Number Theory: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=146
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 17 Apr 2014 07:11:45 GMTThu, 17 Apr 2014 07:11:45 GMTNew applications in the Number Theory categoryhttp://www.mapleprimes.com/images/mapleapps.gifNumber Theory: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=146
Integer Partition Tools
http://www.maplesoft.com/applications/view.aspx?SID=135973&ref=Feed
<p>This application will allow the user to generate integer partitions of a given type. Generators for partitions into odd parts, even parts, parts of restricted size, and parts +/- 1 mod d are included, as well as generators for partitions with a restricted number of parts. Other useful generators are present, along with a means of counting the number of partitions of a given type.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Integer Partition Tools" align="left"/><p>This application will allow the user to generate integer partitions of a given type. Generators for partitions into odd parts, even parts, parts of restricted size, and parts +/- 1 mod d are included, as well as generators for partitions with a restricted number of parts. Other useful generators are present, along with a means of counting the number of partitions of a given type.</p>135973Fri, 20 Jul 2012 04:00:00 ZSamuel PineSamuel PineReversible and palindromic primes
http://www.maplesoft.com/applications/view.aspx?SID=133995&ref=Feed
<p>The application allows to compute the set of all reversible n-digit <br />prime numbers, with n = 2 .. 8, sets of all palindromic n-digit primes, <br />n = 3, 5, 7 and many reversible primes with an arbitrary <br />n = 2, 3, ..., several thousands.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Reversible and palindromic primes" align="left"/><p>The application allows to compute the set of all reversible n-digit <br />prime numbers, with n = 2 .. 8, sets of all palindromic n-digit primes, <br />n = 3, 5, 7 and many reversible primes with an arbitrary <br />n = 2, 3, ..., several thousands.</p>133995Wed, 09 May 2012 04:00:00 ZAndrzej TeodorczukAndrzej TeodorczukCar Talk Puzzler
http://www.maplesoft.com/applications/view.aspx?SID=128825&ref=Feed
<p>National Public Radio in the USA carries Car Talk, a humorous phone-in program in which Tom and Ray Magliozzi (Click and Clack, the Tappet Brothers) diagnose and offer solutions for mysterious auto-related maladies. One of the program's segments is a weekly Puzzler, a logic (or other) mental puzzle begging for a solution. On May 21, 2011, their Puzzler caught my attention. Here’s the synopsis:</p>
<p>A six-digit odometer shows a palindromic number. The car it's in is driven no more than an hour, and again the odometer shows a palindromic number. How far was the car driven?</p>
<p>Intrigued, I decided to use Maple to solve the problem. I also wrote a <a href="http://www.mapleprimes.com/maplesoftblog/128796-Car-Talk-Puzzler" class="plainlink">blog post</a> describing how I solved it.</p><img src="/view.aspx?si=128825/thumb.jpg" alt="Car Talk Puzzler" align="left"/><p>National Public Radio in the USA carries Car Talk, a humorous phone-in program in which Tom and Ray Magliozzi (Click and Clack, the Tappet Brothers) diagnose and offer solutions for mysterious auto-related maladies. One of the program's segments is a weekly Puzzler, a logic (or other) mental puzzle begging for a solution. On May 21, 2011, their Puzzler caught my attention. Here’s the synopsis:</p>
<p>A six-digit odometer shows a palindromic number. The car it's in is driven no more than an hour, and again the odometer shows a palindromic number. How far was the car driven?</p>
<p>Intrigued, I decided to use Maple to solve the problem. I also wrote a <a href="http://www.mapleprimes.com/maplesoftblog/128796-Car-Talk-Puzzler" class="plainlink">blog post</a> describing how I solved it.</p>128825Thu, 15 Dec 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezNumber Theory Integer Types
http://www.maplesoft.com/applications/view.aspx?SID=126158&ref=Feed
<p>Number Theory Integer Types</p>
<p>by Michael Carter</p>
<p>Carl Friedrich Gauss was known as one of the top ten greatest mathematicians. He was known as the prince of mathematics. Gauss is quoted calling mathematics the "queen of the sciences" and calling number theory the "queen of mathematics." Number theory is called "higher arithmetic." Number Theory is the study of whole numbers and sometimes spills over to rational numbers. The whole numbers that is the most fascinating are the primes. We know very little about the primes. Because we know so little about the primes we can make encryption for banks and computer security systems. We also have another fascinating number that we know more about, the composite numbers. The composite numbers are composed of two or more prime's products.</p>
<p>There are approximately 10 fields within Number Theory: Elementary, Analytic, Algebraic, Geometry, Combinatorial, Computational, Arithmetic Algebraic, Arithmetic Topology, Arithmetic Dynamics, and Modular Form. All these fields can take advantage of this maple package.</p>
<p>This Maple package is called inttypes, which is short for Integer Types. The inttypes package provides over 160 whole number types for Number Theorists to explore numbers. This package may be used as a tool for an academic course, Maple programming, or for research. It is also fun to play with these types for exploration and developing intuition about numbers. In addition, to these number theory integer types, this package also offers three very important functions: ithcomposite, ithprimorial, and printtypes. Maple have a function called, ithprime; which is used to find a particular prime based on its order. This inttypes package provides a function that gives a particular composite number based on its order, called ithcomposite. There is also an ithprimorialfunction that provides a particular primordial number based on its order. Finally, we have a function that will search among all the types of this package for a particular number and print its type. Srinivasa Ramanujan's taxicab number, 1729, is a good test for printtypes (see below). The number, 3, is also a good test for the printtypes function. On the Number Theory Types Test: Just change the integer on each of the for-loops for larger tests when testing the number theory types.</p>
<p>All types return a Boolean value of either true or false. It does not return a number. Note in the testing of the types below we can give input but the output is true or false. The input integer is not the output, hence, do not get confused and begin believing that the input is the output.</p>
<p>Finally, there is an official website for referencing and testing the integer types: "The On-Line Encyclopedia of Integer Sequences™ (OEIS™)" located at: http://oeis.org/</p>
<p>I hope you enjoy this package as much as I enjoyed creating it for you.</p>
<p>Cheers.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Number Theory Integer Types" align="left"/><p>Number Theory Integer Types</p>
<p>by Michael Carter</p>
<p>Carl Friedrich Gauss was known as one of the top ten greatest mathematicians. He was known as the prince of mathematics. Gauss is quoted calling mathematics the "queen of the sciences" and calling number theory the "queen of mathematics." Number theory is called "higher arithmetic." Number Theory is the study of whole numbers and sometimes spills over to rational numbers. The whole numbers that is the most fascinating are the primes. We know very little about the primes. Because we know so little about the primes we can make encryption for banks and computer security systems. We also have another fascinating number that we know more about, the composite numbers. The composite numbers are composed of two or more prime's products.</p>
<p>There are approximately 10 fields within Number Theory: Elementary, Analytic, Algebraic, Geometry, Combinatorial, Computational, Arithmetic Algebraic, Arithmetic Topology, Arithmetic Dynamics, and Modular Form. All these fields can take advantage of this maple package.</p>
<p>This Maple package is called inttypes, which is short for Integer Types. The inttypes package provides over 160 whole number types for Number Theorists to explore numbers. This package may be used as a tool for an academic course, Maple programming, or for research. It is also fun to play with these types for exploration and developing intuition about numbers. In addition, to these number theory integer types, this package also offers three very important functions: ithcomposite, ithprimorial, and printtypes. Maple have a function called, ithprime; which is used to find a particular prime based on its order. This inttypes package provides a function that gives a particular composite number based on its order, called ithcomposite. There is also an ithprimorialfunction that provides a particular primordial number based on its order. Finally, we have a function that will search among all the types of this package for a particular number and print its type. Srinivasa Ramanujan's taxicab number, 1729, is a good test for printtypes (see below). The number, 3, is also a good test for the printtypes function. On the Number Theory Types Test: Just change the integer on each of the for-loops for larger tests when testing the number theory types.</p>
<p>All types return a Boolean value of either true or false. It does not return a number. Note in the testing of the types below we can give input but the output is true or false. The input integer is not the output, hence, do not get confused and begin believing that the input is the output.</p>
<p>Finally, there is an official website for referencing and testing the integer types: "The On-Line Encyclopedia of Integer Sequences™ (OEIS™)" located at: http://oeis.org/</p>
<p>I hope you enjoy this package as much as I enjoyed creating it for you.</p>
<p>Cheers.</p>126158Sat, 01 Oct 2011 04:00:00 ZMichael CarterMichael CarterChinese Remainder Theorem
http://www.maplesoft.com/applications/view.aspx?SID=99372&ref=Feed
<p>Let m[1],m[2],...,m[n] be pairwise relatively prime integers</p>
<p> Then the simultaneous congruence</p>
<p>x = r[1] mod m[1]</p>
<p>...</p>
<p>x = r[n] mod m[n]</p>
<p>has a unique solution modulo the product m[1].m[2]- -.-. .m[n]</p><img src="/view.aspx?si=99372/maple_icon.jpg" alt="Chinese Remainder Theorem" align="left"/><p>Let m[1],m[2],...,m[n] be pairwise relatively prime integers</p>
<p> Then the simultaneous congruence</p>
<p>x = r[1] mod m[1]</p>
<p>...</p>
<p>x = r[n] mod m[n]</p>
<p>has a unique solution modulo the product m[1].m[2]- -.-. .m[n]</p>99372Wed, 24 Nov 2010 05:00:00 ZRoland EngdahlRoland Engdahlvan Roomen Problem
http://www.maplesoft.com/applications/view.aspx?SID=96978&ref=Feed
<p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p><img src="/view.aspx?si=96978/maple_icon.jpg" alt="van Roomen Problem" align="left"/><p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p>96978Sat, 18 Sep 2010 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyQUADRATIC FIELDS and CLASS NUMBER FORMULA
http://www.maplesoft.com/applications/view.aspx?SID=34981&ref=Feed
<p>The aim of this document is to give same procedures in order to work explicitely with quadratic fields; in particular the idea of work was born in order to find a useful procedure to compute the class number of a quadratic filed.</p>
<div>Many problems of number theory lead to the important question in the arithmetic of algebraic number fields of decomposition of algebraic numbers into prime factors. We shall define a procedure <em>Dec </em> which returns the decomposition of algebraic numbers into prime factors in a quadratic filed. The problems of factorization are very closely connected with Fermat's (last) theorem. Historically, it was precisely the problem of Fermat's theorem which led Kummer to his fundamental work on the arithmetic of algebraic numbers. </div>
<div>It is well known the important role of the number <em>h </em>of divisor classes of algebraic number filed play in the arithmetic of the field. Thus one would like to have an explicit formula for the number <em>h</em> in terms of simpler values which depend on the filed. Although this has not been accomplished for arbitrary algebraic number fields, for certain fields of great interest, such as quadratic fields, such formulas as been found. </div>
<div>Since all divisors are products of prime divisors and the number of prime divisors is infinite, then to compute the number <em>h</em> in a finite number of steps we must use some infinite processes. This is why, in the determination of <em>h,</em> we shall have to consider infinite products, series and other analytic concepts.</div><img src="/view.aspx?si=34981/0\images\Campi_quadratici_12.gif" alt="QUADRATIC FIELDS and CLASS NUMBER FORMULA" align="left"/><p>The aim of this document is to give same procedures in order to work explicitely with quadratic fields; in particular the idea of work was born in order to find a useful procedure to compute the class number of a quadratic filed.</p>
<div>Many problems of number theory lead to the important question in the arithmetic of algebraic number fields of decomposition of algebraic numbers into prime factors. We shall define a procedure <em>Dec </em> which returns the decomposition of algebraic numbers into prime factors in a quadratic filed. The problems of factorization are very closely connected with Fermat's (last) theorem. Historically, it was precisely the problem of Fermat's theorem which led Kummer to his fundamental work on the arithmetic of algebraic numbers. </div>
<div>It is well known the important role of the number <em>h </em>of divisor classes of algebraic number filed play in the arithmetic of the field. Thus one would like to have an explicit formula for the number <em>h</em> in terms of simpler values which depend on the filed. Although this has not been accomplished for arbitrary algebraic number fields, for certain fields of great interest, such as quadratic fields, such formulas as been found. </div>
<div>Since all divisors are products of prime divisors and the number of prime divisors is infinite, then to compute the number <em>h</em> in a finite number of steps we must use some infinite processes. This is why, in the determination of <em>h,</em> we shall have to consider infinite products, series and other analytic concepts.</div>34981Thu, 17 Dec 2009 05:00:00 ZProf.ssa Marina MarchisioProf.ssa Marina MarchisioMEANS
http://www.maplesoft.com/applications/view.aspx?SID=34930&ref=Feed
<p>Elementary calculations of means with classroom examples, showing proofs of inequalities between means, theoretical and by geometry.</p>
<p>One solutiion for an extremely difficult puzzle illustrating inequality in three dimensions.</p>
<p>Iteration on mixed arithmetic-geometric-harmonic means</p>
<p>Calculation values of elliptic integrals by mixed iteration.</p><img src="/view.aspx?si=34930/means.png" alt="MEANS" align="left"/><p>Elementary calculations of means with classroom examples, showing proofs of inequalities between means, theoretical and by geometry.</p>
<p>One solutiion for an extremely difficult puzzle illustrating inequality in three dimensions.</p>
<p>Iteration on mixed arithmetic-geometric-harmonic means</p>
<p>Calculation values of elliptic integrals by mixed iteration.</p>34930Wed, 09 Dec 2009 05:00:00 ZRoland EngdahlRoland EngdahlFixed Point Iteration
http://www.maplesoft.com/applications/view.aspx?SID=33178&ref=Feed
<p>This worksheet is concerned with finding numerical solutions of non-linear equations in a single unknown. Using MAPLE 12 the <em>fixed-point iteration </em>has been applied to some examples.</p>
<p> </p><img src="/view.aspx?si=33178/maple_icon.jpg" alt="Fixed Point Iteration" align="left"/><p>This worksheet is concerned with finding numerical solutions of non-linear equations in a single unknown. Using MAPLE 12 the <em>fixed-point iteration </em>has been applied to some examples.</p>
<p> </p>33178Thu, 02 Jul 2009 04:00:00 ZProf. Josef BettenProf. Josef BettenSpeed-up calculation of nextprime
http://www.maplesoft.com/applications/view.aspx?SID=5729&ref=Feed
A speed-up calculation of the functions nextprime and prevprime is intended. In some distributions used it was observed similarities to "Prime Number Races" (primes of the form qn+a).
From the "restart:" line, run the worksheet lines with ENTER (some time to wait is required to compute some line), following the text comment lines, or simply read all with vertical scrolling.<img src="/view.aspx?si=5729/nextprime_19_sm.gif" alt="Speed-up calculation of nextprime" align="left"/>A speed-up calculation of the functions nextprime and prevprime is intended. In some distributions used it was observed similarities to "Prime Number Races" (primes of the form qn+a).
From the "restart:" line, run the worksheet lines with ENTER (some time to wait is required to compute some line), following the text comment lines, or simply read all with vertical scrolling.5729Wed, 26 Mar 2008 00:00:00 ZGiulio BonfissutoGiulio BonfissutoFirst Digit Simulation
http://www.maplesoft.com/applications/view.aspx?SID=5562&ref=Feed
The simulations examine the distribution of the first (nonzero) digits in the products of 2, 3 ...
factors created from random numbers.<img src="/view.aspx?si=5562//applications/images/app_image_blank_lg.jpg" alt="First Digit Simulation" align="left"/>The simulations examine the distribution of the first (nonzero) digits in the products of 2, 3 ...
factors created from random numbers.5562Wed, 19 Dec 2007 00:00:00 ZRoland EngdahlRoland EngdahlSn1b Primes
http://www.maplesoft.com/applications/view.aspx?SID=4870&ref=Feed
The worksheet concerns particular primes numbers which can be written down by means of the sequences of ones using positional representation with an arbitrary base.<img src="/view.aspx?si=4870//applications/images/app_image_blank_lg.jpg" alt="Sn1b Primes" align="left"/>The worksheet concerns particular primes numbers which can be written down by means of the sequences of ones using positional representation with an arbitrary base.4870Fri, 16 Feb 2007 00:00:00 ZProf. Czeslaw KoscielnyProf. Czeslaw KoscielnyLes paquet de Cauchy harmoniques ne sont jamais entier
http://www.maplesoft.com/applications/view.aspx?SID=4837&ref=Feed
Thanks to the valuation of an integer we prove that the sum of inverses of consecutive integers is never an integer.<img src="/view.aspx?si=4837/valuation_41.gif" alt="Les paquet de Cauchy harmoniques ne sont jamais entier" align="left"/>Thanks to the valuation of an integer we prove that the sum of inverses of consecutive integers is never an integer.4837Thu, 26 Oct 2006 00:00:00 ZKERNIVINEN SebastienKERNIVINEN SebastienPsedudoprimes
http://www.maplesoft.com/applications/view.aspx?SID=4816&ref=Feed
In order to prove primality of a number, previously we required a list of primes. Eratosthenes, about 230 B.C., created his famous sieve, The Sieve of Eratosthenes. Nowadays we have methods to perform sieve and store a table of primes on a computer system.
Fermat, 1640 , described one characteristic of primes in his famous theorem. The converse of this theorem is however not true. There are composite numbers which comply with this characteristic These numbers are called pseudoprimes.
This application provides examples and programs about primes, pseudoprimes, Carmichael numbers, strong pseudoprimes and primality testing. Only definitions of the concepts are included.<img src="/view.aspx?si=4816//applications/images/app_image_blank_lg.jpg" alt="Psedudoprimes" align="left"/>In order to prove primality of a number, previously we required a list of primes. Eratosthenes, about 230 B.C., created his famous sieve, The Sieve of Eratosthenes. Nowadays we have methods to perform sieve and store a table of primes on a computer system.
Fermat, 1640 , described one characteristic of primes in his famous theorem. The converse of this theorem is however not true. There are composite numbers which comply with this characteristic These numbers are called pseudoprimes.
This application provides examples and programs about primes, pseudoprimes, Carmichael numbers, strong pseudoprimes and primality testing. Only definitions of the concepts are included.4816Tue, 05 Sep 2006 00:00:00 ZRoland EngdahlRoland EngdahlInterface to the QaoS databases of algebraic objects
http://www.maplesoft.com/applications/view.aspx?SID=1668&ref=Feed
<p>The Maple package QaoS contains procedures for querying the QaoS databases for algebraic objects in Berlin. Currently a database of all transitive groups up to degree 30 and a database of more than a million number fields of degree up to 9 are available. There are procedures for accessing invariants of the query results. The results of a query can then be used in Maple.</p><img src="/view.aspx?si=1668/Command line.JPG" alt="Interface to the QaoS databases of algebraic objects" align="left"/><p>The Maple package QaoS contains procedures for querying the QaoS databases for algebraic objects in Berlin. Currently a database of all transitive groups up to degree 30 and a database of more than a million number fields of degree up to 9 are available. There are procedures for accessing invariants of the query results. The results of a query can then be used in Maple.</p>1668Thu, 29 Sep 2005 00:00:00 ZSebastian PauliSebastian PauliCollatz Problem
http://www.maplesoft.com/applications/view.aspx?SID=1665&ref=Feed
The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers n to 3n+1 and even integers n to n/2. The 3x+1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1.<img src="/view.aspx?si=1665/collatz1.JPG" alt="Collatz Problem" align="left"/>The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers n to 3n+1 and even integers n to n/2. The 3x+1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1.1665Tue, 13 Sep 2005 00:00:00 ZMuharrem AktümenMuharrem AktümenElliptic Curve Arithmetic over the Real Numbers
http://www.maplesoft.com/applications/view.aspx?SID=4511&ref=Feed
Elliptic curve arithmetic has useful applications in cryptography. Many texts treat the material in an algebraic way or provide only very few geometric illustrations. An exploration of the geometry of elliptic curve arithmetic gives a much deeper insight into the topic. This worksheet explores some basic concepts in elliptic curve groups over the real numbers.It is meant as a companion to a course or seminar in elliptic curves<img src="/view.aspx?si=4511//applications/images/app_image_blank_lg.jpg" alt="Elliptic Curve Arithmetic over the Real Numbers" align="left"/>Elliptic curve arithmetic has useful applications in cryptography. Many texts treat the material in an algebraic way or provide only very few geometric illustrations. An exploration of the geometry of elliptic curve arithmetic gives a much deeper insight into the topic. This worksheet explores some basic concepts in elliptic curve groups over the real numbers.It is meant as a companion to a course or seminar in elliptic curves4511Fri, 11 Jun 2004 14:01:27 ZJudith KoellerJudith KoellerBill Clinton, Bertie Ahern, and Digital Signatures
http://www.maplesoft.com/applications/view.aspx?SID=4463&ref=Feed
This worksheet is a talk given by John Cosgrave at the Annual International Conference on Technology in Collegiate Mathematics in 2003. In this, he explains the difference between private- key and public-key cryptography, demonstrate the Pohlig-Hellman private-key cryptographic method, demonstrate a simplified element of the renowned Rivest-Shamir-Adleman (RSA) public-key cryptographic method, and illustrate the relevance of the unsolved fast factorisation problem to the security of the RSA method.<img src="/view.aspx?si=4463//applications/images/app_image_blank_lg.jpg" alt="Bill Clinton, Bertie Ahern, and Digital Signatures" align="left"/>This worksheet is a talk given by John Cosgrave at the Annual International Conference on Technology in Collegiate Mathematics in 2003. In this, he explains the difference between private- key and public-key cryptography, demonstrate the Pohlig-Hellman private-key cryptographic method, demonstrate a simplified element of the renowned Rivest-Shamir-Adleman (RSA) public-key cryptographic method, and illustrate the relevance of the unsolved fast factorisation problem to the security of the RSA method.4463Thu, 11 Dec 2003 09:51:50 ZJohn CosgraveJohn CosgraveInteger root extraction and perfect-power detection via p-adic Newton-Hensel lifting
http://www.maplesoft.com/applications/view.aspx?SID=1390&ref=Feed
The computation problem of root extraction is as follows: Given positive integers n and r we want to find a positive integer x such that
n = x<sup>r</sup>. If no such x exists, then we do not care about getting an approximate solution.<p>This worksheet can serve as an introduction to p-adic methods, as a review of Newton's method, or as a discussion of the number theoretic problem of perfect-power detection.<img src="/view.aspx?si=1390//applications/images/app_image_blank_lg.jpg" alt="Integer root extraction and perfect-power detection via p-adic Newton-Hensel lifting" align="left"/>The computation problem of root extraction is as follows: Given positive integers n and r we want to find a positive integer x such that
n = x<sup>r</sup>. If no such x exists, then we do not care about getting an approximate solution.<p>This worksheet can serve as an introduction to p-adic methods, as a review of Newton's method, or as a discussion of the number theoretic problem of perfect-power detection.1390Thu, 09 Jan 2003 12:55:26 ZCarl DeVoreCarl DeVoreBlinding RSA
http://www.maplesoft.com/applications/view.aspx?SID=4142&ref=Feed
In this worksheet we look at how to get a valid signature on a message the owner of the key-pair (Bob) was not willing to sign.
Imagine the following: Marvin (evil) wants Bob (good) to sign a message M. Bob refuses. Now Marvin picks a random Number r and computes an incospicuous Message M'=r^e*M mod N , N being the RSA-Modulus and e Bob's public key. He then asks Bob to sign the Message which he does. Thus Marvin receives the signature S' of M'. Recalling that S'=(M')^d mod N he can now simply compute S=S'/r mod N and yield the valid signature of M.
Since we don't use a one-way-hash (MD5, SHA-1) but the simple S=M^d mod N here this is far from being a real threat to RSA
<img src="/view.aspx?si=4142//applications/images/app_image_blank_lg.jpg" alt="Blinding RSA" align="left"/>In this worksheet we look at how to get a valid signature on a message the owner of the key-pair (Bob) was not willing to sign.
Imagine the following: Marvin (evil) wants Bob (good) to sign a message M. Bob refuses. Now Marvin picks a random Number r and computes an incospicuous Message M'=r^e*M mod N , N being the RSA-Modulus and e Bob's public key. He then asks Bob to sign the Message which he does. Thus Marvin receives the signature S' of M'. Recalling that S'=(M')^d mod N he can now simply compute S=S'/r mod N and yield the valid signature of M.
Since we don't use a one-way-hash (MD5, SHA-1) but the simple S=M^d mod N here this is far from being a real threat to RSA
4142Wed, 10 Oct 2001 16:58:21 ZMartin MonerjanMartin Monerjan