1,142 research outputs found
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
Four primality testing algorithms
In this expository paper we describe four primality tests. The first test is
very efficient, but is only capable of proving that a given number is either
composite or 'very probably' prime. The second test is a deterministic
polynomial time algorithm to prove that a given numer is either prime or
composite. The third and fourth primality tests are at present most widely used
in practice. Both tests are capable of proving that a given number is prime or
composite, but neither algorithm is deterministic. The third algorithm exploits
the arithmetic of cyclotomic fields. Its running time is almost, but not quite
polynomial time. The fourth algorithm exploits elliptic curves. Its running
time is difficult to estimate, but it behaves well in practice.Comment: 21 page
A faster pseudo-primality test
We propose a pseudo-primality test using cyclic extensions of . For every positive integer , this test achieves the
security of Miller-Rabin tests at the cost of Miller-Rabin
tests.Comment: Published in Rendiconti del Circolo Matematico di Palermo Journal,
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