chinese Mathematics : Rebecca And Tommy
    The chinese remainder theorem (TaYen). It was not until 1247 thatQin Jiushao (c 1202-1261) published a general method for solving  
    http://www.roma.unisa.edu.au/07305/remain.htm
    
    Extractions: It was not until 1247 that Qin Jiushao (c 1202-1261) published a general method for solving systems of linear congruence's in his book called ' Shushu jiuzhang (Mathematical Treatise in Nine Sections)' (Katz, 1992, p188). A book clearly influenced by the old chiu chang suan shu , as were a majority of Chinese mathematical works. Before this time only specific problems had been solved, by people such as Shu Zi (late third century). This method became known as the Ta-Yen. The basic format of problems it was to solve was ; N = a(mod b) = c(mod d) = ... This meant find N such that when divided by b gives a remainder of a and when divided by d gives a remainder of c. Throughout this page we will use the example

56. Chinese Remainder Problem
    Unfortunately, Problem 26 is the only problem that illustrates thechinese remainder theorem in the Sun Tzu Suan Ching. As such, we  
    http://www.math.sfu.ca/histmath/China/3rdCenturyBC/CRP1.html
    
    Extractions: Aside: Writing N r1 (mod m1) [this means N is congruent to r1 modulo m1] means that N divided by m1 leaves r1 as the remainder. The goal here is to find the smallest positive integer satisfying the congruences states above. Now that you know what a Chinese Remainder Problem is, you must be wondering why or what has this particular kind of problem to do with Chinese Mathematical History. The reason why it is called the Chinese Remainder Problem is because the earliest versions of these congruence problems occured in early Chinese mathematical works. The earliest of such works that contains the Chinese Remainder Problem is the Sun Tzu Suan Ching (also known as Sunzi suanjing) written in approximately late third century by Sun Zi . Problem 26 (also known as the problem of Master Sun) in the third volume of the Sun Tzu Suan Ching offers the earliest recorded Chinese Remainder Problem. Problem 26 is as stated below:

57. Www.cs.nott.ac.uk/~lad/challenges/IWC008.txt
    chinese remainder theorem Summary Needs many lemmas(some of whose proofs are also challenging) An Existential Witness has to  
    http://www.cs.nott.ac.uk/~lad/challenges/IWC008.txt
    
    Extractions: Springer-Verlag.Chinese Remainder Theorem - Summary: Needs many lemmas (some of whose proofs are also challenging) An Existential Witness has to be Provided Definitions: allcongruent (num, list) -> bool allcongruent(x, nil) = true allcongruent(x, y::z) = allcongruent(x, z) and (rem(x, first(y))=rem(second(y), first(y))) allpositive - list -> bool true if all members of the list are greater than 0. allprime2 - list -> bool allprime2(nil) = true allprime2(y::z) = prime2list(x, first(y)) and allprime2(x, z) prime2 is true iff its argumens are relatively prime prime2list (num, list) -> bool prime2list(x, nil) = true prime2list(x, y::z) = prime2(x, first(y)) and prime2list(x, z) products1 - product of list. rem - returns the remainder of x by y Theorem: forall l:(list (pair nat)). exists x:nat. (allpositive(l) and allprime2(l)) -> allcongruent(x,l) forall l:(list (pair nat)). forall x,y:nat. ((allpositive(l) and allprime2(l)) and allcongruent(x,l) and allcongruent(y,l)) -> (mod (x - y),products(l)) Comments Actually two proof, existance and uniqueness. Proved in RRL by Zhang and Hua and there is a good deal of comment of the proof in their CADE-11 paper on the subject. They provide by hand the existential witness needed for the existance part of the proof. Source Proving the Chinese Remainder Theorem by the Cover Set Induction, H. Zhang and X. Hua, CADE-11, D. Kapur (ed), 1992.

58. 0.5.14 Chinese Remainder Theorem
    Prime Number Up 0.5 Miscellaneous Algorithms Previous 0.5.13 Horner'sRule 0.5.14 chinese remainder theorem. Scott Gasch 199907-09  
    http://www.fearme.com/misc/alg/node139.html

59. [quant-ph/9911051] Large Numbers, The Chinese Remainder Theorem, And The Circle
    201250 GMT (12kb) Large Numbers, the chinese remainder theorem,and the Circle of Fifths. Authors SA Fulling Comments 9 pages  
    http://arxiv.org/abs/quant-ph/9911051

60. Chinese Remainders
    The chinese remainder theorem is not particularly easy to understand it justsays that under certain conditions (ie the divisors have no common factors  
    http://www.delphiforfun.org/Programs/chinese_remainders.htm
    
    Extractions: Home Introduction About Delphi Feedback ... Site Search (bottom of page) Available Now (Click a level below to expand program list if it is collapsed - MS Internet Explorer only) 10 Easy Pieces Bouncing Ball A Card Trick? Cards #1 ... Multiple of 12? 15 Puzzle #1 The 9321 Problem Aliquot Sums (Friendly If not Perfect) Alphametics ... Word Ladder 15 Puzzle #2 Akerue Brute Force Countdown Plus! ... WordStuff 2 Contact Feedback Send an e-mail with your comments about this program (or anything else). Search WWW Search delphiforfun.org Problem Description What is the smallest number that can be divided by 6, leaving a remainder of 5; by 5 leaving 4; and by 4 leaving 3? Source: The Mensa (c) Puzzle Calendar for Oct 18. 2001; Here's an introduction to Chinese Remainder problems. They're called "Chinese Remainder" because the problem and the theorem which defines when they can be solved were both known to the early Chinese scholars. The earliest known example of this type of problem was published in the 3rd, 4th or 5th century AD by Chinese scholar, Sun Zi, in a book titled "

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