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"The authors propose using NAME_1 operator theory to analyze the security of public-key cryptographic protocols like Diffie-Hellman key exchange. They transform the nonlinear modulo operation into a linear system for easier analysis, but acknowledge that this method does not significantly reduce the computational complexity of brute-force attacks. The paper presents a fresh viewpoint on traditional cryptographic algorithms and connects them with chaotic dynamics.

Notations include definitions of natural numbers, integers, nonnegative integers, ranges of integers, symbols A^{dagger} and ln(z), complex exponential function, and polar forms. The equation z = exp(w) = exp(w+i2πℓ) holds for all ℓ∈ℤ and implies that if z is given in polar form r⋅exp(iθ) with r>0 and θ∈ℝ, then all complex logarithms of z are of the form ln(r)+i(θ+2πℓ).

An integer m is called a quadratic residue modulo p (abbreviated by mRp) if there exists an integer x such that m = x^{2} mod p. Two integers p_{1}, p_{2} are relatively prime or coprime if their greatest common divisor is equal to 1. All integers from {1,...,p} which are relatively prime to p form the multiplicative group Z_{p}^{*} of integers modulo p. Additionally, we use γ_{1}^{γ_{2}} as an abbreviation of γ_{1}≡γ_{2}modp.

A number m is a primitive root modulo p if for any integer c coprime to p there exists an exponent k∈ℕ such that m^{k}≡1modp. Equivalently, m is called a generator of Z_{p}^{*}."

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