2,029 research outputs found
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
Factoring Safe Semiprimes with a Single Quantum Query
Shor's factoring algorithm (SFA), by its ability to efficiently factor large
numbers, has the potential to undermine contemporary encryption. At its heart
is a process called order finding, which quantum mechanics lets us perform
efficiently. SFA thus consists of a \emph{quantum order finding algorithm}
(QOFA), bookended by classical routines which, given the order, return the
factors. But, with probability up to , these classical routines fail, and
QOFA must be rerun. We modify these routines using elementary results in number
theory, improving the likelihood that they return the factors.
The resulting quantum factoring algorithm is better than SFA at factoring
safe semiprimes, an important class of numbers used in cryptography. With just
one call to QOFA, our algorithm almost always factors safe semiprimes. As well
as a speed-up, improving efficiency gives our algorithm other, practical
advantages: unlike SFA, it does not need a randomly picked input, making it
simpler to construct in the lab; and in the (unlikely) case of failure, the
same circuit can be rerun, without modification.
We consider generalizing this result to other cases, although we do not find
a simple extension, and conclude that SFA is still the best algorithm for
general numbers (non safe semiprimes, in other words). Even so, we present some
simple number theoretic tricks for improving SFA in this case.Comment: v2 : Typo correction and rewriting for improved clarity v3 : Slight
expansion, for improved clarit
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