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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… (More)

- Eric Bach
- 2010

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 0(log m). This has been generalized by Lagañas. Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split… (More)

- Eric Bach, Anne Condon, Elton Glaser, Celena Tanguay
- J. Comput. Syst. Sci.
- 1996

- Eric Bach, Jonathan P. Sorenson
- Math. Comput.
- 1996

Let E/K be an abelian extension of number fields, with E 6= Q. Let ∆ 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))(log ∆ + 2n)2. (2)… (More)

- Eric Bach
- SIAM J. Comput.
- 1988

- Eric Bach, René Peralta
- Math. Comput.
- 1996

We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β) be the asymptotic probability that a random integer n is semismooth with respect to nβ and nα. We present new recurrence relations for G and… (More)

- Eric Bach, Jeffrey Shallit
- 26th Annual Symposium on Foundations of Computer…
- 1985

This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials f(X) known to have the following property: given a multiple of f(p), we can quickly split any composite number that has p as a prime divisor. For example -- taking f(X) to be X- 1 -- a multiple of p - 1 will suffice to easily factor any… (More)

- Eric Bach
- Inf. Comput.
- 1991

Pollard’s “rho” method for integer factorization iterates a simple polynomial map and produces a nontrivial divisor of n when two such iterates agree modulo this divisor. Experience and heuristic arguments suggest that a prime divisor p should be detected in O(J) steps, but this has never been proved. Indeed, nothing seems to be have been rigorously proved… (More)

- Eric Bach
- IEEE Trans. Information Theory
- 1998

We show that the random number generator of Marsaglia and Zaman produces the successive digits of a rational -adic number. (The -adic number system generalizes -adic numbers to an arbitrary integer base .) Using continued fractions, we derive an efficient prediction algorithm for this generator.

This thesis addresses theoretical and practical aspects of the dynamic detecting and debugging of race conditions in shared-memory parallel programs. To reason about race conditions, we present a formal model that characterizes actual, observed, and potential behaviors of the program. The actual behavior precisely represents the program execution, the… (More)