Elementary Number Theory: Primes, Congruences, and Secrets: A Computational ApproachSpringer Science & Business Media, 28.10.2008 - 168 Seiten This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem. |
Inhalt
1 | |
The Ring of Integers Modulo n | 21 |
Publickey Cryptography | 48 |
Quadratic Reciprocity | 69 |
Continued Fractions | 93 |
Elliptic Curves | 123 |
Answers and Hints | 148 |
References | 155 |
Andere Ausgaben - Alle anzeigen
Elementary Number Theory: Primes, Congruences, and Secrets: A Computational ... William Stein Keine Leseprobe verfügbar - 2009 |
Elementary Number Theory: Primes, Congruences, and Secrets: A Computational ... William Stein Keine Leseprobe verfügbar - 2010 |
Elementary Number Theory: Primes, Congruences, and Secrets: A Computational ... William Stein Keine Leseprobe verfügbar - 2009 |
Häufige Begriffe und Wortgruppen
abelian group algebraic number algorithm B-power smooth bijection binary Chinese Remainder Theorem compute congruent number conjecture coprime cryptography cyclic define Definition Diffie-Hellman digits discrete log problem divides divisible element of Z/pZ elliptic curve encrypt equation field sieve finite follows form 4x Gauss sum gcd(a greatest common divisor group homomorphism implement implies infinitely many primes Lemma Math Mathematics Mersenne prime Michael and Nikita mod q multiple natural numbers odd prime output partial convergents perfect square Pollard polynomial positive integer prime factorization prime numbers primitive root modulo product of primes proof of Theorem Proposition 4.2.1 prove public-key Quadratic Reciprocity Law quadratic residue quotient rational number real numbers residues modulo Riemann Hypothesis ring RSA cryptosystem SAGE Example secret key Section sequence set of residues solution Springer Science+Business Media square modulo subgroup Suppose Z/nZ