61. Citation symposium on Computer architecture toc 1993 , San Diego, California, United StatesThe chinese remainder theorem and the prime memory system Also published http://portal.acm.org/citation.cfm?id=165172&dl=ACM&coll=portal&CFID=11111111&CF

62. The Chinese Remainder Theorem Factorization The chinese remainder theorem. This is an application of Euclid'stheorem (although the ancient Chinese proved it differently). http://vorpal.mcs.drexel.edu/course/founds/numbertheory/node8.html

Next: Properties of the Euler Up: Notes on Number Theory Previous: Unique Factorization The Chinese Remainder Theorem This is an application of Euclid's theorem (although the ancient Chinese proved it differently). Let be two relatively prime integers (i.e., ) and suppose Then there exists a unique solution to these two equations modulo Proof: Since , we can find integers and such that Just set and reduce it mod . If we had any other solution to these equations mod , say , then would satisfy so would be a multiple of and . Since and have no factors in common, the smallest multiple of both of them is , so would have to be a multiple of and Example: Say and and suppose our two equations are Then so we set which solves the problem. Next: Properties of the Euler Up: Notes on Number Theory Previous: Unique Factorization Justin R. Smith 2001-05-18

63. Chinese Remainder Theorem chinese remainder theorem. Dustin Davis CS 2310 SLCC 9/15/02 Pleasefill the necessary values in the form,then click OK button. http://school.davisvillage.com/cs2310/assignments/3.html

Chinese Remainder Theorem Please fill the necessary values in the form,then click OK button. x ( mod x ( mod x ( mod click to get the solution.

64. [pac] Chinese Remainder Theorem pac chinese remainder theorem. Previous message pac ANNOUNCECryptRSA 1.33; Next message pac chinese remainder theorem; http://lists.vipul.net/pipermail/pac/2001-April/000034.html

[pac] Chinese Remainder Theorem Benjamin Trott ben@rhumba.pair.com Fri, 06 Apr 2001 21:17:21 -0700 Previous message: [pac] [ANNOUNCE] Crypt::RSA 1.33 Next message: [pac] Chinese Remainder Theorem Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Vipul In another message (in the "A few bugs/wishes") thread you said: > They are not used yet. These parameters can be used instead of decryption exponent `d' for faster decryption using the Chinese Remainder Theorum. This is also on TODO. Previous message: [pac] [ANNOUNCE] Crypt::RSA 1.33 Next message: [pac] Chinese Remainder Theorem Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

65. [pac] Chinese Remainder Theorem pac chinese remainder theorem. Previous message pac chinese remainder theorem;Next message pac Re Using keys not generated by your module http://lists.vipul.net/pipermail/pac/2001-April/000037.html

[pac] Chinese Remainder Theorem Vipul Ved Prakash mail@vipul.net Fri, 6 Apr 2001 15:00:39 -0700 Previous message: [pac] Chinese Remainder Theorem Next message: [pac] Re: Using keys not generated by your module ... Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Vipul In another message (in the "A few bugs/wishes") thread you said: I have some code that implements the CRT (from Net::SSH::Perl::Util). It lying around, as well. 'u', I believe, is just: where mod_inverse is my($a, $n) = @_; my $m = Mod(1, $n); lift($m / $a); Here's the code: my %params = @_; my $p2 = mod_exp($I % $p, $d % ($p-1), $p); my $q2 = mod_exp($I % $q, $d % ($q-1), $q); my $r = (($q2 - $p2) * $u) % $q; return $p2 + ($p * $r); my($a, $exp, $n) = @_; my $m = Mod($a, $n); lift($m ** $exp); Hope this is useful. Excellent! Unless you want to send me a patch, I'll integrate this with :: http://www.vipul.net/ PGP Fingerprint d5f78d9fc694a45a00ae086062498922 Previous message: [pac] Chinese Remainder Theorem Next message: [pac] Re: Using keys not generated by your module ...

66. EECS 658 Problem Set #2 Fall 1999 30, 1999. THIS WEEK chinese remainder theorem, Euclidian algorithm. Apply thechinese remainder theorem to derive the Lagrange interpolation formula. http://www.eecs.umich.edu/~aey/eecs658/probset2.html

EECS 658_PROBLEM SET #2_Fall 1999 ASSIGNED: Sept. 23, 1999. READ: Convolutions handout given out on Thursday Sept. 23. DUE DATE: Sept. 30, 1999. THIS WEEK: Chinese remainder theorem, Euclidian algorithm. A very simple application of the polynomial Chinese remainder theorem Show that the solution to M(z)X(z)=1 mod(z-a) is X(z)=1/M(a), provided M(a) isn't 0. This is useful for the polynomial Chinese remainder theorem solution procedure. Show you can formulate the interpolation problem X(z i )=c i as: X(z)=c i mod(z-z i ), i=1...n. Apply the Chinese remainder theorem to derive the Lagrange interpolation formula. Apply Winograd y +y z+y +h z+h +u z+u HINTS: (1) z You can use the latter hint to avoid the Euclidian algorithm for polynomials, if desired. We wish to multiply (3458)(2992) using residue number systems (illustrative example). Use as moduli: 2,3,5,7,11,13,17,19; their product is about 10 million (large enough). Compute the residues of 3458 and 2992 for each modulus (total of 16 numbers). Compute the residues of the product for each modulus (total of 8 numbers). Use the Chinese remainder theorem to compute (3458)(2992). Confirm this is right.

67. Math Resources From Grau-Hall Scientific Arapahulian Rope Computer Big Math Attack The chinese remainder theorem The ChineseRemainder Theorem from University of Colorado - Denver Fields Medal http://www.grauhall.com/math.htm

grauhall grauhall.com Search this site! Type in one or more words in the white rectangle above, and then click on "Search." Powered by Atomz.com Teacher's Resources: Math e p i Click once on any underlined text to go to that site: Arapahulian Rope Computer Big Math Attack The Chinese Remainder Theorem The Chinese Remainder Theorem - from University of Colorado - Denver Fields Medal Winners Handbook of Applied Cryptography - downloadable copy Hilbert Functions and the Chinese Remainder Theorem KdV Institute - math links Kids Math Syvum Book - children's educational software from Syvum Technologies Lawrence Hall of Science - math and science education links Los Alamos National Laboratory - math e-print archive MacTutor History of Math Virtual Math Lab - from KCOS TV Math Spans All Dimensions , from the Joint Policy Board for Mathematics Merit Software - "DollarSkills" and "Word Problem Shape-Up" software Quicksilver Educational Games - Math At Work Robot Repair - a math sight from Dr. Gene Oldfield Softseek - math software, shareware and freeware SpeedDrillMath - math software Syber Saver - educational software and games for kids, including "Math Blaster"

68. CHALLENGING PROOFS BY MATHEMATICAL INDUCTION 006 Two definition of Even are equivalent. 007 - All numbersare odd or even. 008 - chinese remainder theorem http://cs-www.cs.yale.edu/homes/carsten/challenges/challenges.html

Challenging Proofs By Mathematical Induction Maintainer: Department of Computer Science Yale University Home Challenges This collection of challenging examples has been assembled for researchers who are working on inductive theorem provers for the purpose of provoding a body of test examples. The description of the example problems is purposely kept informal, in order not to intervene with the representation of a problem in a particular theorem prover. Major contributions to this corpus have come from Louise Dennis, University of Edinburgh Dieter Hutter, DFKI If you want to contribute to the corpus, please send mail to Carsten Schuermann . New additions will be annouced on the challenges mailing list. You may subscribe to the mailing list by sending email to majordomo@twelf.org with "subscribe challenge" in the body (the header might stay empty). Everybody is invited to submit challenges. 001 - Arithmetic Geometric Mean First Order Version of the Arithmetic/Geometric Mean Higher Order Version of the Arithmetic/Geometric Mean (Version 1) Higher Order Version of the Arithmetic/Geometric Mean (Version 2) 002 - Length of the joined list of two lists of even length is even ... 003 - A member of a list is a member of that list joined to another 004 - Rotate Length Rotate Length (Version I) Rotate Length (Version II) 005 - Binomial Theorems Binomial Theorems (Version I) Binomial Theorems (Version II) 006 - Two definition of Even are equivalent 007 - All numbers are odd or even 008 - Chinese Remainder Theorem

69. Untitled Lenore Blum, B6511, mablum@cityu.edu.hk. ALGEBRA, ALGORITHMS and APPLICATIONSPrototype example The chinese remainder theorem (CRT). http://moscow.cityu.edu.hk/~maweb/history/discrete_mathematics/outline.html

Course Outline Module Title: Discrete Mathematics II Module Code: MA2102 Semester A, 1997 Lecturer: Prof. Lenore Blum mablum@cityu.edu.hk ALGEBRA, ALGORITHMS and APPLICATIONS Prototype example: The Chinese Remainder Theorem (CRT) Review: Mathematical Reasoning Methods of Proof Set Theory, Relations, Functions Fundamentals: Algorithms, Number Theory Algorithms: The growth of functions Complexity of algorithms Computer operations with integers Complexity of integer operations Number Theory: The Integers and Division Divisibility Congruences Prime numbers Greatest Common Divisors and Prime Factorization The Euclidean algorithm Continued Fractions The fundamental theorem of arithmetic Fermat numbers, primality tests and factorization methods The Chinese Remainder Theorem and applications Applications of Number Theory: Coding and Cryptography Round-robin tournaments, Computer file storage OTHER MATERIAL may include: Turing Machines, Finite Automata, Boolean Algebra and Applications Matrices and Graphs Abstract Algebra: Groups, Rings, Fields More Number Theory and Applications Main Text: Elementary Number Theory and its Applications 3rd edition, by Kenneth H. Rosen, Addison Wesley Publishing Co., 1993.

70. URS Symposium 2001 Abstract Winners On the chinese remainder theorem andIts Applications. David Tello and Carmelo Tapia. In a book similar http://www.uic.edu/orgs/urs/abstracts/2001/10.html

Why An Undergraduate Research Symposium? Past Winners 2002 Abstract Winners Boooklet (PDF) 2001 Abstract Winners ... Pictures Undergraduate Research Symposium 2001 Abstract Winners On the Chinese Remainder Theorem and Its Applications David Tello and Carmelo Tapia In a book similar to that of the "Arithmetic in Nine Sections," (1257 AD) written by the Chinese mathematician, Sun-tzã, we encounter the first Chinese problem in indeterminate analysis. The problem says: "There are things of an unknown number which when divided by 3 leaves 2, by 5 leave 3, and by 7 leave 2. What is the (smallest) number?" This problem is considered to be the beginnings of the famous Chinese Remainder Theorem of Elementary Number Theory. In our process of extending the Chinese Remainder Theorem to polynomials, we found that in the particular case when the divisors are different prime polynomials of degree 1, the algorithm for finding the desired polynomial is the LaGrange Interpolation Formula found in Numerical Analysis. To return to the 2001 Abstract Winners, click here.

71. Untitled The result is a problem called the chinese remainder theorem. It is because of SunTsu’sproposal, that this problem is called the chinese remainder theorem. http://eiffel.ps.uci.edu/cyu/p231C/Projects/outlines00/shean.html

72. Chinese Remainder Theorem 23Oct-01 chinese remainder theorem http://www.cs.appstate.edu/~blk/cs5110/ch31/ch31_part5.htm

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73. Error Analysis Of Approximate Chinese-Remainder-Theorem Decoding pp. 13441348 Error Analysis of Approximate chinese-remainder-theorem Decoding. http://www.computer.org/tc/tc1995/t1344abs.htm

p p. 1344-1348 Error Analysis of Approximate Chinese-Remainder-Theorem Decoding Behrooz  Parhami, Ching Yu  Hung Abstract —Approximate Chinese-remainder-theorem decoding of residue numbers is a useful operation in residue arithmetic. The decoding yields an approximation to ( X ... Computation errors, computer arithmetic, residue numbers, RNS representation, scaled decoding. 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

74. The Chinese Remainder Problem It has given rise to such terms as chinese remaindering and chineseremainder theorem. There even is a book entirely devoted to it. http://members.tripod.com/~Probability/diophan/chinese.htm

The Chinese Remainder Problem There is a number which divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What is the number? Sun Tsu Suan Ching (4th century AD) The above is a classic problem that has had considerable impact. It has given rise to such terms as "Chinese remaindering" and "Chinese remainder theorem." There even is a book entirely devoted to it. The problem usually is formulated in modular number notation but here we shall use the equivalent arbitrary integer constant notation. N = 3t + 2 N = 5u + 3 N = 7v + 2 This is a system of three linear Diophantine equations in four unknowns. In solving the system algebraically the usual procedure is to substitute the first equation into the second, resulting in a single equation in two unknowns. This is solved for t or u in terms of a new arbitrary integer variable. This solution for N then is substituted into the third equation which is then solved for v. Once v is known, then N can be calculated from the last equation. Let's do it!

75. A Possible New Quantum Algorithm: Arithmetic With Large Integers Via The Chinese computation is more likely to be of this type. Arithmetic by the Chineseremainder theorem is a highly parallel procedure The idea. http://citeseer.nj.nec.com/337637.html

A Possible New Quantum Algorithm: Arithmetic with Large Integers via the Chinese Remainder Theorem (Make Corrections) S. A. Fulling Home/Search Context Related View or download: tamu.edu/~stephen.fulling/. chinqc.ps Cached: PS.gz PS PDF DjVu ... Help From: tamu.edu/~stephen.fulling/./qc (more) Homepages: S.Fulling HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: kind, in which the constructive interference that builds up the answer takes place at the level of classical waves or signals. Arguably, eventual general-purpose quantum computation is more likely to be of this type. Arithmetic by the Chinese remainder theorem is a highly parallel procedure [36]. The idea is that the fixed word size of a standard computer can be transcended by doing all addition, subtraction, and multiplication modulo each of a set of pairwise relatively prime integers... (Update) Active bibliography (related documents): More All Large Numbers, the Chinese Remainder Theorem, and the Circle of.. - Fulling (2001) (Correct) ... (Correct) Similar documents based on text: More All On Rigidity And The Albanese Variety For Parallelizable Manifolds - Winkelmann (1997) (Correct) ... What We Should Have Learned From G. H. Hardy About Quantum Field.. - Fulling

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