Rigorous Mathematical Proofs and Group-Theoretic Constructions
Zero-knowledge cash systems are built upon several fundamental mathematical concepts:group theory, cryptographic entropy, homomorphic properties, and zero-knowledge proof systems. Each component provides essential security guarantees.
The construction relies on a group-theoretic entropy foundation where initial randomness is bound to a hard discrete-logarithm group, evolved through algebra-preserving maps, and validated with succinct zero-knowledge proofs.
This mathematical framework ensures computational soundness while hiding every intermediate value through carefully designed cryptographic primitives and proof systems.
Cyclic groups, generators, and discrete logarithm problem
Rényi entropy, cryptographic randomness, and sampling
Structure-preserving maps and algebraic operations
Pedersen commitments and binding/hiding properties
Completeness, soundness, and zero-knowledge properties
Cryptographic assumptions and post-quantum considerations
Group Structure: G = ⟨g⟩ where |G| = q (prime) ensures every element can be expressed as gk
Entropy Preservation: Hα(si) ≥ Hmin for initial randomness pool
Homomorphic Evolution: φH : Si → Si+1 preserving group law
Commitment Properties: C(m,r) = gmhr with perfect hiding and computational binding
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