Computational Aspects of NUCOMP

In 1989, Shanks introduced the NUCOMP algorithm [10] for computing the reduced composite of two positive definite binary quadratic forms of discriminant ∆. Essentially by applying reduction before composing the two forms, the intermediate operands are reduced from size O(∆) to O(∆) in most cases and at worst to O(∆). Shanks made use of this to extend the capabilities of his hand-held calculator to computations involving forms with discriminants with as many as 20 decimal digits, even though his calculator had only some 10 digits precision. Improvements by Atkin (described in [3], [4]) have also made NUCOMP very effective for computations with forms of larger discriminant. Although there is nothing in Shanks’ original description which suggests that NUCOMP is only applicable to positive definite forms, for years there were no documented applications in any other setting. Recently, van der Poorten [13] has shown that, with very little extra effort, NUCOMP can also be applied to computations in the infrastructure involving indefinite binary quadratic forms. This opens the door to practical improvements in real quadratic field-based applications such as regulator computation and key exchange protocols in the infrastructure. Until now NUCOMP has been applied exclusively to computations in number fields. However, Cantor’s algorithm [2, 12] for adding reduced divisors on hyperelliptic curves (equivalently ideal multiplication in function fields) is virtually identical to the composition and reduction algorithms for binary quadratic forms; the main difference being that coefficients of the binary quadratic forms are polynomials over a finite field rather than integers. Thus, there is no reason to believe that NUCOMP cannot also be applied in function fields. Intuitively, one would expect that applying NUCOMP will reduce the degrees of the intermediate operands from O(2g) to at most O(3g/2), where g is the genus of the hyperelliptic curve or function field. Furthermore, by combining van der Poorten’s ideas [13] we can also apply NUCOMP to computations in the infrastructure of a real quadratic function field.