21. Math_class: Number Theory 101 (Chinese Remainder Theorem) I think that the story of the name of the chinese remainder theorem is, by far, The chinese remainder theorem. Given n1, n2, nr which are positive http://www.csh.rit.edu/~pat/math/series/nt/20020926/

: Number Theory 101 (Chinese Remainder Theorem) I said, in the last lesson, that we would get into factoring during this lesson. I had forgotten, at the time, that I wanted to hit on the Chinese Remainder Theorem. So, factoring will have to wait until next class. In other news, I entirely blew creating a class for two weeks ago. And, last week, I was sick to the point of inertness for 80% of the week. My apologies for blowing the class two weeks ago. I hope the content is interesting enough to bring y'all back after this unscheduled hiatus. Greatest Common Divisor (Continued) One more interesting thing to note about the Greatest Common Divisor of two numbers (at least one of which is non-zero). The Greatest Common Divisor of two numbers is the smallest positive number which can be written as a linear combination of the two numbers. It shouldn't come as a surprise to you that the proof of this takes advantage of the Well-Ordering Principle. Just about any mathematical statement containing "smallest number" requires the Well-Ordering Principle. The proof creates a set of all positive numbers a * u + b * v where u and v are integers. It shows that the set is non-empty (one way to do this is to let

22. The Chinese Remainder Theorem The chinese remainder theorem. In this section, we see how to solve simple Theorem 1.7.31 (chinese remainder theorem) Let $ n_1 1$ and $ n_2 1$ be http://web.usna.navy.mil/~wdj/book/node37.html

Next: General Chinese remainder theorem Up: Congruences Previous: Second method (due to Contents Index The Chinese remainder theorem In this section, we see how to solve simple simultaneous congruences modulo . This will be applied to the study of the Euler -function. Theorem 1.7.31 (Chinese remainder theorem) Let and be relatively prime integers. Then has a simultaneous solution . Furthermore, if are two solutions to ( ) then Example 1.7.32 and proof if and only if , for some . Therefore, the truth of the existence claim above is reduced to finding an integer such that . Since , there are integers such that , so . This implies , where . Thus a solution exists. To prove uniqueness , let and . Subtracting, we get and . Since , the result follows. Subsections General Chinese remainder theorem Next: General Chinese remainder theorem Up: Congruences Previous: Second method (due to Contents Index David Joyner 2002-08-23

23. General Chinese Remainder Theorem Theorem 1.7.33 (chinese remainder theorem, general version) Let $ n_1 1,n_2 1, ,n_k 1 be pairwise relatively prime integers. Let $ n=n_1n_2 n_k$ . Then http://web.usna.navy.mil/~wdj/book/node38.html

Next: An application to Euler's Up: The Chinese remainder theorem Previous: The Chinese remainder theorem Contents Index General Chinese remainder theorem Theorem 1.7.33 (Chinese remainder theorem, general version) Let be pairwise relatively prime integers. Let . Then has a simultaneous solution . Furthermore, if are two solutions to ( ) then This follows from the case proven above using mathematical induction. The details are left as an exercise. We give a different proof below. proof : As runs over all integers , the -tuples form a collection of distinct -tuples in . (Exercise: show why they are distinct.) On the other hand, there are distinct, -tuples with . Therefore, each -tuple must equal one of the , for David Joyner 2002-08-23

24. Sun_Zi This, of course, is important for it is a problem which is solved using thechinese remainder theorem. It is the earliest known occurrence of this type of http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Sun_Zi.html

Sun Zi Born: about 400 in China Died: about 460 in China Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing Nothing is known about Sun Zi except his text Sunzi suanjing (Sun Zi's Mathematical Manual). Dating this is made more difficult since it is not known how much the text was changed or added to over time. Let us first look at the various theories about the date. In the 17 th century Sun Zi was identified with Sun Wu, a famous military expert of the sixth century BC who wrote Sun Zi's art of war. The Ruan Yuan in his Chouren zhuan or Biographies of astronomers and mathematicians (1799) certainly realised that references in certain problems in the Sunzi suanjing meant that the identification with Sun Wu was incorrect. He placed Sun Zi around 250 BC but knew that there was still problems with this dating which he said would have to be studied later. Indeed such studies did take place and Dai Zhen, an 18 th century scholar, historian and mathematician, stated that it was impossible for the Sunzi suanjing to have been written before about 50 BC.

25. Hilbert Functions And The Chinese Remainder Theorem: Open Problems The chinese remainder theorem (CRT) tells us that H R R/I1 x xR/In issurjective with kernel I=I1 In; ie, there is a solution, and given any http://www.math.unl.edu/~bharbour/UNLregionwstalk.html

Hilbert Functions and the Chinese Remainder Theorem: Open Problems Notes: This 20 minute talk was given February 27, 1999, at the UNL Regional Workshop. For this talk let k be any algebraically closed field. Preliminary Problems Problem 1 : Given points x , ... , x n of k and values v , ... , v n of k, find all polynomials f(x) in k[x] such that f(x i ) = v i for all i. Solution : There exists a unique solution, f L (x), of degree at most n-1; it is given by the Lagrange interpolation formula: Let g(x) = (x-x )...(x-x n Let g i (x) = g(x)/(x-x i ) [This is a polynomial.] Then f L (x) = (v /g (x ))g (x) + ... + (v n /g n (x n ))g n (x) The complete set of solutions is f L (x) + (g); i.e., f L (x) + p(x)g(x), where p(x) is any polynomial. [In the usual notation, (g) is the ideal generated by g(x).] Restatement of Problem 1 and Solution : The problem is to find all f(x) conguent mod I j to v j , for all j, where I j is the ideal (x-x j ) of the ring R=k[x]. I.e., given an element v = (v , ... , v n ) of R/I x ... x R/I n , find all elements f of R mapping to v under the homomorphism H : f -> (f , ... , f

26. MATLAB Central File Exchange - Chinese Remainder Theorem For Polynomials of the book by Neal Koblitz describing the chinese remainder theorem for Integers.Blahut also describes the chinese remainder theorem for Integers in P 70. http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=5841&ob

27. MATLAB Central File Exchange - Chinese Remainder Theorem For Integers The answer is the chinese remainder theorem for Integers refer NK/P21 Refer chinese remainder theorem in the books by http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=5851&ob

28. Chinese Remainder Theorem -- Facts, Info, And Encyclopedia Article chinese remainder theorem. Categories Theorems, Commutative algebra, The chinese remainder theorem is the name for several related results in (Click http://www.absoluteastronomy.com/encyclopedia/c/ch/chinese_remainder_theorem.htm

Chinese remainder theorem [Categories: Theorems, Commutative algebra, Modular arithmetic] The Chinese remainder theorem is the name for several related results in (Click link for more info and facts about abstract algebra) abstract algebra and (Click link for more info and facts about number theory) number theory Simultaneous congruences of integers The original form of the theorem, contained in a book by the Chinese mathematician (Click link for more info and facts about Qin Jiushao) Qin Jiushao published in 1247, is a statement about simultaneous congruences (see (Click link for more info and facts about modular arithmetic) modular arithmetic ). Suppose n n k are positive (Any of the natural numbers (positive or negative) or zero) integers which are pairwise (Click link for more info and facts about coprime) coprime (meaning (Click link for more info and facts about gcd) 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 Furthermore, all solutions

29. RSA Speedup With Chinese Remainder Theorem Immune Against Hardware Fault Cryptan This article considers the problem of how to prevent the fast RSA signature anddecryption computation with residue number system (or called the CRTbased http://doi.ieeecomputersociety.org/10.1109/TC.2003.1190587

Search: Advanced Search Home Digital Library Site Map ... April 2003 (Vol. 52, No. 4)   pp. 461-472 RSA Speedup with Chinese Remainder Theorem Immune against Hardware Fault Cryptanalysis Sung-Ming Yen Seungjoo Kim Seongan Lim Sang-Jae Moon Full Article Text: DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2003.1190587 Abstract About References Back to Top References R.L. Rivest,, A. Shamir, and L.A. Adleman,, "A Method for Obtaining Digital Signatures and Public Key Cryptosystems," Comm. ACM, vol. 21, pp. 120-126, 1978. T. ElGamal, A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms ... vol. 31, no. 4, pp. 469-472, 1985. Proc. Second USENIX Workshop Electronic Commerce, pp. 1-11, 1996. Pre-proc. 1997 Security Protocols Workshop, Apr. 1997. pp. 37-51, 1997. Pre-proc. 1997 Security Protocols Workshop, Pre-proc. 1997 Symp. Cryptography and Information Security, 29 Jan.-1 Feb. 1997. An earlier version was presented at the rump session of ASIACRYPT '96.

30. Making Mathematics: Mathematics Tools: The Chinese Remainder Theorem The chinese remainder theorem states that for relatively prime m1, m2, For a discussion and some sample problems, see chinese remainder theorem. http://www2.edc.org/makingmath/mathtools/remainder/remainder.asp

Home Mathematics Tools The Chinese Remainder Theorem The Chinese Remainder Theorem states that for relatively prime m , m , ... , there is a unique solution (mod m m ...) to the system of congruences x = a (mod m x = a (mod m For a discussion and some sample problems, see Chinese Remainder Theorem Making Mathematics Home Mathematics Projects Students ... Parents Translations of mathematical formulas for web display were created by

31. Chinese Remainder Theorem Author hasinoff What is the chinese remainder theorem as it applies to solvingequations The chinese remainder theorem if (Mi,Mj) = 1 for i ! http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH056.HTM

Ask A Scientist Mathematics Archive Chinese remainder theorem Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

32. Chinese Remainder Theorem Re chinese remainder theorem by Henno Brandsma (Feb 15, 2005). Re Re ChineseRemainder Theorem by mat12 (Feb 15, 2005). Re Re Re Chinese Remainder http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraic_topologist;task=show_msg;

33. Re: Chinese Remainder Theorem Re chinese remainder theorem by Henno Brandsma (Feb 15, 2005) In replyto chinese remainder theorem , posted by Mat12 on Feb 14, 2005 http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraic_topologist;task=show_msg;

34. Chinese Remainder Theorem - Definition Of Chinese Remainder Theorem In Encyclope The chinese remainder theorem is any of a number of related results in abstractalgebra and number theory.Contents showTocToggle( show , hide )1 http://encyclopedia.laborlawtalk.com/Chinese_remainder_theorem

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law 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 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 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

35. The Chinese Remainder Theorem The chinese remainder theorem states that if you have Q numbers N1 to NQ which The chinese remainder theorem says that every integer from 0 to 14 will http://www.disappearing-inc.com/C/chineseremainder.html

Cyclopedia Cryptologia The Chinese Remainder Theorem The chinese remainder theorem states that if you have Q numbers N to N Q which have no factors in common, then any integer greater than or equal to and less than the product of all numbers N can be uniquely represented by a series consisting of the remainders of division by the numbers N. To walk through a simple example of how this works, let's say that N is 3 and N is 5. The Chinese remainder theorem says that every integer from to 14 will have a unique set of remainders modulo these two N's. And in fact, that's 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. 7 has a remainder of 1 modulo 3 and a remainder of 2 modulo 5.

36. The Chinese Remainder Theorem We prove the chinese remainder theorem and Thue s Theorem as well as severaluseful number theory propositions. MML Identifier WSIERP_1 http://mizar.uwb.edu.pl/JFM/Vol9/wsierp_1.html

Journal of Formalized Mathematics Volume 9, 1997 University of Bialystok Association of Mizar Users The Chinese Remainder Theorem Andrzej Kondracki AMS Management Systems Poland, Warsaw Summary. MML Identifier: The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Bibliography 1] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics 2] Grzegorz Bancerek. The ordinal numbers Journal of Formalized Mathematics 3] Grzegorz Bancerek. Joining of decorated trees Journal of Formalized Mathematics 4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics 5] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics 6] Czeslaw Bylinski. The sum and product of finite sequences of real numbers Journal of Formalized Mathematics 7] Katarzyna Jankowska. Transpose matrices and groups of permutations Journal of Formalized Mathematics 8] Andrzej Kondracki. Basic properties of rational numbers Journal of Formalized Mathematics 9] Jaroslaw Kotowicz and Yatsuka Nakamura.

37. The Chinese Remainder Theorem Recall that the chinese remainder theorem from elementary number theory In terms of rings, the chinese remainder theorem asserts that the natural map http://modular.fas.harvard.edu/papers/ant/html/node31.html

Next: Discrimannts, Norms, and Finiteness Up: Chinese Remainder Theorem Previous: Chinese Remainder Theorem Contents Index The Chinese Remainder Theorem Recall that the Chinese Remainder Theorem from elementary number theory asserts that if are integers that are coprime in pairs, and are integers, then there exists an integer such that for each . In terms of rings, the Chinese Remainder Theorem asserts that the natural map is an isomorphism. This result generalizes to rings of integers of number fields. Lemma If and are coprime ideals in , then Proof . The ideal is the largest ideal of that is divisible by (contained in) both and . Since and are coprime, is divisible by , i.e., . By definition of ideal , which completes the proof. Remark This lemma is true for any ring and ideals such that . For the general proof, choose and such that . If then so , and the other inclusion is obvious by definition. Theorem (Chinese Remainder Theorem) Suppose are ideals of such that for any . Then the natural homomorphism induces an isomorphism Thus given any then there exists such that for Proof . First assume that we know the theorem in the case when the are powers of prime ideals. Then we can deduce the general case by noting that each

38. Chinese Remainder Theorem In this section we will prove the chinese remainder theorem for rings of integers,deduce several surprising and useful consequences, then learn about http://modular.fas.harvard.edu/papers/ant/html/node30.html

Next: The Chinese Remainder Theorem Up: Classical Viewpoint Previous: Essential Discriminant Divisors Contents Index Chinese Remainder Theorem In this section we will prove the Chinese Remainder Theorem for rings of integers, deduce several surprising and useful consequences, then learn about discriminants, and finally norms of ideals. We will also define the class group of and state the main theorem about it. The tools we develop here illustrate the power of what we have already proved about rings of integers, and will be used over and over again to prove other deeper results in algebraic number theory. It is essentially to understand everything we discuss in this chapter very well. Subsections The Chinese Remainder Theorem William Stein 2004-05-06

39. 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

40. Chinese Remainder Theorem chinese remainder theorem (CRT) The following problem was posed by Sunzi SunTsu (4th century AD) in the book Sunzi Suanjing There are certain things http://www.chinapage.org/math/crt.html

Chinese Remainder Theorem Chinese Remainder Theorem (CRT) The following problem was posed by Sunzi [Sun Tsu] (4th century AD) in the book Sunzi Suanjing: There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? The answer is hidden in the following song: Math Page

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