Quantum computers can break the RSA and El Gamal public-key cryptosystems, since they can factor integers and extract discrete logarithms. If we believe that quantum computers will someday become a reality, we would like to have post-quantum cryptosystems which can be implemented today with classical computers, but which will remain secure even in the presence of quantum attacks.In this article we show that the McEliece cryptosystem over well-permuted, well-scrambled linear codes resists precisely the attacks to which the RSA and El Gamal cryptosystems are vulnerablenamely, those based on generating and measuring coset states. This eliminates the approach of strong Fourier sampling on which almost all known exponential speedups by quantum algorithms are based. Specifically, we show that the natural case of the Hidden Subgroup Problem to which the McEliece cryptosystem reduces cannot be solved by strong Fourier sampling, or by any measurement of a coset state. We start with recent negative results on quantum algorithms for Graph Isomorphism, which are based on particular subgroups of size two, and extend them to subgroups of arbitrary structure, including the automorphism groups of linear codes. This allows us to obtain the first rigorous results on the security of the McEliece cryptosystem in the face of quantum adversaries, strengthening its candidacy for post-quantum cryptography.
We study the behavior of the A* search algorithm when coupled with a heuristic h satisfying (1-epsilon1)h*
Abstract. We study the problem of local search on a graph. Given a real-valued black-box function f on the graph's vertices, this is the problem of determining a local minimum of f -a vertex v for which f (v) is no more than f evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires Ω(2 n/2−o(1) ) queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to Ω(2 n/2 /n 2 ) and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are Ω " √ N d log N "and Ω, respectively.
We study the problem of \emph{local search} on a graph. Given a real-valued black-box function $f$ on the graph's vertices, this is the problem of determining a local minimum of $f$---a vertex $v$ for which $f(v)$ is no more than $f$ evaluated at any of $v$'s neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires $\Omega(2^{n/2 - o(n)} )$ queries to determine a local minima on the $n$-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to $\Omega(2^{n/2}/n^2)$ and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph $G$ of $N$ vertices and diameter $d$, the randomized and quantum query complexities for local search on $G$ are $\Omega\left({\sqrt{N}}/{d\log N}\right)$ and $\Omega\left({\sqrt[4]{N}}/{\sqrt{d\log N}}\right)$, respectively.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.