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CHAPTER 4 - Congruences
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4.1 Introduction to Congruences

Page 141

Modular arithmetic is discussed at the Cut-the-Knot site at
http://www.cut-the-knot.com/blue/Modulo.html (Modular Arithmetic)

Biographical information about Karl Friedrich Gauss can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Gauss.html (Gauss)

 

Page 144

 

A calculator for performing modular arithmetic can be found at http://www.math.csusb.edu/faculty/susan/modular/modcalc.html (A Clock (Modular) Arithmetic Calculator)

 

You can also do modular arithmetic using the tool at http://www.numbertheory.org/php/arithmeticm.html  (Modular arithmetical operations on integers)

 

 

Page 147

You can find source code for a C program implementing the algorithm for modular exponentiation by repeated squaring and multiplying at
http://www.mindspring.com/~pate/crypto/chap02.html

 

Tools for carrying out modular exponentiation can be found at http://science.slc.edu/~msiff/old-courses/crypto01/modexp.php

 

 

 

 

4.2 Linear Congruences

 
Page 153

An applet for solving linear congruences can be found at http://www.math.temple.edu/~renault/cryptology/congruences.html (Linear Congruences)

 

You can find the solution to linear congruences using the tool at http://www.numbertheory.org/php/cong.html (Solving a linear congruence)

 

Page 155

You can find source code for a C program that computes modular inverses at
http://www.mindspring.com/~pate/crypto/chap02.html

 

You can compute modular inverses on-line at http://www.math.csusb.edu/faculty/susan/modular/modcalc.html

 

 

 

4.3 The Chinese Remainder Theorem

Page 158

You can learn more about the Chinese remainder theorem at
http://www.cut-the-knot.com/blue/chinese.html (Chinese Remainder Theorem)

 

You can solve simultaneous congruences using the Chinese remainder theorem using the tools at http://www.numbertheory.org/php/chinese2.html and http://www.numbertheory.org/php/chinesea.html

 

Applets for solving simultaneous congruences using the Chinese remainder theorem can be found at http://www.math.temple.edu/~renault/cryptology/congruences.html (Chinese Remainder Theorem Applet) and at http://linguistlist.org/~zheng/courseware/remainder.html (Chinese Remainder Theorem)
 
Biographical information about Ch'in Chiu-Shao can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Ch'in.html (Ch'in)

Page 166

You can calculate your biorhythms using the program at
http://www.facade.com/biorhythm (Biorhythm)

 

4.4 Solving Polynomial Congruences

Page 170

Biographical information about Kurt Hensel can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Hensel.html (Hensel)

 

4.5 Systems of Linear Congruences

 

Page 174

 

An applet for solving systems of congruences of the type discussed in this section can be found at http://linguistlist.org/~zheng/courseware/linear.html (Systems of Linear Congruence)

Page 183

 
An excellent starting point for exploring Magic squares on the Web is
http://forum.swarthmore.edu/alejandre/magic.square.html (Suzanne Alejandre: Magic Square)

 

 

4.6 Factoring Using the Pollard Rho Method

Page 184

Source code for the Pollard rho factorization method can be found at
http://www.mindspring.com/~pate/course/chap08.html