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One-dimensional quantum walks
We define and analyze quantum computational variants of random walks on one-dimensional lattices, including the Hadamard walk, a quantum analog of the symmetric random walk. Expand
Crossed and nested dependencies in German and Dutch
We show that clause-final verbal clusters in Standard Dutch and German differ strikingly in the kinds of dependencies they normally permit between verbs and their arguments, with Dutch preferring crossed dependencies and German nested. Expand
Explicit bounds for primality testing and related problems
Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omitsExpand
Algorithmic Number Theory
  • E. Bach, J. Shallit
  • Mathematics, Computer Science
  • Lecture Notes in Computer Science
  • 1 September 1996
Explicit bounds for primes in residue classes
We prove the following theorems, assuming the Extended Riemann Hypothesis: There is a degree-1 prime p of K such that (p/E/K) generates the same group as σ, satisfying Np ≤ (1 + o(1))(log Δ) 2 . Expand
Discrete Logarithms and Factoring
This note discusses the relationship between the two problems of the title. We present probabilistic polynomial-time reduction that show: 1) To factor n, it suffices to be able to compute discreteExpand
Analytic methods in the analysis and design of number-theoretic algorithms
This book makes a substantial contribution to the understanding of a murky area of number theory that is important to computer science, an area relevant to the design and analysis of number-theoreticExpand
Realistic analysis of some randomized algorithms
  • E. Bach
  • Mathematics, Computer Science
  • STOC '87
  • 1987
Pseudo-random numbers can be used to estimate the failure probability of probabilistic algorithms for finding square roots modulo a prime. Expand
Toward A Theory of Pollard's Rho Method
  • E. Bach
  • Computer Science, Mathematics
  • Inf. Comput.
  • 2 February 1991
This paper shows that for fixed k , this probability is at least ( 2 k ) p + O(p −3 2 ) as p → ∞. Expand