Chaotification of 1D Maps by Multiple Remainder Operator Additions—Application to B-Spline Curve Encryption

Abstract

In this work, a chaotification technique is proposed for increasing the complexity of chaotic maps. The technique consists of adding the remainder of multiple scalings of the map’s value for the next iteration, so that the most- and least-significant digits are combined. By appropriate parameter tuning, the resulting map can achieve a higher Lyapunov exponent value, a result that was first proven theoretically and then showcased through numerical simulations for a collection of chaotic maps. As a proposed application of the transformed maps, the encryption of B-spline curves and patches was considered. The symmetric encryption consisted of two steps: a shuffling of the control point coordinates and an additive modulation. A transformed chaotic map was utilised to perform both steps. The resulting ciphertext curves and patches were visually unrecognisable compared to the plaintext ones and performed well on several statistical tests. The proposed work gives an insight into the potential of the remainder operator for chaotification, as well as the chaos-based encryption of curves and computer graphics.

A Novel Conservative Chaotic System Involved in Hyperbolic Functions and Its Application to Design an Efficient Colour Image Encryption Scheme

“…Chaotic systems can be found in different areas such as weather forecasting [35], telecommunications [36], and biological modelling [37], and can be classified into different categories by physical (number of dimensions), dynamical (number of wings), and algebraic features (number of equilibrium points).…”

A Family of 1D Chaotic Maps without Equilibria

“…The PRBG generates 16 bits in each iteration, by hashing the value of the chaotic map through a multiplication and a modulo function. The bit generation in each iteration is given as (10 12 x k , 2 16 ) , 16), (22) where de2bi denotes the binary representation of the obtained decimal number, rem is the remainder operator, and B = {B 1 , B 2 , . .…”

“…However, the newly created systems are often computationally complicated (e.g., they require the use of appropriate numerical methods to generate solutions), or it is difficult to precisely determine for which parameters chaos occurs. Furthermore, in the literature, works on the so-called chaotification can be found, that is, methods of constructing new systems or improving the properties of already existing chaotic systems [16][17][18].…”