41. Chinese Remainder Theorem chinese remainder theorem. What is this number x ? This is the famous Chinese remaindertheorem. Following is my own unique method for solving this puzzle. http://www.geocities.com/dirkie6/chinese.html

Chinese remainder theorem A certain number x is divided by 3 and a remainder of 2 results. The same number x is divided by 5 and a remainder 4 results. The same number x is divided by 7 and a remainder of 3 results. What is this number x ? This is the famous Chinese remainder theorem. Following is my own unique method for solving this puzzle. Let the dividers all be prime. Let x be the number we are looking for.So for n dividers and n remainders we have: x == a mod p (equation 1) x == a mod p (equation 2) x == a mod p (equation 3) x == a mod p (equation 4) x == a n mod p n (equation n) Now we start by multiplying equation 1 by p and equation 2 by p We then have x p == a p mod( p p ) and x p == a p mod( p p But GCD(p , p ) =1 , so we can find a r and s so that rp + sp Therefore we have xrp == ra p mod( p p ) and xsp == sa p mod( p p ) and thus xrp + xsp == ra p + sa p mod( p p So x == ra p + sa p mod( p p So x == b mod ( p p ) where b = ra p + sa p Next we look at x == b mod ( p p ) and x == a mod p We do the same as above and get x == b mod ( p p p In the end we will have x == b n mod ( p p p .....p n So x = b n will then be the number we are looking for.

42. 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

Search Next: 0.5.15 Large Prime Number Up: 0.5 Miscellaneous Algorithms Previous: 0.5.13 Horner's Rule 0.5.14 Chinese Remainder Theorem Scott Gasch

43. Mathenomicon.net : Reference : Chinese Remainder Theorem Mathenomicon.net, chinese remainder theorem. An early result in number theory. Theorem2.1 (chinese remainder theorem) Let be pairwise coprime (that is, ). http://www.cenius.net/refer/display.php?ArticleID=chineseremaindertheorem_ency

44. Mathenomicon.net : Reference : Chinese Remainder Theorem Mathenomicon.net, chinese remainder theorem. noun. Home. About this Site.News. Reference. AZ Index. Subject Index. Random Article. Bookshop. Search. http://www.cenius.net/refer/display.php?ArticleID=chineseremaindertheorem

45. Chinese Remainder Theorem Definition of chinese remainder theorem, possibly with links to more informationand implementations. chinese remainder theorem. (algorithm). http://www.guides.sk/CRCDict/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 Algorithms, Data Structures, and Problems home page. If you have suggestions, corrections, or comments, please get in touch with Paul E. Black paul.black@nist.gov Entry modified Fri Jul 16 10:09:06 1999. HTML page formatted Wed Dec 22 09:34:57 1999. This page's URL is http://hissa.nist.gov/dads/HTML/chineseRmndr.html

46. Chinese Remainder Theorem A Mechanical Proof of the chinese remainder theorem. David M. Russinoff.This paper (ps, pdf), which was presented at ACL2 Workshop http://nitro.xyzdns.net/~russ/david/papers/crt.html

A Mechanical Proof of the Chinese Remainder Theorem David M. Russinoff This paper ( ps pdf ), which was presented at ACL2 Workshop 2000 (see slides: ps pdf ), describes an ACL2 proof of the Chinese Remainder Theorem: If m m k are pairwise relatively prime moduli and a a k are natural numbers, then there exists a natural number x that simultaneously satisfies x a i (mod m i i k The entire proof is contained in the single event file crt.lisp , except that it depends on some lemmas from the author's library of floating-point arithmetic . In order to certify this file (after obtaining and certifying the library), first replace each of the two occurrences of " /u/druss/ " with the path to the directory under which your copy of the library resides. A second event file, summary.lisp , which contains the definitions and main lemmas involved in the proof, may then be certified.

47. Abstract: Time Efficient Chinese Remainder Theorem Algorithm For Full-field Frin Time efficient chinese remainder theorem algorithm for fullfield fringephase analysis in multi-wavelength interferometry. Catherine http://www.opticsinfobase.org/abstract.cfm?id=79263

48. Inverse Chinese Remainder Theorem - Technology Services Physics Help and Math Help Physics Forums Mathematics Number Theory inversechinese remainder theorem. Click Here inverse chinese remainder theorem. http://www.physicsforums.com/archive/t-15301

Physics Help and Math Help - Physics Forums Mathematics Number Theory View Thread : inverse chinese remainder theorem inverse chinese remainder theorem juan avellaneda hi all im new on the forum I wonder if is possible to find a method that proofs that a number IS NOT a solution of a set of congruences Maybe using the chinese remainder theorem best regards japam Register Now! Free! Talk Science! Muzza Well, simply plugging in the number in the congruences and seeing it they become true should work, right? Register Now! Free! Talk Science! juan avellaneda thats a solution but i was thinking in other possibility, a mathematical condition involving the gcd or MCM, for example im sure that it exists but i dont know how to probe that Register Now! Free! Talk Science! Hurkyl I can't imagine any way that could be simpler than plugging your number into the equation. Is there something unusual about your problem that this is not a desirable technique? Register Now! Free! Talk Science! juan avellaneda this method is disadvantageous when you have a big number , or a big number of congruences to check

49. Inverse Chinese Remainder Theorem - Physics Help And Math Help - Physics Forums inverse chinese remainder theorem. ) In general, you can use the ChineseRemainder theorem. Hurkyl. Reply With Quote. http://www.physicsforums.com/showthread.php?t=15301

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

bg="white"; Chinese Remainder Theorem Solve the following linear congruences x a i (mod m i x

51. 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

Prerequisites for Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography 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. Recommended prerequisite books: This book: (Read review) Suggested mathematical background in: - Linear Algebra - Group Theory Suggested computer language experience: N/A

52. Chinese Remainder Theorem a topic from mathhistory-list chinese remainder theorem. post amessage on this topic post a message on a new topic 13 Oct 1997 http://mathforum.org/epigone/math-history-list/yulplushun

a topic from math-history-list Chinese Remainder Theorem post a message on this topic post a message on a new topic 13 Oct 1997 Chinese Remainder Theorem , by Stacy Langton 18 Sep 2000 Chinese Remainder Theorem , by Antreas P. Hatzipolakis 13 Oct 1997 Re: Chinese Remainder Theorem , by Antreas P. Hatzipolakis 13 Oct 1997 Re: Chinese Remainder Theorem , by Milo Gardner 13 Oct 1997 Re: Chinese Remainder Theorem , by Kermit Rose 13 Oct 1997 Re: Chinese Remainder Theorem , by Antreas P. Hatzipolakis 13 Oct 1997 Re: Chinese Remainder Theorem , by Avinoam Mann The Math Forum

53. Leibnitz's Formula (was: Re: Chinese Remainder Theorem) By Antreas P. Hatzipolak Leibnitz s Formula (was Re chinese remainder theorem) by AntreasP. Hatzipolakis. reply to this message post a message on a new http://mathforum.org/epigone/math-history-list/cloxthouger

Leibnitz's Formula (was: Re: Chinese Remainder Theorem) by Antreas P. Hatzipolakis reply to this message post a message on a new topic Back to math-history-list Subject: Leibnitz's Formula (was: Re: Chinese Remainder Theorem) Author: xpolakis@hol.gr Date: http://users.hol.gr/~xpolakis/ The Math Forum

54. CHINESE REMAINDER THEOREM METHOD FOR FAST DECRYPTION TUTORIAL USING THE chinese remainder theorem (CRT) FOR FAST DECRYPTION TUTORIALBy Konrad Walus University of Calgary. chinese remainder theorem. http://cedarsgw.leeds.ac.uk/settle/u3a/chinese.html

USING THE CHINESE REMAINDER THEOREM (CRT) FOR FAST DECRYPTION TUTORIAL By Konrad Walus University of Calgary Edited by David Holdsworth An important calculation in the RSA encryption scheme is the modular exponentiation M = E d (mod n ). This is performed each time a part of the message is encrypted/decrypted. Both d and n are very large integers therefore this operation is very computationaly intensive. We must therefore find alternatives to the binary method for modular exponentiation. In this tutorial we will consider the use of the Chinese Remainder Theorem as a method for computing the modular exponentiation. The basic advantage with using the Chinese Remainder Theorem is that it allows us to split up the large modulo exponentiation into two much smaller exponentiations, one over p and one over q . These two moduli are the prime factors of n and are known. As well we can further reduce the size of the problem by using Fermat's Little Theorem as described below. This method was first proposed by Quisquater and Couvreur. Chinese Remainder Theorem Definition: A residue is the remainder of a division by a number called a modulus (i.e residue of 5/7 is 2).

55. Chinese Remainder Theorem chinese remainder theorem. In General. The chinese remainder theorem is actuallyvalid for many values, as hinted in the very first few paragraphs. http://cedarsgw.leeds.ac.uk/settle/u3a/chinese2.html

Chinese Remainder Theorem Definition: A residue is the remainder of a division by a number called a modulus (i.e residue of 5/7 is 2). Definition: A residue representation of a number x r r r k m m m k r r r k x x r i r j i j ). Such moduli are known as pairwise relatively prime(M. Shahkarami and G.A. Jullien, 2002, University of Calgary). For the case of RSA cryptography, we are only interested in having 2 factors, commonly called p and q . Their product is commonly called n p q In this case the Chinese Remainder Theorem says that if we know the residues: M p = M mod p M q = M mod q we can calculate: M mod n M p y q q M q y p p where n is the modulus p q , and y q q ) mod p y q is often called the inverse of q modulo p , and is thus often written ( q mod p We can prove that this value of M has the correct residues. M mod p M p y q q ) mod p because the second term is a multiple of p M p mod p y q q ) mod p M p and similarly for M q It only remains to prove that there is only one value of M mod n Imagine that there are two values of M M and M , each less than n M mod p M p and M mod p M p Thus: M M ) mod p In other words M M is divisible by p , and by a similar argument it is also divisible by q . Because p and q are prime, this means that

56. Combined Random Number Generator Via The Generalized Chinese Remainder Theorem Combined random number generator via the generalized chinese remainder theorem. Thispaper analyzes the combined MRG via the chinese remainder theorem. http://portal.acm.org/citation.cfm?id=632946&dl=ACM&coll=portal&CFID=11111111&CF

57. The Chinese Remainder Theorem And Its Application In A High-speed RSA Crypto Chi The chinese remainder theorem and its application in a highspeed RSA cryptochip. Full text, Full text available on the Publisher sitePublisher Site. http://portal.acm.org/citation.cfm?id=784729&dl=ACM&coll=portal&CFID=11111111&CF

58. 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

The Chinese Remainder Theorem (Ta-Yen) 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 N = 10(mod 12) = 0(mod 11) = 0(mod 10) = 4(mod 9) = 6(mod 8) = 0(mod 7) = 4(mod 6) In simple terms the method goes like this : 1. Find the least common multiple of the moduli. In our example the moduli are 12,11,10,9,8,7 and 6. How was this done? Reduce all moduli to a multiplication of prime numbers or their powers, unless they are already prime or a power of a prime alone. x x 3, 11 = 11 (already a prime), 10 = 2

59. Chinese Remainder Theorem chinese remainder theorem. The classical chinese remainder theorem is the followingresult. Now we can prove the chinese remainder theorem. Proof (CRT). http://www.math.boun.edu.tr/instructors/karabudak/math483/notes/CRT.htm

Chinese Remainder Theorem The classical Chinese Remainder Theorem is the following result. Theorem 1 [CRT] Let m ,m ,...,m k be pairwise relatively prime integers, that is gcd(m i ,m j ) = 1 whenever i j. Then, for any a ,a ,...,a k Z , the system of equations x a mod m x a mod m x a k mod m k has a solution, and the solution is unique modulo m = m m ...m k . That is, if x and x are solutions, then we have x = x mod m. It is easy to see that if x is a solution of these equations, so is x = x+jm for any j Z This result has a long history. Its earliest known formulation is in the Chinese mathematical text Sunzi suan ching Mathematical work of Sunzi ). Various historians date this text between 1st and 5th centuries AD. The proof is based on the crucial fact that if an integer is relatively prime to several other integers, then it is also relatively prime to their product. Proof (CRT). Let n i j i m j . Then gcd(m i ,n i ) = 1 = s i m i +t i n i for some s i ,t i Z . Let u i = t i n i . Then u i s i m i = 1 mod m i and u i = mod m j for j i. Then

60. RSA Speedup With Chinese Remainder Theorem Immune Against Hardware Fault Cryptan pp. 461472 RSA Speedup with chinese remainder theoremImmune against Hardware Fault Cryptanalysis. http://csdl.computer.org/comp/trans/tc/2003/04/t0461abs.htm

p p. 461-472 RSA Speedup with Chinese Remainder Theorem Immune against Hardware Fault Cryptanalysis Sung-Ming Yen, Seungjoo Kim, Seongan Lim, Sang-Jae Moon ... Chinese Remainder Theorem (CRT), cryptography, denial of service attack, factorization, fault detection, fault infective CRT, fault tolerance, hardware fault cryptanalysis, physical cryptanalysis, residue number system, side channel attack. The full text of IEEE Transactions on Computers is available to members of the IEEE Computer Society who have an online subscription and an web account

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