On the use of Egyptian fractions for stream ciphers

Abstract

Within the scope of the mutual interference between modern number theory and chaos-based cryptography, and as it is well-known and already explored for the decimal or the continued fraction expansion of irrational numbers, interesting random-like behaviors seem to be hidden in Egyptian fraction expansions, thus suggesting new chaos-based encryption systems. In fact, at a practical level, and as will be shown through the present study, concatenation of involved denominators and binary expansion generally lead to pseudo-random streams which satisfactorily pass the NIST statistical test suite for randomness. Then, to a certain extent, and for cryptographic purposes, this can be considered as a new tool to complete the conventional class of already-in-use pseudo-random number generators. Some mathematical issues, however, have to be clarified, as for example the chaoticity of the process obtained after converting these denominators to fractional parts of a wandering sequence of real numbers in the unit interval. To the best of our knowledge, and from a cryptographic point of view, previous works on the subject are limited to statistical tests (and subsequent cryptographic analysis) of the resulting one-time pads or stream ciphers, but no rigorous study has been conducted to justify these methods within chaos theory. Considering Egyptian expansions as a working example and using the term “chaotic” in the sense of Devaney, this is what our proposal is aimed at in the present work.