We show that a 2-variable integer program defined by m constraints involving coefficients with at most s bits can be solved with O(m+s log m) arithmetic operations or with O(m+logm log s)M(s) bit operations, where M(s) is the time needed for s-bit integer multiplication.

We show that a positive definite integral ternary form can be reduced with O(M(s)log s) bit operations, where s is the binary encoding length of the form and M(s) is the bit-complexity of s-bit integer multiplication.

We show that a shortest vector of a 2-dimensional integral lattice with respect to the ` -norm can be computed with a constant number of extended-gcd computations, one common-convergent computation and a constant number of arithmetic operations. It follows that in two dimensions, a fast basis-reduction algorithm can be solely based on Schnhage's classical algorithm on the fast computation of continued fractions and the reduction algorithm of Gau. Keywords: Algorithms, computational geometry, number theoretic algorithms 1

This paper gives a survey on cryptographic primitives based on class groups of imaginary quadratic orders (IQ cryptography, IQC). We present IQC versions of several well known cryptographic primitives, and we explain, why these primitives are secure if one assumes the hardness of the underlying problems. We give advice on the selection of the cryptographic parameters and show the impact of this advice on the eciency of some IQ cryptosystems.