27 research outputs found
Applying Grover's algorithm to AES: quantum resource estimates
We present quantum circuits to implement an exhaustive key search for the
Advanced Encryption Standard (AES) and analyze the quantum resources required
to carry out such an attack. We consider the overall circuit size, the number
of qubits, and the circuit depth as measures for the cost of the presented
quantum algorithms. Throughout, we focus on Clifford gates as the
underlying fault-tolerant logical quantum gate set. In particular, for all
three variants of AES (key size 128, 192, and 256 bit) that are standardized in
FIPS-PUB 197, we establish precise bounds for the number of qubits and the
number of elementary logical quantum gates that are needed to implement
Grover's quantum algorithm to extract the key from a small number of AES
plaintext-ciphertext pairs.Comment: 13 pages, 3 figures, 5 tables; to appear in: Proceedings of the 7th
International Conference on Post-Quantum Cryptography (PQCrypto 2016
A Novel Approach for Ellipsoidal Outer-Approximation of the Intersection Region of Ellipses in the Plane
In this paper, a novel technique for tight outer-approximation of the
intersection region of a finite number of ellipses in 2-dimensional (2D) space
is proposed. First, the vertices of a tight polygon that contains the convex
intersection of the ellipses are found in an efficient manner. To do so, the
intersection points of the ellipses that fall on the boundary of the
intersection region are determined, and a set of points is generated on the
elliptic arcs connecting every two neighbouring intersection points. By finding
the tangent lines to the ellipses at the extended set of points, a set of
half-planes is obtained, whose intersection forms a polygon. To find the
polygon more efficiently, the points are given an order and the intersection of
the half-planes corresponding to every two neighbouring points is calculated.
If the polygon is convex and bounded, these calculated points together with the
initially obtained intersection points will form its vertices. If the polygon
is non-convex or unbounded, we can detect this situation and then generate
additional discrete points only on the elliptical arc segment causing the
issue, and restart the algorithm to obtain a bounded and convex polygon.
Finally, the smallest area ellipse that contains the vertices of the polygon is
obtained by solving a convex optimization problem. Through numerical
experiments, it is illustrated that the proposed technique returns a tighter
outer-approximation of the intersection of multiple ellipses, compared to
conventional techniques, with only slightly higher computational cost