Principal Ideal Testing in $\mathcal{O}$

Let i be any given $\mathcal{O}$-ideal. In this chapter we will denote by P the problem of determining whether or not i is a principal $\mathcal{O}$-ideal. Of course, as we can easily find α and a reduced $\mathcal{O}$-ideal b such that b = (α)i, we may consider this problem as applying, instead, to a given reduced $\mathcal{O}$-ideal b. Also, since a principal ideal is invertible, we may also assume that b (or i) is invertible. An extended version of P is the problem of determining, once we know that b is principal, a generator β of b. By our observations in Chapter 12, all we really need to determine β is some q, b ∈ $\mathbb{Q}$ such that ¦log₂β − b¦ < q, where q is small, say q ≤ 10. Both of these problems can be solved by the index-calculus method described in §13.4, but because we need the truth of unproved hypotheses to be sure our result is correct, particularly when b has been declared not principal, this technique is conditional. In this chapter we will develop methods for solving P that are either unconditional or are only conditional concerning the runtime; the answer is still mathematically correct (a Las Vegas algorithm). The problem with these processes is that they are unfortunately of exponential complexity.