While position integrity is crucial when positioning systems are to be used for safety-critical operations such as in aviation applications, current positioning algorithms are generally optimised for accuracy instead. Even when they are combined with fault detection and exclusion schemes, these algorithms still give sub-optimal integrity. This paper promotes a new method of algorithm design that takes integrity rather than accuracy as the parameter to optimise. The new class of high-integrity positioning algorithms described aims to improve integrity with both current and new systems not by improving the physical infrastructure, but by using clever algorithmic optimisation in the receiver. A small simulation example shows that the integrity and availability of un-augmented GPS for non-precision approach can indeed be improved substantially.