ENI6MA a Cryptographic Engine for Zero-Knowledge Proofs Across Modalities

This architecture enables secure identity, data integrity, and authorization—without storing secrets, revealing credentials, or requiring specialized hardware. ENI6MA functions as a layer-zero trust substrate for post-quantum ecosystems, redefining authentication as an epistemic act, not a secret exchange.

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RWP Accumulator with Lower Dimensional Witnesses

The Rosario-Wang Accumulator cypher employs a novel approach to cryptographic accumulators, leveraging the Holographic Morphism with private languages. This cypher uses a set of enumerated alphabets with equal cardinality to create witnesses, ensuring efficient and secure membership proofs.

Witnesses in the cypher are lower-dimensional references that are expanded after determining the holographic morphism. This morphism identifies a subset of the projective alphabet sets containing character members of another alphabet, whose members are distributed equally across the new lower-dimensional projective set. The higher-dimensional set contains a larger number of members, representing the individual members of the secret key, from which the witness is provided in any $i^{\text{th}}$ round of a sigma protocol.

Key Definitions

The Accumulator cypher defines the projective set $\Pi$ as a set containing elements arranged in a lower-dimensional space, which expands to identify specific subsets through holographic morphisms. The holographic morphism $\mathcal{H}$ is a hidden morphism linking two alphabets that have no intersection. This morphism is used to expand lower-dimensional witnesses. The lower-dimensional representation process maps higher-dimensional members of an alphabet into lower-dimensional projective sets.

$$\text{pk}_\chi = \text{HoloMorphism}(\nu_{i_k}) \quad \text{for} \quad \chi \in \Xi$$ $$\Pi = \{\nu_{i_1}, \nu_{i_2}, \ldots, \nu_{i_r}\}$$

Interactive Cypher Setup

In the setup phase, the projective set $\Pi$ is defined to model the domain $M$. A synonym relation to the emergent set is established via a hidden morphism.

During the accumulation phase, the short representation $\text{Acc}_\Xi$ of the set $X$ is created, partitioning it into subsets that form the basis of the projective set:

$$\text{Acc}_\Xi = \bigl\{\Pi = (\nu_{i_1}, \nu_{i_2}, \ldots, \nu_{i_r}),\; \Omega = (\nu_{j_1}, \nu_{j_2}, \ldots, \nu_{j_r})\bigr\}$$

For membership holomorphic witness responses, a lower-dimensional reference is created using the Holographic Morphism, then expanded to identify a subset of the projective alphabet sets.

For non-membership witnesses, lower-dimensional references are generated and expanded to verify non-inclusion in the set $X$:

$$\text{pk}_\chi = \text{HoloMorphism}(\nu_{i_k}) \quad \text{for} \quad \chi \in \Xi$$

The verification process utilizes the projective morphism to trace the exact path from the lower-dimensional witness to the accumulated representation $\text{Acc}_\Xi$:

$$\text{Verify}\!\bigl(\text{resp}(\chi, \text{pk}_\chi, \chi), \text{Acc}_\Xi\bigr) = \begin{cases} \text{True} & \text{if valid membership} \\ \text{False} & \text{otherwise} \end{cases}$$

The primary equations and formulations in the RWP Accumulator cypher include the accumulator representation, membership holomorphic witness responses, non-membership holomorphic witness responses, and the sigma protocol for witness verification.

Example

Consider a set $X = \{\chi_1, \chi_2, \chi_3, \chi_4\}$. The accumulator represents this set in projective form:

$$\text{Acc}_X = \bigl\{\Pi = (v_1, v_2, v_3, v_4),\; \Omega = (v_5, v_6, v_7, v_8)\bigr\}$$

For membership witness $\chi_1$:

$$\text{pk}_{\chi_1} = \text{HoloMorphism}(v_1)$$

Verification:

$$\text{Verify}\!\bigl(\text{resp}(\chi_1, \text{pk}_{\chi_1}, \chi), \text{Acc}_X\bigr)$$

Holographic Witness Accumulation

In setup, $\Pi$ models the domain $\mathcal{M}$. The accumulate phase creates the short representation $\text{Acc}_\Xi$ of the set $\Xi$, partitioning it into projective subsets:

$$\text{Acc}_\Xi = \bigl\{ \Pi = (\nu_{i_1}, \ldots, \nu_{i_r}),\; \Theta = (\nu_{j_1}, \ldots, \nu_{j_r}) \bigr\}$$

Public Key Generation

Membership witnesses:

$$\text{pk}_\xi = \mathcal{H}(\nu_{i_k}) \quad \text{for} \quad \xi \in \Xi$$

Non-membership witnesses:

$$\text{pk}_{\xi'} = \mathcal{H}(\nu_{j_k}) \quad \text{for} \quad \xi' \in \mathcal{M} \setminus \Xi$$

Verification

$$\text{Verify}\!\bigl(\text{resp}(\xi, \text{pk}_\xi, \chi), \text{Acc}_\Xi\bigr) = \begin{cases} \text{True} & \text{if valid membership} \\ \text{False} & \text{otherwise} \end{cases}$$

Equations and Formulations

1. Accumulator Representation

   $$\text{Acc}_\Xi = \bigl\{ \Pi = (\nu_{i_1}, \ldots, \nu_{i_r}),\; \Theta = (\nu_{j_1}, \ldots, \nu_{j_r}) \bigr\}$$

2. Membership Public Key Generation

   $$\text{pk}_\xi = \mathcal{H}(\nu_{i_k}) \quad \text{for} \quad \xi \in \Xi$$

3. Non-Membership Public Key Generation

   $$\text{pk}_{\xi'} = \mathcal{H}(\nu_{j_k}) \quad \text{for} \quad \xi' \in \mathcal{M} \setminus \Xi$$

4. Sigma Protocol for Witness Verification

   Challenge: $\chi \in \{0,1\}^n$ Response:

   $$\text{resp}(\xi, \text{pk}_\xi, \chi) = \text{Expand}\bigl(\mathcal{H}(\nu_{i_k}), \chi\bigr)$$

   Verification:

   $$\text{Verify}\!\bigl(\text{resp}(\xi, \text{pk}_\xi, \chi), \text{Acc}_\Xi\bigr) = \begin{cases} \text{True} & \text{if valid membership} \\ \text{False} & \text{otherwise} \end{cases}$$

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[Bellare1999]⁸ [Benaloh1993]⁹ [Pointcheval1996]¹⁰

🔬 Cryptographic Foundations

ENI6MA introduces a universal symbolic language:

• A unified symbol set $L = \bigcup_i \Sigma_i$ spans multiple modalities—text, tone, color, gesture.
• A morphism-invariant commitment array encodes a secret word across any medium.
• Each symbol maps to an eigen-signature in a Hilbert manifold $\mathbb{M}$, enabling projection, verification, and recovery across media: light, sound, haptics, or RF.

Above this geometric layer operates the Rosario–Wang Protocol:

• Each session generates a nonce-shuffled alphabet $X^R$ and a round-specific projection $\Pi_{e^R}(P) \subset \mathbb{M}$.
• The prover must show correct subset membership of their projection across $n$ randomized rounds, producing a single-bit accumulator $\Lambda$ for verifier validation.
• This interaction is stateless, zero-knowledge, and replay-resistant—even when transcripts are fully observed.

The result is a zero-knowledge password proof (ZKPP)—a cognitive authentication primitive that is:

• Fully entropy-synchronized via symmetric nonce lifecycles (AVAILABLE → RESERVED → USED),
• Immune to phishing, replay, or brute-force attacks,
• Secure against quantum adversaries due to combinatorial projection hardness.

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🔐 Security Model

Threat Class	Defense Mechanism	Result
Replay Attacks	Entropy-gated morphisms $\Pi_{e^R}$, non-reusable nonces	Observed transcripts become invalid
Phishing / Keylogging	Symbol shuffling $\Sigma(A) \rightarrow X^R$ per round	Recorded keystrokes don’t replay secrets
Quantum Attacks	Hilbert lattice traversal over multiple combinatorial spaces	Grover and Shor ineffective
Storage Breach	Stateless commitments, no persistent secrets	Server-side breaches yield nothing usable

Mathematically, soundness and zero-knowledge are formalized via:

• Entropy-accumulated membership proofs,
• Roundwise projection foliations over disjoint symbolic subspaces,
• Probabilistic guarantees of security: $\Pr[\text{forgery}] \le (|\Sigma|^{-1})^{nk}$.

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🌐 Applications Across Ecosystems

ENI6MA’s symbolic core and modular deployment model allow it to anchor secure communication, identity, and verification in:

🔗 Enterprise & IAM

• Replace MFA codes with in-memory cognitive proofs
• Integrate via SSO and Zero-Trust identity providers (Okta, Azure AD)

🌍 IoT & Embedded Systems

• Stateless zero-knowledge attestation on constrained hardware
• Supports gesture pads, LED rings, ultrasonic emitters

📡 Blockchain & Web3

• Post-quantum wallets with portable seed commitments
• ENI6MA proofs substitute for VRF/VRF+PoS-based block eligibility

🧠 Human Authentication & E-ID

• Memory-based passwords resistant to shoulder-surfing
• Cognitive symbol challenges rendered via QR, NFC, or audio

🎛️ Content Integrity & AI Provenance

• Symbolic eigen-signatures watermark neural networks
• Embed proofs in weights or activations without degrading performance

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🧩 Protocol Stack & Reference Architecture

Layer	Role	Patent Anchor
Commitment Engine	Builds $C = (s_k, \Sigma_j, t_k)$ from user-secret + nonce	Claim 1, Axiom 3
Projection Driver	Generates session-specific slice of manifold $X^R \subset \mathbb{M}$	Eq. 10, Claim 8
Verifier Module	Sensory matching $\|\mathbf{v} - \varphi(s)\| < \varepsilon$	Eq. 12–13

The system requires no vault, no shared secrets, and no device registration. Each login projects a new slice of the manifold, verified via entropic gating and human-memory input.

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References

1. FD. Rosario & L. Wang PhD, Proof of Information Entanglement, provisional patent USPTO 2025.
2. S. Goldwasser, S. Micali & C. Rackoff, "The Knowledge Complexity of Interactive Proofs", SIAM J. Comput. 18(1), 1989.
3. E. Ben-Sasson et al., "Post-Quantum Zero-Knowledge Proof Systems", J. Cryptology 35(4), 2022.