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Mathematical Foundations
Rigorous mathematical proofs and group‑theoretic constructions
This section provides rigorous mathematical treatment of zero‑knowledge cash systems, including formal proofs, group‑theoretic constructions, and cryptographic security analysis.
1.1 Mathematical Prerequisites
Zero‑knowledge cash systems rely on group theory, cryptographic entropy, homomorphic properties, and zero‑knowledge proofs. Each provides essential guarantees for privacy and soundness.
1.2 Core Mathematical Framework
We anchor randomness in a hard discrete‑log group, evolve it through algebra‑preserving maps, and validate transitions with succinct zero‑knowledge proofs.
This framework yields computational soundness while hiding every intermediate value via commitments and proofs.
Key Mathematical Properties
Group Structure: G = ⟨g⟩ with |G| = q prime ensures each element can be expressed as g^k
Entropy Preservation: H_α(s_i) ≥ H_min for the initial randomness pool
Homomorphic Evolution: φ_H: S_i → S_(i+1) preserving the group law
Commitment Properties: C(m, r) = g^m h^r with perfect hiding and computational binding