The logistic map is a chaotic mapping. Although several studies have examined logistic maps over real domains with infinite/finite precisions, there has been little analysis of the logistic map over integers. Focusing on differences between the logistic map over the real domain with infinite precision and the logistic map over integers with finite precision, we herein show the characteristic properties of the logistic map over integers and discuss the sequences generated by the map.

In ordinary digital signature schemes, anyone can verify signatures with signer's public key. However it is not necessary for anyone to be convinced a justification of signer's dishonorable message such as a bill. It is enough for a receiver only to convince outsiders of signature's justification if the signer does not execute a contract. On the other hand there exist messages such as official documents which will be first treated as limited verifier signatures but after a few years as ordinary digital signatures. We will propose a limited verifier signature scheme based on Horster-Michels-Petersen's authenticated encryption schemes, and show that our limited verifier signature scheme is more efficient than Chaum-Antwerpen undeniable signature schemes in a certain situation. And we will propose a convertible limited verifier signature scheme based on our limited verifier signature scheme, and show that our convertible limited verifier signature scheme is more efficient than Boyar-Chaum-Damg rd-Pedersen convertible undeniable signature schemes in a certain situation.

Because of its simple structure, many reports on the logistic map have been presented. To implement this map on computers, finite precision is usually used, and therefore rounding is required. There are five major methods to implement rounding, but, to date, no study of rounding applied to the logistic map has been reported. In the present paper, we present experimental results showing that the properties of sequences generated by the logistic map are heavily dependent on the rounding method used and give a theoretical analysis of each method. Then, we describe why using the map with a floor function for rounding generates long aperiodic subsequences.