. 2022 Sep 9;22(18):6838. doi: 10.3390/s22186838

Table 1.

Mathematical definitions of Chebyshev chaotic maps.

Mathematical Definitions	Descriptions
Chebyshev polynomial	$Chebyshev polynomial T_{n} (x) : \to [- 1, 1]$ is a polynomial in x of degree n, defined as $T_{n} {(x) = \cos (ncos}^{- 1} (x))$ .
Recurrent relation	$T_{n} {(x) = 2 x T}_{n - 1} {(x) - T}_{n - 2} (x)$ $for any n \geq 2$ $, T_{0} (x) = 1$ $, and T_{1} (x) = x$ .
Semi-group property	$T_{r} {(T}_{s} {(x)) = T}_{r s} {(x) = T}_{s} {(T}_{r} (x))$ $for any (s, r) \in Z$ $and s \in [- 1, 1]$ . Chebyshev polynomial restricted to interval [–1, 1] is a well-known chaotic map for all n > 1, which has a unique continuous invariant measure with positive Lyapunov exponent ln n. For n = 2, Chebyshev maps reduces to well-known logistic maps.
Extended Chebyshev polynomials	Zhang [34] proved that the semi-group property holds for Chebyshev polynomials defined on interval $(- \infty, + \infty)$ , and extended Chebyshev polynomials is defined as $T_{n} {(x) = (2 xT}_{n - 1} {(x) - T}_{n - 2} (x)) \mod N$ $, where n \geq 2$ $, x \in (- \infty, + \infty)$ , and N is a large prime number. Semi-group property holds, and extended Chebyshev polynomials also commute as $T_{r} (T_{s} {(x)) \mod N = T}_{r s} {(x) \mod N = T}_{s} (T_{r} (x)) \mod N$ .
Chaotic maps-based discrete logarithm problem (CMDLP)	Given two elements x and y, it is computationally infeasible to find the integer n such that $T_{n} (x) \mod N = y$ .
Chaotic maps-based Diffie-Hellman problem (CMDHP)	$Given three elements x, T_{r} (x) \mod N$ $, and T_{s} (x) \mod N$ $, it is computationally infeasible to compute T_{r s} (x) \mod N$ .