Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach
Springer 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.
Was andere dazu sagen - Rezension schreiben
Es wurden keine Rezensionen gefunden.
Andere Ausgaben - Alle anzeigen
Elementary Number Theory: Primes, Congruences, and Secrets: A Computational ...
Keine Leseprobe verfügbar - 2010
abelian group algorithm B-power smooth bijection binary Chinese Remainder Theorem composite compute congruent number conjecture coprime cryptography cyclic decryption define Definition Diffie-Hellman Diffie-Hellman key exchange digits discrete log problem divides divisible element of Z/pZ elliptic curve y2 encrypt Exercise field sieve finite follows form 4x G Z/pZ Gauss sum gcd(a greatest common divisor group homomorphism implement implies infinitely many primes Lemma Mersenne prime Michael and Nikita multiple natural numbers number theory odd prime output partial convergents perfect square Pollard’s method polynomial positive integer primality test 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 r(nB 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