34,756 research outputs found

    Quantum Algorithm for the Collision Problem

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    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

    The NISQ Complexity of Collision Finding

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    Collision-resistant hashing, a fundamental primitive in modern cryptography, ensures that there is no efficient way to find distinct inputs that produce the same hash value. This property underpins the security of various cryptographic applications, making it crucial to understand its complexity. The complexity of this problem is well-understood in the classical setting and Θ(N1/2)\Theta(N^{1/2}) queries are needed to find a collision. However, the advent of quantum computing has introduced new challenges since quantum adversaries \unicode{x2013} equipped with the power of quantum queries \unicode{x2013} can find collisions much more efficiently. Brassard, H\"oyer and Tapp and Aaronson and Shi established that full-scale quantum adversaries require Θ(N1/3)\Theta(N^{1/3}) queries to find a collision, prompting a need for longer hash outputs, which impacts efficiency in terms of the key lengths needed for security. This paper explores the implications of quantum attacks in the Noisy-Intermediate Scale Quantum (NISQ) era. In this work, we investigate three different models for NISQ algorithms and achieve tight bounds for all of them: (1) A hybrid algorithm making adaptive quantum or classical queries but with a limited quantum query budget, or (2) A quantum algorithm with access to a noisy oracle, subject to a dephasing or depolarizing channel, or (3) A hybrid algorithm with an upper bound on its maximum quantum depth; i.e., a classical algorithm aided by low-depth quantum circuits. In fact, our results handle all regimes between NISQ and full-scale quantum computers. Previously, only results for the pre-image search problem were known for these models by Sun and Zheng, Rosmanis, Chen, Cotler, Huang and Li while nothing was known about the collision finding problem.Comment: 40 pages; v2: title changed, major extension to other complexity model

    Quantum Query Complexity of Multilinear Identity Testing

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    Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring R=∠r1,...,rkR=\angle{r_1,...,r_k} where {r1,r2,...,rk}\{r_1,r_2,...,r_k\} is an additive generating set for RR and a multilinear polynomial f(x1,...,xm)f(x_1,...,x_m) over RR also accessed as a black-box function f:Rm→Rf:R^m\to R (where we allow the indeterminates x1,...,xmx_1,...,x_m to be commuting or noncommuting), we study the problem of testing if ff is an \emph{identity} for the ring RR. More precisely, the problem is to test if f(a1,a2,...,am)=0f(a_1,a_2,...,a_m)=0 for all ai∈Ra_i\in R. We give a quantum algorithm with query complexity O(m(1+α)m/2kmm+1)O(m(1+\alpha)^{m/2} k^{\frac{m}{m+1}}) assuming k≄(1+1/α)m+1k\geq (1+1/\alpha)^{m+1}. Towards a lower bound, we also discuss a reduction from a version of mm-collision to this problem. We also observe a randomized test with query complexity 4mmk4^mmk and constant success probability and a deterministic test with kmk^m query complexity.Comment: 12 page

    Low-gate Quantum Golden Collision Finding

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    International audienceThe golden collision problem asks us to find a single, special collision among the outputs of a pseudorandom function. This generalizes meet-in-the-middle problems, and is thus applicable in many contexts, such as cryptanalysis of the NIST post-quantum candidate SIKE. The main quantum algorithms for this problem are memory-intensive, and the costs of quantum memory may be very high. The quantum circuit model implies a linear cost for random access, which annihilates the exponential advantage of the previous quantum collision-finding algorithms over Grover's algorithm or classical van Oorschot-Wiener. Assuming that quantum memory is costly to access but free to maintain, we provide new quantum algorithms for the golden collision problem with high memory requirements but low gate costs. Under the assumption of a two-dimensional connectivity layout, we provide better quantum parallelization methods for generic and golden collision finding. This lowers the quantum security of the golden collision and meet-in-the-middle problems, including SIKE

    Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

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    The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently
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