2,199 research outputs found
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
We present two algorithms to compute the endomorphism ring of an ordinary
elliptic curve E defined over a finite field F_q. Under suitable heuristic
assumptions, both have subexponential complexity. We bound the complexity of
the first algorithm in terms of log q, while our bound for the second algorithm
depends primarily on log |D_E|, where D_E is the discriminant of the order
isomorphic to End(E). As a byproduct, our method yields a short certificate
that may be used to verify that the endomorphism ring is as claimed.Comment: 16 pages (minor edits
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Quantum Probabilistic Subroutines and Problems in Number Theory
We present a quantum version of the classical probabilistic algorithms
la Rabin. The quantum algorithm is based on the essential use of
Grover's operator for the quantum search of a database and of Shor's Fourier
transform for extracting the periodicity of a function, and their combined use
in the counting algorithm originally introduced by Brassard et al. One of the
main features of our quantum probabilistic algorithm is its full unitarity and
reversibility, which would make its use possible as part of larger and more
complicated networks in quantum computers. As an example of this we describe
polynomial time algorithms for studying some important problems in number
theory, such as the test of the primality of an integer, the so called 'prime
number theorem' and Hardy and Littlewood's conjecture about the asymptotic
number of representations of an even integer as a sum of two primes.Comment: 9 pages, RevTex, revised version, accepted for publication on PRA:
improvement in use of memory space for quantum primality test algorithm
further clarified and typos in the notation correcte
A one line factoring algorithm
We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package
Entanglement and its Role in Shor's Algorithm
Entanglement has been termed a critical resource for quantum information
processing and is thought to be the reason that certain quantum algorithms,
such as Shor's factoring algorithm, can achieve exponentially better
performance than their classical counterparts. The nature of this resource is
still not fully understood: here we use numerical simulation to investigate how
entanglement between register qubits varies as Shor's algorithm is run on a
quantum computer. The shifting patterns in the entanglement are found to relate
to the choice of basis for the quantum Fourier transform.Comment: 15 pages, 4 eps figures, v1-3 were for conference proceedings (not
included in the end); v4 is improved following referee comments, expanded
explanations and added reference
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