Quantum computation presupposes that an extremely reliable probabilistic result found “logarithmically fast” by a quantum computer is in practice preferable to an accurate (but exponential time) result. The quantum computer is still in the making (and no clue to its time of arrival is given in this paper), but applications in cryptography, such as the primality factorization problem, have created a sense of anticipation [1].

Another important problem (addressed here) is that of tallying points on a curve over a field of q = pⁿ elements, which leads to the zeta function whose coefficients embody these tallies for each n. These points generate the divisor group, which is as important to curve cryptography as the residue group of a prime is to the RSA method. This paper is a status report on the probabilistic methods used in this group problem, beginning with a summary of the basic algebra geometry, and going on into computational complexity, with genus and field both variable. One typical device is the result that random elements of a divisor group determine the group probabilistically. The state-of-the-art theory is very futuristic, and cannot even be summarized.