Group-Theoretic Entropy and Mathematical Foundations
Let G be a cyclic group of prime order q with generators g, h. The initial randomness pool is defined as:
where Hα denotes the Rényi entropy of order α > 1. Sampling from S0 produces elements indistinguishable from uniform under the discrete-log assumption.
Formal results on random walks in groups yield asymptotic lower bounds:
This guarantees that an adversary gains negligible advantage even after observing polynomially many epochs.
A generalization of Shannon entropy that measures the uncertainty in a probability distribution. For order α > 1, it provides stronger security guarantees than standard entropy measures.
The computational hardness assumption that finding x given g^x in a cyclic group G is computationally infeasible. This forms the basis for many cryptographic protocols.
The group G = ⟨g⟩ where |G| = q (prime) ensures that every element can be expressed as gk for some k ∈ ℤq.
The Rényi entropy Hα provides a measure of randomness that is preserved under group operations, ensuring cryptographic security.
Elements sampled from S0 are computationally indistinguishable from uniform random elements in G, providing perfect hiding properties.
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