181 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
Introduction to Quantum Information Processing
As a result of the capabilities of quantum information, the science of
quantum information processing is now a prospering, interdisciplinary field
focused on better understanding the possibilities and limitations of the
underlying theory, on developing new applications of quantum information and on
physically realizing controllable quantum devices. The purpose of this primer
is to provide an elementary introduction to quantum information processing, and
then to briefly explain how we hope to exploit the advantages of quantum
information. These two sections can be read independently. For reference, we
have included a glossary of the main terms of quantum information.Comment: 48 pages, to appear in LA Science. Hyperlinked PDF at
http://www.c3.lanl.gov/~knill/qip/prhtml/prpdf.pdf, HTML at
http://www.c3.lanl.gov/~knill/qip/prhtm
Concrete quantum cryptanalysis of binary elliptic curves
This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2 (n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2 (n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography
Concrete quantum cryptanalysis of binary elliptic curves
This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2 (n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2 (n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography
Quantum-accelerated algorithms for generating random primitive polynomials over finite fields
Primitive polynomials over finite fields are crucial for various domains of
computer science, including classical pseudo-random number generation, coding
theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient
classical algorithm for generating random primitive polynomials over finite
fields remains an ongoing challenge. In this paper, we show how to solve this
problem efficiently through hybrid quantum-classical algorithms, and designs of
the specific quantum circuits to implement them are also presented. Our
research paves the way for the rapid and real-time generation of random
primitive polynomials in diverse quantum communication and computation
applications
Improved quantum circuits for elliptic curve discrete logarithms
We present improved quantum circuits for elliptic curve scalar
multiplication, the most costly component in Shor's algorithm to compute
discrete logarithms in elliptic curve groups. We optimize low-level components
such as reversible integer and modular arithmetic through windowing techniques
and more adaptive placement of uncomputing steps, and improve over previous
quantum circuits for modular inversion by reformulating the binary Euclidean
algorithm. Overall, we obtain an affine Weierstrass point addition circuit that
has lower depth and uses fewer gates than previous circuits. While previous
work mostly focuses on minimizing the total number of qubits, we present
various trade-offs between different cost metrics including the number of
qubits, circuit depth and -gate count. Finally, we provide a full
implementation of point addition in the Q# quantum programming language that
allows unit tests and automatic quantum resource estimation for all components.Comment: 22 pages, to appear in: Int'l Conf. on Post-Quantum Cryptography
(PQCrypto 2020
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