Eric Bach

One-dimensional quantum walks

We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the <italic>Hadamard walk</italic>. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range <italic>[-t/\sqrt 2, t/\sqrt 2]</italic> after <italic>t</italic> steps, which is in sharp contrast to the classical random walk, which has distance <italic>O(\sqrt t)</italic> from the origin with high probability. With an absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is <italic>2/&pgr</italic>, and with an additional absorbing boundary at location <italic>n</italic>, the probability that the walk exits to the left actually increases, approaching <italic>1/\sqrt 2</italic> in the limit. In the classical case both values are 1.

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 omits a number that is O(log2 m) . This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log2m .

One-dimensional quantum walks with absorbing boundaries

In this paper we analyze the behavior of quantum random walks. In particular, we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the one-dimensional case. We compute these probabilities both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain domain, and can be used to compute several additional interesting properties, such as the time dependence of absorption. The eigenfunction method has the distinct advantage that it can be extended to arbitrary dimensionality. We outline the solution of the absorption probability problem of a (D-1)-dimensional wall in a D-dimensional space.

How to generate factored random numbers

This paper presents an efficient method for generating a random integer with known factorization. When given a positive integer N, the algorithm produces the prime factorization of an integer x drawn uniformly from

Explicit bounds for primes in residue classes

{N / 2} < x \leqq N

Factor refinement

. The expected running time is that required for

Toward a theory of Pollard's rho method

O(\log N)

Asymptotic semismoothness probabilities

prime tests on integers less than or equal to N.If there is a fast deterministic algorithm for primality testing, this is a polynomial-time process. The algorithm can also be implemented with randomized primality testing; in this case, the distribution of correctly factored outputs is uniform, and the possibility of an incorrectly factored output can in practice be disregarded.

Realistic analysis of some randomized algorithms

Let E/K be an abelian extension of number fields, with E ¬= Q. Let Δ and n denote the absolute discriminant and degree of E. Let σ denote an element of the Galois group of E/K. We prove the following theorems, assuming the Extended Riemann Hypothesis: (1) There is a degree-1 prime p of K such that (p/E/K) = σ, satisfying Np ≤ (1+ o(1))(logΔ + 2n) 2 . (2) 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 . (3) For K = Q, there is a prime p such that (p/E/Q) = σ, satisfying P ≤ (1 + o(1))(log Δ) 2 . In (1) and (2) we can in fact take p to be unramified in K/Q. A special case of this result is the following. (4) If gcd(m,q) = 1, the least prime p? m (mod q) satisfies p ≤ (1 + o(1))(φ(q)log q) 2 . It follows from our proof that (1)-(3) also hold for arbitrary Galois extensions, provided we replace σ by its conjugacy class . Our theorems lead to explicit versions of (1)-(4), including the following: the least prime p? m (mod q) is less than 2(q log q) 2 .

Comments on search procedures for primitive roots

Suppose we have obtained a partial factorization of an integer m, say m = ml1722 . . . mj. Can we efficiently “refine” this factorization of m to a more complete factorization m= rI n2ien, I<i<k -where all the ni 2 2 are pairwise relatively prime, and k 2 2? A procedure to find such refinements can be used to convert a method for splitting integers into one that produces complete factorizations, to combine independently generated factorizations of a composite number, and to parallelize the generalized Chinese remainder algorithm. We apply Sleator and Tarjan’s formulation of amortized analysis to prove the surprising fact that our factor refinement algorithm takes O((logm)2) bit operations, the same as required for a single gtd. This is our main result, and appears to be the first application of amortized techniques to the analysis of a numbertheoretic algorithm. We also characterize the output of our factor refinement algorithm, showing that the result of factor refinement is actually a natural generalization of the greatest common divisor. Finally, we also show how similar results can be obtained for polynomials. As an application, we give +Research supported by NSF grants DCR-8504485 and DCR8552596. ++Research supported by NSF grant CCR-8809573 and a Walter Burke award. +++Research supported by NSF grant CCR-8817400, the Wisconsin Alumni Research Foundation, and a Walter Burke award. *Department of Computer Science, Dartmouth College, Hanover, NH 03755. #Computer Sciences Department, University of Wisconsin, 1210 W. Dayton, Madison, WI 53706. algorithms to produce relatively torizations and normal bases. prime squarefree fac