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
A Type of Hyperelliptic Continued Fraction
TL;DR: In this paper, a continued fraction was proposed to calculate the torsion of divisors D - D- on a hyperelliptic curve of genus = 2, where D is an effective divisor of degree 2 and D- denotes the image of D under the hyperellipisic involution.
10
Algebraic Shift Register Sequences: List of figures
Mark Goresky,A. Klapper +1 more
TL;DR: The paper explores algebraic shift register sequences, their construction, and security measures. It covers various sequence types, including linear feedback shift registers, d-FCSRs, and maximal period sequences. The paper also discusses register synthesis and security measures related to these sequences.
10
A Low-Memory Algorithm for Point Counting on Picard Curves
TL;DR: A low memory algorithm is given that is efficient enough to handle Picard curves over finite prime fields ${\mathbb F}_{p}$, where p is a prime with 58 bits.
10
Model checking QCTL plus on quantum Markov chains
TL;DR: In this paper , the authors propose a more expressive logic QCTL+ (QCTL plus), which extends QCTl by allowing the conjunction in path formulas and the negation in the top level of path formulas.
10
•Dissertation
The integer chebyshev problem: computational explorations
Alan Meichsner
- 01 Jan 2009
TL;DR: The integer Chebyshev polynomials of degree at most n with integer coefficients having minimal supremum norm on a domain D and then analyzing the limiting value of the nth root of the supremum norms is called the integer Chebyhev constant for D as discussed by the authors.
10
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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