Factoring Numbers in 0(log n) Arithmetic Steps

Abstract

In this paper we show that a non-trivial factor of a composite number n can be found by performing arithmetic steps in a number proportional to the number of bits in n, and thus there are extremely short straight-line factoring programs. However, this theoretical result does not imply that natural numbers can be factored in polynomial time in the Turing-Machine model of complexity, since the numbers operated on can be as big as 2^cn^2, thus requiring exponentially many bit operations.