Read e-book online Algorithmic number theory: lattices, number fields, curves PDF

By J.P. Buhler, P. Stevenhagen

ISBN-10: 0521808545

ISBN-13: 9780521808545

Quantity conception is without doubt one of the oldest and such a lot beautiful components of arithmetic. Computation has consistently performed a task in quantity concept, a job which has elevated dramatically within the final 20 or 30 years, either as a result creation of contemporary pcs, and thanks to the invention of bizarre and strong algorithms. therefore, algorithmic quantity concept has progressively emerged as a massive and distinctive box with connections to computing device technology and cryptography in addition to different parts of arithmetic. this article presents a complete advent to algorithmic quantity idea for starting graduate scholars, written through the top specialists within the box. It comprises numerous articles that disguise the basic themes during this zone, resembling the basic algorithms of basic quantity thought, lattice foundation aid, elliptic curves, algebraic quantity fields, and techniques for factoring and primality proving. moreover, there are contributions pointing in broader instructions, together with cryptography, computational category box idea, zeta services and L-series, discrete logarithm algorithms, and quantum computing.

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Extra resources for Algorithmic number theory: lattices, number fields, curves and cryptography

Example text

The set of congruence classes containing integers that are coprime to n, under the operation of multiplication. n/ is the Euler-phi function — the number of positive integers less than n that are coprime to n. Finally, we let a mod n denote the class of ‫=ޚ‬n‫ ޚ‬that contains a. We will tolerate the conflict with earlier usage because the meaning can be disambiguated from context: if a mod n is an integer then the remainder is intended, and if a mod n lies in ‫=ޚ‬n‫ ޚ‬then the congruence class is intended.

Heuristically) have a 50/50 chance of dividing different factors. x y; n/ will be a proper factor of n. If this fails, try again by choosing another random y. After k choices, the probability that n remains unfactored is 2 k . Thus FACTORING and M ODULAR S QUARE ROOTS are in practice equivalent in difficulty. R EMARK 8. Replacing square roots by eth roots for e > 2 leads to a problem closely related to the RSA cryptosystem, perhaps the most famous of all publickey cryptographic systems. ‫=ޚ‬n‫ޚ‬/ .

Stevenhagen, “The number field sieve”, pp. 83–100 in Surveys in algorithmic number theory, edited by J. P. Buhler and P. Stevenhagen, Math. Sci. Res. Inst. Publ. 44, Cambridge University Press, New York, 2008. [Vardi 1998] I. Vardi, “Archimedes’ cattle problem”, Amer. Math. Monthly 105:4 (1998), 305–319. [Vollmer 2002] U. Vollmer, “An accelerated Buchmann algorithm for regulator computation in real quadratic fields”, pp. 148–162 in Algorithmic Number Theory, ANTSV, edited by C. Fieker and D. R.

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Algorithmic number theory: lattices, number fields, curves and cryptography by J.P. Buhler, P. Stevenhagen

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