Wrap any math symbols here which are not already wrapped with a $ and fix any improper LaTeX of the following:

"The authors propose applying NAME_1 operator theory to analyze the security of public-key cryptographic protocols, such as the Diffie-Hellman key exchange. They transform the nonlinear modulo operation into a linear system to facilitate analysis, but they acknowledge that this method does not significantly reduce the computational complexity of brute-force attacks. The paper presents a novel perspective 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, implying 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 p1, p2 are relatively prime or coprime if their greatest common divisor is equal to 1. All integers from {1, ..., p} that are relatively prime to p form the multiplicative group Zp* of integers modulo p. We also use γ1≢2 as an abbreviation for γ1 ≡ γ2 mod p.

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 ≡ 1 mod p. Alternatively, m is called a generator of Zp*."

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