945 research outputs found
Claw Finding Algorithms Using Quantum Walk
The claw finding problem has been studied in terms of query complexity as one
of the problems closely connected to cryptography. For given two functions, f
and g, as an oracle which have domains of size N and M (N<=M), respectively,
and the same range, the goal of the problem is to find x and y such that
f(x)=g(y). This paper describes an optimal algorithm using quantum walk that
solves this problem. Our algorithm can be generalized to find a claw of k
functions for any constant integer k>1, where the domains of the functions may
have different size.Comment: 12 pages. Introduction revised. A reference added. Weak lower bound
delete
Quantum Algorithm for the Collision Problem
In this note, we give a quantum algorithm that finds collisions in arbitrary
r-to-one functions after only O((N/r)^(1/3)) expected evaluations of the
function. Assuming the function is given by a black box, this is more efficient
than the best possible classical algorithm, even allowing probabilism. We also
give a similar algorithm for finding claws in pairs of functions. Furthermore,
we exhibit a space-time tradeoff for our technique. Our approach uses Grover's
quantum searching algorithm in a novel way.Comment: 8 pages, LaTeX2
Random Oracles in a Quantum World
The interest in post-quantum cryptography - classical systems that remain
secure in the presence of a quantum adversary - has generated elegant proposals
for new cryptosystems. Some of these systems are set in the random oracle model
and are proven secure relative to adversaries that have classical access to the
random oracle. We argue that to prove post-quantum security one needs to prove
security in the quantum-accessible random oracle model where the adversary can
query the random oracle with quantum states.
We begin by separating the classical and quantum-accessible random oracle
models by presenting a scheme that is secure when the adversary is given
classical access to the random oracle, but is insecure when the adversary can
make quantum oracle queries. We then set out to develop generic conditions
under which a classical random oracle proof implies security in the
quantum-accessible random oracle model. We introduce the concept of a
history-free reduction which is a category of classical random oracle
reductions that basically determine oracle answers independently of the history
of previous queries, and we prove that such reductions imply security in the
quantum model. We then show that certain post-quantum proposals, including ones
based on lattices, can be proven secure using history-free reductions and are
therefore post-quantum secure. We conclude with a rich set of open problems in
this area.Comment: 38 pages, v2: many substantial changes and extensions, merged with a
related paper by Boneh and Zhandr
Span programs and quantum algorithms for st-connectivity and claw detection
We introduce a span program that decides st-connectivity, and generalize the
span program to develop quantum algorithms for several graph problems. First,
we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries
to the n x n adjacency matrix to decide if vertices s and t are connected,
under the promise that they either are connected by a path of length at most d,
or are disconnected. We also show that if T is a path, a star with two
subdivided legs, or a subdivision of a claw, its presence as a subgraph in the
input graph G can be detected with O(n) quantum queries to the adjacency
matrix. Under the promise that G either contains T as a subgraph or does not
contain T as a minor, we give O(n)-query quantum algorithms for detecting T
either a triangle or a subdivision of a star. All these algorithms can be
implemented time efficiently and, except for the triangle-detection algorithm,
in logarithmic space. One of the main techniques is to modify the
st-connectivity span program to drop along the way "breadcrumbs," which must be
retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure
Quantum All-Subkeys-Recovery Attacks on 6-round Feistel-2* Structure Based on Multi-Equations Quantum Claw Finding
Exploiting quantum mechanisms, quantum attacks have the potential ability to
break the cipher structure. Recently, Ito et al. proposed a quantum attack on
Feistel-2* structure (Ito et al.'s attack) based onthe Q2 model. However, it is
not realistic since the quantum oracle needs to be accessed by the adversary,
and the data complexityis high. To solve this problem, a quantum
all-subkeys-recovery (ASR) attack based on multi-equations quantum claw-finding
is proposed, which takes a more realistic model, the Q1 model, as the scenario,
and only requires 3 plain-ciphertext pairs to quickly crack the 6-round
Feistel-2* structure. First, we proposed a multi-equations quantum claw-finding
algorithm to solve the claw problem of finding multiple equations. In addition,
Grover's algorithm is used to speedup the rest subkeys recovery. Compared with
Ito et al.'s attack, the data complexity of our attack is reduced from O(2^n)
to O(1), while the time complexity and memory complexity are also significantly
reduced.Comment: 18 pages, 4 figure
Parallelizing quantum circuit synthesis
We present an algorithmic framework for parallel quantum circuit synthesis using meet-in-the-middle synthesis techniques. We also present two implementations thereof, using both threaded and hybrid parallelization techniques.
We give examples where applying parallelism offers a speedup on the time of circuit synthesis for 2- and 3-qubit circuits. We use a threaded algorithm to synthesize 3-qubit circuits with optimal T -count 9, and 11, breaking the previous record of T-count 7. As the estimated runtime of the framework is inversely proportional to the number of processors, we propose an implementation using hybrid parallel programming which can take full advantage of a computing cluster’s thousands of cores. This implementation has the potential to synthesize circuits which were previously deemed impossible due to the exponential runtime of existing algorithms
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