Integer Machines

Abstract

Computations with integers are at the core of a long-standing tradition in both computer science and number theory. A common feature is a measure of magnitude of an integer x given by the number of zeros and ones necessary to write the binary expansion of x (i.e.,$\left\lceil {\log (|{\text{x}}| + 1)} \right\rceil$). The relevant complexity measure is the bit cost. In the first section of this chapter, we show that machines over ℤ with bit cost and over ℤ₂ with unit cost are polynomially equivalent. Thus, we refer to either setting as classical.