A Course in Number Theory and Cryptography

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Springer Science & Business Media, Sep 2, 1994 - Mathematics - 235 pages
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. . . both Gauss and lesser mathematicians may be justified in rejoic ing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. - G. H. Hardy, A Mathematician's Apology, 1940 G. H. Hardy would have been surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography (secret codes). Less than a half-century after Hardy wrote the words quoted above, it is no longer inconceivable (though it hasn't happened yet) that the N. S. A. (the agency for U. S. government work on cryptography) will demand prior review and clearance before publication of theoretical research papers on certain types of number theory. In part it is the dramatic increase in computer power and sophistica tion that has influenced some of the questions being studied by number theorists, giving rise to a new branch of the subject, called "computational number theory. " This book presumes almost no background in algebra or number the ory. Its purpose is to introduce the reader to arithmetic topics, both ancient and very modern, which have been at the center of interest in applications, especially in cryptography. For this reason we take an algorithmic approach, emphasizing estimates of the efficiency of the techniques that arise from the theory.
 

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Contents

Some Topics in Elementary Number Theory
1
2 Divisibility and the Euclidean algorithm
12
3 Congruences
19
4 Some applications to factoring
27
Finite Fields and Quadratic Residues
31
1 Finite fields
33
2 Quadratic residues and reciprocity
42
Cryptography
54
Primality and Factoring
125
1 Pseudoprimes
126
2 The rho method
138
3 Fermat factorization and factor bases
143
4 The continued fraction method
154
5 The quadratic sieve method
160
Elliptic Curves
167
2 Elliptic curve cryptosystems
178

2 Enciphering matrices
65
Public Key
83
2 RSA
92
3 Discrete log
97
4 Knapsack
111
5 Zeroknowledge protocols and oblivious transfer
117
3 Elliptic curve primality test
187
4 Elliptic curve factorization
191
Answers to Exercises
200
Index
231
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