Compact representation in real quadratic congruence function fields

Abstract

A real quadratic congruence function field $K = \mathbb{F}_q \left( x \right)\left( {\sqrt D } \right)$typically contains many elements α of large height $H(\alpha ) = \max \left\{ {|\alpha |,|\bar \alpha |} \right\}$and small norm (in absolute value) $|N(\alpha )| = |\alpha \bar \alpha |$. A prominent example for this kind of behavior is the fundamental unit ηκ whose norm has absolute value 1, but whose height is often exponential in ¦D¦. Hence it requires exponential time to even write down ηκ, let alone perform computations on ηκ In this paper, we present a shorter representation for elements α in any quadratic order $\mathcal{O} = \mathbb{F}_q \left[ x \right]\left[ {\sqrt \Delta } \right]$of K. This representation is analogous to the one for quadratic integers developed by Buchmann, Thiel, and Williams, and is polynomially bounded in log ¦N(α)¦, log deg H(α), and. log¦Δ¦. For the fundamental unit ηκ of K, such a representation requires O((log ¦D¦)²) bits of storage. We show how to perform arithmetic with compact representations and prove that the problems of principal ideal testing, ideal equivalence, and the discrete logarithm problem for ideal classes belong to the complexity class NP.