Canonical Modular Logarithm and Minimal Primitive Roots

We define a canonical modular logarithm function f on the multiplicative group of integers modulo k², denoted (Z mod k² Z)×, for any prime k > 2. Using this function, we show that among the generators modulo k, at least φ(k−1)/2 remain generators modulo kⁿ when k > 3, and that at least one such generator exists when k = 3. We determine the general behavior of the sequence of multiplicative orders modulo kⁿ for any integer coprime to k, and present several basic properties of the function f. Additional extensions and algebraic considerations are also proposed.