Open AccessBook
A Course in Computational Algebraic Number Theory
Henri Cohen
- 01 Jan 1993
3.2K
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
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Abstract: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
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Citations
Optimization of the MOVA undeniable signature scheme
Jean Monnerat,Yvonne Anne Oswald,Serge Vaudenay +2 more
- 28 Sep 2005
TL;DR: In this article, the authors present an overview of the expected performance of the MOVA scheme depending on the group homomorphism and demonstrate that the latter provides a signature generation which is three times faster than RSA.
Solvability of norm equations over cyclic number fields of prime degree
TL;DR: An algorithm is described which takes as input α and the minimal polynomial of α over Q, and determines if α is a norm of an element of L, and it is shown that, if the authors ignore the time needed to obtain a complete factorization of α and a completefactorization of the discriminant of α, then the algorithm runs in timePolynomial in the size of the input.
Area-Time Efficient Implementation of the Elliptic Curve Method of Factoring in Reconfigurable Hardware for Application in the Number Field Sieve
Kris Gaj,Soonhak Kwon,Patrick Baier,Paul Kohlbrenner,Hoang Le,Mohammed Khaleeluddin,Ramakrishna Bachimanchi,Marcin Rogawski +7 more
TL;DR: Results indicate that low-cost families of FPGAs, such as Spartan-3 and Spartans-3E, offer at least an order of magnitude improvement over the same generation of microprocessors in terms of the performance to cost ratio, without the use of embedded FPGA resources,such as embedded multipliers.
Patent
Elliptic curve encryption method and system
Hiroyuki Kurumatani
- 23 Feb 1999
TL;DR: In this paper, the authors proposed an elliptic curve-based encryption scheme for doubling and multiplication modulo operation using a prime having a form of p = Abn + B (Step 303) where 0 < A < 2w; 0 < B < 2W; B = 2w, A, b, n and B are positive integers.
13
Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type
Benjamin Bakker,Andrei Jorza +1 more
TL;DR: In this article, the authors classified the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action.
13
References
Modular multiplication without trial division
TL;DR: A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.
•Book
Advanced Topics in the Arithmetic of Elliptic Curves
Joseph H. Silverman
- 01 Jan 1994
TL;DR: In this article, the authors continue the study of elliptic curves by presenting six important, but somewhat more specialized topics: Elliptic and modular functions for the full modular group.
2.2K
Improved methods for calculating vectors of short length in a lattice, including a complexity analysis
U. Fincke,Michael Pohst +1 more
TL;DR: In this paper, the authors show that searching through an ellipsoid is in many cases much more efficient than enumerating all vectors of Z'.. in a suitable box.
Lattice basis reduction: improved practical algorithms and solving subset sum problems
Claus-Peter Schnorr,M. Euchner +1 more
TL;DR: Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC1 + computer.
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