Rosario Hologram

The Rosario-Wang architecture implements a holographic morphism model that achieves information-theoretic security via symmetric entanglement, wherein a dual-layer private key structure supports zero-knowledge proofs even over adversarial or compromised channels. This construction ensures that no partial disclosure or observation of the communication reveals knowledge about the witness, due to the tightly coupled, non-factorable entanglement of symbol layers within the morphic space.


🔐 Symmetric Holographic Morphism ( Entangled Proof of Knowledge )

The Rosario-Wang architecture presents a novel zero-knowledge proof system based on holographic morphism and dual-layer private keying, designed to ensure information-theoretic security even over compromised channels. The system models information as projections within a separable Hilbert space $\mathcal{H}$, encoding a private sequence $P = {p_1, p_2, \ldots, p_n} \subset A$, where $A$ is a static alphabet, through dynamically shuffled challenge spaces. Each authentication round $R$ uses a shuffling function $\Sigma: A \rightarrow X^R$, generating a distinct challenge alphabet $X^R$, thereby preventing repetition and enabling non-determinism.

Security derives from a dual-layer private key structure. The first layer comprises a cognitive-symbolic commitment: the sequence $P$ exists only in the prover’s memory and is never transmitted. The second layer consists of a projection morphism $\mathcal{M}: P \mapsto \Pi(X^R)$, which embeds the sequence into a round-specific morphic space. Crucially, the two layers are independently non-invertible—without access to $P$, the projection appears statistically random, and without $\mathcal{M}$, the sequence lacks semantic context. This design enforces entanglement in the cryptographic sense: partial knowledge of either layer yields no meaningful leakage.

The core primitive of this architecture is the holographic morphism $\mathcal{H}$, defined as a modality-preserving bijection between distinct representational layers:

\[\mathcal{H}: \mathcal{L}_i \rightarrow \mathcal{L}_j \quad \text{such that} \quad \forall \ell_i \in \mathcal{L}_i, \; \exists \ell_j = f(\ell_i, \theta),\]

where $\mathcal{L}_i, \mathcal{L}_j$ represent separate encoding spaces (e.g., letters, colors, spatial orientations), $\theta$ is a private transformation parameter, and $f$ is a non-linear, structure-preserving function. This mechanism introduces symbolic entanglement: recovery of a partial component (e.g., a color) provides no advantage without the correct coupling to its corresponding elements. Formally, the probability of reconstructing the full state from an intercepted partial morphism satisfies

\[\Pr[\text{recover full state} \mid \text{partial view}] \approx \frac{1}{|\mathcal{L}_i|^k},\]

where $k \geq 3$ in typical implementations involving triple entanglement across symbol, color, and rotation spaces.

To validate the prover’s knowledge, the system employs an accumulator $\Lambda$ that aggregates individual verification results across $n$ rounds:

\[\Lambda = \bigwedge_{R=1}^n M(p_i, x_i^R),\]

where $M(p_i, x_i^R)$ denotes whether symbol $p_i$ maps to the expected subset $x_i^R \subset X^R$ under the morphism. A probabilistic variant models uncertainty or noise in the challenge-response channel:

\[\Lambda_{\text{new}} = \prod_{R=1}^n \Pr(M(p_i, x_i^R) = \text{true} \mid x_i^R, R).\]

This construction supports zero-knowledge operation: no verifier or adversary ever observes the witness $P$ or projection key $\mathcal{M}$, only the result of the verifier function over public challenge alphabets $X^R$.

The projection function $\Pi$ maps the sequence $P$ into a foliated submanifold $\mathcal{M}_X \subset \mathcal{H}$, such that any external observer can at most view a low-dimensional “slice” of the projection space. This design ensures semantic indistinguishability and prevents reverse engineering of the morphism.

Rosario-Wang system provides a zero-knowledge authentication protocol that is both quantum-resistant and secure under passive interception. The key insight lies in treating symbol spaces as entangled morphisms within a higher-dimensional encoding framework—each authentication round behaves as a non-repeating projection into a different facet of a morphic lattice, tightly binding cognitive and mathematical primitives. This makes the protocol suitable for secure identity verification, cognitive ZKPs, and cryptographic applications requiring modality-agnostic commitments.


1. Hilbert Space Projections

Let $\mathcal{H}$ denote a separable Hilbert space. Information is encoded via manifold projections:

2. Two-Layer Private Key Structure

Each layer is non-derivable from the other:

This dual secrecy layer reflects entanglement—alteration or observation of one layer collapses the entire verification ability.

3. Holographic Morphism (ℋ)

Defined as:

\[\mathcal{H} : \mathcal{L}_i \rightarrow \mathcal{L}_j \text{ such that } \forall \ell_i \in \mathcal{L}_i, \exists \ell_j \in \mathcal{L}_j \text{ where } \ell_j = f(\ell_i, \theta)\]

Where:

This yields perfect security under symmetric entanglement: guessing any one component (e.g., a color) gives no advantage without its entangled counterparts (e.g., orientation and symbol), due to:

\[\Pr[\text{recover full state} \mid \text{partial morphism observed}] \approx \frac{1}{|\mathcal{L}_i|^k}\]

with $k$ ≄ 3 for triple-entanglement (e.g., symbol-color-rotation).


🛡 Security Guarantees Over Compromised Channels

The security of the Rosario-Wang proof is retained via:

a. Zero-Knowledge Witnessing

The proof $\Lambda$ is an accumulation:

\[\Lambda = \bigwedge_{R=1}^n M(p_i, x_i^R)\]

with probabilistic version:

\[\Lambda_{\text{new}} = \prod_{R=1}^n \Pr(M(p_i, x_i^R) = \text{true} \mid x_i^R, R)\]

No private component $P$ is transmitted. An attacker intercepts only the public morphism challenge $X^R$ and does not learn $P$ or $\mathcal{M}$.

b. Indistinguishability via Manifold Foliation

The projection:

\[\Pi : P \mapsto \text{foliated manifold } \mathcal{M}_X \subset \mathcal{H}\]

ensures that the observer sees only one “slice” of a high-dimensional morphic surface—insufficient to invert or learn private mappings.


🔁 Summary of Security Pattern

Component Function Role in Security
$P$ Symbolic witness Hidden input (memory-only)
$\mathcal{M}$ Projection key Private morphism of $P$
$\mathcal{H}$ Holographic morphism Entanglement map across alphabets
$\Lambda$ Accumulator Logical product of valid subset hits
$X^R$ Challenge space Public morphic projection
$M(p_i, x_i^R)$ Verifier condition True iff mapping exists in correct foliation

🧠 Key Insight

The system’s holographic morphism acts like a private, non-invertible lens: a non-linear, multidimensional encoding function that entangles symbolic content across multiple modalities. Together with a dual-layer private key architecture and zero-knowledge accumulation, it offers quantum-resistant, interception-safe authentication even across compromised channels.

References

  1. Trapdoor one-way functions / permutations – Diffie–Hellman proposal for public-key crypto (1976) and the RSA construction (1977). These are keyed maps $f : X → Y$ that anyone can evaluate, but only someone holding a secret “trapdoor’’ can invert. When RSA chose modular exponentiation as f, it created the archetypal private map: encryption is public, decryption is private. (en.wikipedia.org, en.wikipedia.org)

  2. Pseudorandom function (PRF) families – Goldreich-Goldwasser-Micali (GGM) construction, Crypto ’84/’86. A PRF family ${F_K}$ makes each keyed instance computationally indistinguishable from a random map on its domain. This gives confidentiality even under chosen-input probing, because without K the mapping leaks no structure. (wisdom.weizmann.ac.il, crypto.stanford.edu)

  3. Pseudorandom permutations (PRP) & Feistel/block-cipher designs Feistel network (Horst Feistel, “Lucifer”, 1971 → DES 1977); Luby–Rackoff proof that 3–4 Feistel rounds from a PRF yield a PRP (1988). A PRP is a bijective private map; knowing the key lets you run the inverse permutation instantly, while outsiders only see an apparently random shuffle of the space. DES-style S-boxes are concrete keyed substitution tables that hide those permutations’ structure. (en.wikipedia.org, en.wikipedia.org, omereingold.wordpress.com)

  4. Oblivious pseudorandom functions (OPRFs) – concept refined by Bellare & Rogaway (mid-1990s) and widely implemented with elliptic-curve OPRF protocols (e.g., EC-OPRF 2017). An OPRF lets one party obtain $F_K(x)$ without learning K, while the key holder learns nothing about x. The output looks random to both except for what the protocol reveals, so the mapping stays private to both sides. (eprint.iacr.org, en.wikipedia.org)

  5. Private-Function Evaluation (PFE) – first explicit frameworks by CrĂ©peau–Kilian (1993) and Ostrovsky-Wray (1997). Here the function itself is secret; parties jointly evaluate $y = f(x)$ so that neither learns more than their rightful share. Internally, PFE relies on garbled circuits and OT to turn a hidden Boolean circuit into a temporary private map. (jis-eurasipjournals.springeropen.com, eprint.iacr.org)

  6. Function Secret Sharing (FSS) – Boyle, Gilboa & Ishai, CCS 2016. FSS splits a keyed function $F_K$ into additive “shares’’ such that each server holds an apparently random map; only by combining outputs do you recover F_K(x). It is a distributed, key-split form of private map ideal for rate-limited or threshold settings. (tzin.bgu.ac.il)