5,485 research outputs found
Implicit Factoring: On Polynomial Time Factoring Given Only an Implicit Hint
Abstract. We address the problem of polynomial time factoring RSA moduli N1 = p1q1 with the help of an oracle. As opposed to other ap-proaches that require an oracle that explicitly outputs bits of p1, we use an oracle that gives only implicit information about p1. Namely, our or-acle outputs a different N2 = p2q2 such that p1 and p2 share the t least significant bits. Surprisingly, this implicit information is already suffi-cient to efficiently factor N1, N2 provided that t is large enough. We then generalize this approach to more than one oracle query. Key words: Factoring with an oracle, lattices
Strengths and Weaknesses of Quantum Computing
Recently a great deal of attention has focused on quantum computation
following a sequence of results suggesting that quantum computers are more
powerful than classical probabilistic computers. Following Shor's result that
factoring and the extraction of discrete logarithms are both solvable in
quantum polynomial time, it is natural to ask whether all of NP can be
efficiently solved in quantum polynomial time. In this paper, we address this
question by proving that relative to an oracle chosen uniformly at random, with
probability 1, the class NP cannot be solved on a quantum Turing machine in
time . We also show that relative to a permutation oracle chosen
uniformly at random, with probability 1, the class cannot be
solved on a quantum Turing machine in time . The former bound is
tight since recent work of Grover shows how to accept the class NP relative to
any oracle on a quantum computer in time .Comment: 18 pages, latex, no figures, to appear in SIAM Journal on Computing
(special issue on quantum computing
A deterministic version of Pollard's p-1 algorithm
In this article we present applications of smooth numbers to the
unconditional derandomization of some well-known integer factoring algorithms.
We begin with Pollard's algorithm, which finds in random polynomial
time the prime divisors of an integer such that is smooth. We
show that these prime factors can be recovered in deterministic polynomial
time. We further generalize this result to give a partial derandomization of
the -th cyclotomic method of factoring () devised by Bach and
Shallit.
We also investigate reductions of factoring to computing Euler's totient
function . We point out some explicit sets of integers that are
completely factorable in deterministic polynomial time given . These
sets consist, roughly speaking, of products of primes satisfying, with the
exception of at most two, certain conditions somewhat weaker than the
smoothness of . Finally, we prove that oracle queries for
values of are sufficient to completely factor any integer in less
than deterministic
time.Comment: Expanded and heavily revised version, to appear in Mathematics of
Computation, 21 page
Five Quantum Algorithms Using Quipper
Quipper is a recently released quantum programming language. In this report,
we explore Quipper's programming framework by implementing the Deutsch's,
Deutsch-Jozsa's, Simon's, Grover's, and Shor's factoring algorithms. It will
help new quantum programmers in an instructive manner. We choose Quipper
especially for its usability and scalability though it's an ongoing development
project. We have also provided introductory concepts of Quipper and
prerequisite backgrounds of the algorithms for readers' convenience. We also
have written codes for oracles (black boxes or functions) for individual
algorithms and tested some of them using the Quipper simulator to prove
correctness and introduce the readers with the functionality. As Quipper 0.5
does not include more than \ensuremath{4 \times 4} matrix constructors for
Unitary operators, we have also implemented \ensuremath{8 \times 8} and
\ensuremath{16 \times 16} matrix constructors.Comment: 27 page
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