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CHAPTER 11 - Quadratic Residues
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11.1 Quadratic Residues and Nonresidues

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

 

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You can find the smallest quadratic nonresidue of a prime using the tool at http://www.numbertheory.org/php/leastqnr.html

 

 

11.2 The Law of Quadratic Reciprocity

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Close to 200 different proofs of the law of quadratic reciprocity are listed by Franz Lemmermeyer. He lists the author, year, and method of each proof, and provides references and links to reviews in Mathematical Reviews and Zentralblatt. Go to
http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html (Proofs of the Law of Quadratic Reciprocity)
An excellent account of Eisenstein's simplification of Gauss's third proof of the law of quadratic reciprocity can be found in "Gauß, Eisenstein, and the "third" proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel" by Reihard Laubenbacher - David J. Pengelley. This article originally appeared in the Mathematical Intelligencer in 1994 and can be accessed at http://www.math.nmsu.edu/~history/schauspiel/schauspiel.html (Gauß, Eisenstein, and the "third" proof of the ...)

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

 

 

11.3 The Jacobi Symbol

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You can compute Jacobi systems using the tool at http://www.numbertheory.org/php/jacobi.html

 

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

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

 

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

 

11.4 Euler Pseudoprimes

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Additional information about Euler pseudoprimes and related types of pseudoprimes can be found at
http://mathworld.wolfram.com/Euler-JacobiPseudoprime.html

 

For a table of Euler pseudoprimes not exceeding 10000 for prime base not exceeding 67 see http://encyclopedia.thefreedictionary.com/Euler-Jacobi%20pseudoprime

 

 

11.5 Zero-Knowledge Proofs
 
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You can learn more about zero-knowledge proofs and interactive proofs at
http://www.rsasecurity.com/rsalabs/faq/2-1-8.html (What are interactive proofs and zero-knowledge proofs?)