• Some bounding techniques are investigated in this section in order to serve as a tool for the analysis that follows. We first state the union upper bound that is widely used in the literature. • Then, some better bounds which are applicable to AWGN channels are also discussed here; they are Gallager bound and sphere upper bound.

Distance-Bounding is used to defeat relay attacks. For wireless payment systems, the payment terminal is not always online. So, the protocol must rely on a public key for the prover (payer). We propose a generic transformation of a (weakly secure) symmetric distance bounding protocol which has no postverification into wide-strong-private and secure public-key distance bounding.

We derive a bounding circle for a fractal image specified by an iterated function system. The radius of the bounding circle is smaller than those from previously published material. The bounding circle is important in fractal design and plotting software as it enables a fractal image to be scaled correctly to fit the screen of a digital computer.