Verification Protocol

Eni6ma Technology and the Rosario-Wang Proof/Cypher is Patent Pending. USPTO 2024. Copyright 2024 All right reserved. Eni6ma.org - Dylan Rosario

Table of Symbolic Keys

1. Component (Π)

   • DESC: [Method] Interactive multi-round proof ceremony for sequence authentication.

2. Alphabet (A)

   • DESC: [Member] Static set of words from which sequences are generated.

3. Shuffled Alphabet $X^R$

   • DESC: [Member] Randomly ordered alphabet derived from A for each round R.

4. Sequence (P)

   • DESC: [Member] Indexed array of members selected from A for verification.

5. Subset of $X^R$ ( $x_i^R$ )

   • DESC: [Variable] Specific subset of $X^R$ targeted in each verification round R.

6. Number of Subsets (n)

   • DESC: [Variable] Total subsets in X, correlating to the rounds in the verification process.

7. Index of $P$ (δ)

   • DESC: [Variable] Index within P being verified in the current round R.

8. Witness (Ω)

   • DESC: [Function] Indicator provided by the authenticator for the subset $x_i^R$ containing element $p_n$.

9. Random Distribution per Round (Σ)

   • DESC: [Function] Random organization of A into $X^R$, and its division into subsets for each round R.

10. Membership Verification per Round (Μ)

    • DESC: [Condition] Verification condition applied per round to confirm the presence of $p_n$ in $x_i^R$.

11. Accumulation of Verification Results (Λ)

    • DESC: [Constraint] Aggregates the verification results from each round, requiring all true for Κ.

12. Round (R)

    • DESC: [Member] Each cycle of the protocol where a new Σ is generated, and Μ is verified against $x_i^R$.

13. Proof of Knowledge (Κ)

    • DESC: [Result] The final validation, confirming the authenticator's knowledge of sequence P.

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Axioms and Lemmas

Axioms

1. Axiom of Initialization:

   • Equation: P = {p_1, p_2, ..., p_n}, $X^R = Σ(A)$ for $R = 1, 2, ..., n$.

   • Expression: This axiom establishes the protocol $Π$ begins with a predefined sequence $P$, consisting of elements $p_1$to $p_n$, and a static alphabet A. For each verification round R, a shuffled version of A, denoted as $X^R$, is produced through the shuffling function Σ, ensuring fresh and unpredictable challenges in each round.

2. Axiom of Random Distribution:

   • Equation: $X^R = Σ(A)$ , ensuring $X^R \neq X^{R'}$ for $R \neq R'$ .

   • Expression: This asserts that each round $R$ of the protocol generates a uniquely shuffled alphabet $X^R$ from A, via Σ. The condition $X^R \neq X^{R'}$ for differing rounds ensures that the sequence of challenges is non-repetitive and unpredictable, fundamental for securing the verification process.

3. Axiom of Sequence Verification:

   • Equation: Existence of $x_i^R$ such that $p_i \in x_i^R$ for each $p_i$ in P.

   • Expression: For every element $p_i$ within the sequence P, there exists a targeted subset $x_i^R$ within the shuffled alphabet $X^R$ where $p_i$ can be found and verified. This axiom underscores the protocol's capacity to pinpoint and verify individual sequence elements.

4. Axiom of Completeness:

   • Equation: $\forall p_i \in P, \exists x_i^R \in X^R : Μ(p_i, x_i^R) = \text{true}$ .

   • Expression: Signifies that for all elements $p_i$ within P, there must be a subset $x_i^R$ in $X^R$ for which the verification condition $Μ$ returns true, ensuring every element of $P$ is verified throughout the protocol's execution.

5. Axiom of Non-collision:

   • Equation: Unique generation of $X^R$ and $x_i^R$, preventing collisions.

   • Expression: Guarantees that the generation process for $X^R$ and its subsets $x_i^R$ produces unique configurations, ensuring the integrity of the verification process by avoiding identical shuffles or subsets across different rounds.

Lemmas

1. Lemma of Witness Validity:

   • Equation: $Ω(p_i) \rightarrow x_i^R \land Μ(p_i, x_i^R) = \text{true}$ .

   • Expression: States that if the witness function Ω accurately identifies the subset $x_i^R$ for an element $p_i$, and the verification $Μ$ confirms $p_i$'s presence in $x_i^R$, then $p_i$'s verification for that round is deemed valid.

2. Lemma of Comprehensive Verification:

   • Equation: $\bigwedge_{R=1}^{n} Μ(p_i, x_i^R) \Rightarrow P \text{ authenticated}$ .

   • Expression: Indicates that the sequence $P$ is fully authenticated against the shuffled alphabet $X^R$ if, for all rounds R, the verification condition $Μ$ for each $p_i$ within its respective subset $x_i^R$ holds true.

3. Lemma of Accumulative Proof:

   • Equation: $Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R) \Rightarrow Κ = \text{true}$.

   • Expression: Explains that the final proof of knowledge $Κ$ is validated if the accumulator $Λ$, which aggregates all verification outcomes $Μ$ across rounds $R$, is true. This encapsulates the protocol's integrity by affirming the sequence $P$'s authentication through cumulative verification success.

4. Lemma of Dynamic Adaptability:

   • Equation: Adaptation to varying $|P|$ and $|A|$ without loss of $Π$ integrity.

   • Expression: Asserts that the protocol $Π$ can flexibly adjust to different sizes of the sequence $P$ and alphabet $A$ without compromising its verification integrity or security, demonstrating $Π$'s scalability and adaptability.

5. Lemma of Security Enhancement:

   • Equation: Enhanced security through unpredictability, $Σ(A) \rightarrow X^R$.

   • Expression: Highlights that security against cryptographic and brute-force threats is significantly enhanced by the unpredictability factor introduced through the shuffling function $Σ$, creating a dynamic and secure verification environment.

Constraints with Expressive Statements

1. Constraint of Round Completeness:

   • Equation: $\forall R, Μ(p_i, x_i^R) \text{ must complete}$.

   • Expression: This constraint mandates that in every round $R$, the verification process $Μ$ for each element $p_i$ within its designated subset $x_i^R$ must be fully executed, ensuring no part of the verification cycle is left incomplete.

2. Constraint of Subset Uniqueness:

   • Equation: $x_i^R \neq x_j^{R'}$ for $R \neq R'$ or $i \neq j$.

   • Expression: To maintain the integrity of the verification process, each subset $x_i^R$ generated for a round $R$ must be unique. This prevents any potential overlap or repetition of subsets across different rounds, reinforcing the security and robustness of $Π$.

3. Constraint of Proof Consistency:

   • Equation: $Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R) \Rightarrow Κ = \text{true}$.

   • Expression: The validation of the proof of knowledge $Κ$ hinges on the consistent truth of all verification outcomes $Μ$ aggregated in $Λ$. This ensures that $Κ$ is declared true only if every element $p_i$ of $P$ is successfully verified across all rounds $R$.

4. Constraint of Witness Integrity:

   • Equation: $Ω(p_i) \text{ must be verifiable against } P$.

   • Expression: The witness $Ω$, indicating where an element $p_i$ should be found within $X^R$, must be reliably linked to the prover's knowledge of the sequence $P$. This guards against misleading or incorrect indications that could compromise the verification integrity.

5. Constraint of Verification Transparency:

   • Equation: $Μ(p_i, x_i^R) \land Κ \text{ must be externally verifiable}$.

   • Expression: The process underscores the necessity for both the verification outcomes $Μ$ and the final proof $Κ$ to be transparent and amenable to external verification. This openness fosters trust and verifiability in the authentication process implemented by $Π$.

Principles with Expressive Statements

• Principle of Sequential Integrity:

  • Equation: Orderly $Μ(p_i, x_i^R)$ preserves $P$ integrity.

  • Expression: The orderly execution of verification $Μ$ for elements within $P$, following the sequence integrity, ensures the robustness of the authentication process, guaranteeing that each step follows logically from the previous one without breaches in logical continuity.

• Principle of Protocol Security:

  • Equation: $Σ(A) \land \bigwedge_{R=1}^{n} Μ(p_i, x_i^R) \land Λ \Rightarrow \text{secure } Π$.

  • Expression: The security of the protocol $Π$ is reinforced through the combination of unpredictable shuffling ($Σ$), thorough verification ($Μ$) across all rounds, and the cumulative confirmation ($Λ$) of these verifications, creating a robust defense against unauthorized access and manipulation.

• Principle of Verifiability:

  • Equation: External verification of $Μ \land Κ$.

  • Expression: Emphasizes the protocol's capacity for its verification steps and the final proof to be validated by third parties, enhancing the overall credibility and trustworthiness of $Π$.

• Principle of Non-repudiation:

  • Equation: $Λ$ documents verification, preventing denial.

  • Expression: The documentation and aggregation of verification results in $Λ$ serve as a solid foundation for non-repudiation, ensuring that once an authentication claim is made, it cannot be denied or disputed.

Systemic Implications with Expressive Statements

• Implication of Continuity:

  • Equation: $Π$ mechanism for re-verification or secure termination after failure.

  • Expression: This ensures that $Π$ includes mechanisms to either allow for the re-verification of elements upon failure or to securely terminate the session, safeguarding the integrity of the process and preventing potential security breaches.

• Implication of Evolution:

  • Equation: $Π$ updates to cryptographic methods without negating past validations.

  • Expression: $Π$ is designed to be future-proof, permitting updates and enhancements to its cryptographic methodologies without invalidating previously authenticated sequences. This adaptability ensures that $Π$ remains relevant and secure in the face of evolving cryptographic landscapes.

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Rosario-Wang Proof Protocol

Initialization of the Protocol (Π)

1. Protocol Initialization:

   • Let $Π$ denote the entire proof of knowledge protocol.

   • $A$ represents the static alphabet from which sequences are generated.

   • $P = {p_1, p_2, \ldots, p_n}$ is the sequence to be authenticated, with $p_i$ being the $i$-th element of $P$.

Preparation of the Alphabet and Shuffled Alphabet

2. Alphabet Preparation and Shuffling:

   • $X^R$ represents the shuffled alphabet derived from $A$ for round $R$, where $R = 1, 2, \ldots, n$.

   • The shuffling process per round is defined by $Σ(A) \rightarrow X^R$, ensuring each $X^R$ is a unique permutation of $A$.

Generation of the Sequence and its Verification

3. Sequence Generation and Subset Selection:

   • For each round $R$, a subset $x_i^R \subseteq X^R$ is targeted for verification. Here, $i$ corresponds to the targeted index within $P$ for that round.

   • The selection of $x_i^R$ for a given $p_i \in P$ is guided by a witness $Ω$, which indicates the appropriate subset $x_i^R$ where $p_i$ should be found.

Verification Process

4. Verification and Witness:

   • The verification condition for round $R$ is denoted as $Μ(p_i, x_i^R)$, checking if $p_i$ is present within $x_i^R$.

   • The witness $Ω(p_i) \rightarrow x_i^R$ links $p_i$ to its corresponding subset $x_i^R$ for verification.

Result Accumulation and Proof of Knowledge

5. Result Accumulation and Conclusion:

   • The accumulation of verification results across all rounds $R$ is captured by $Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R)$, where $Λ$ is true iff all instances of $Μ$ are true.

   • The final proof of knowledge, $Κ$, is validated iff $Λ$ is true, denoted mathematically as $Κ \Leftrightarrow Λ$.

Notational Summary

• $Π$: The multi-round proof of knowledge ceremony.

• $A$: The static alphabet.

• $P$: The sequence to be authenticated.

• $X^R$: The shuffled alphabet for round $R$.

• $x_i^R$: The subset of $X^R$ targeted in round $R$.

• $Σ$: The random distribution function that generates $X^R$ from $A$.

• $Ω$: The witness function that indicates the subset $x_i^R$ for verification of $p_i$.

• $Μ$: The verification condition for a member $p_i$ in subset $x_i^R$.

• $Λ$: The accumulator of verification results across rounds.

• $Κ$: The final proof of knowledge, affirming the authenticity of $P$.

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1. Protocol and Sequence Declaration:

   • $Π$: Proof of knowledge protocol.

   • $P = {p_1, p_2, \ldots, p_n}$: Sequence to be authenticated.

2. Alphabet and Shuffling:

   • $A$: Static alphabet.

   • $X^R = Σ(A)$: Shuffled alphabet $X$ for round $R$, obtained by applying the shuffling function $Σ$ to A.

3. Subset Selection and Verification:

   • $x_i^R$ : Subset of $X^R$ targeted in round $R$ for verifying element $p_i$.

   • $Ω(p_i) \rightarrow x_i^R$: Witness function indicating the subset $x_i^R$ where $p_i$ is expected to be found for verification.

4. Verification Condition:

   • $Μ(p_i, x_i^R)$: Verification condition for round $R$, checking if $p_i$ is present within $x_i^R$.

5. Result Accumulation and Proof of Knowledge:

   • $Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R)$: Accumulator of verification results across all rounds $R$, where $Λ$ is true if and only if all instances of $Μ$ are true.

   • $Κ \Leftrightarrow Λ$: The final proof of knowledge $Κ$ is validated if and only if $Λ$ is true.

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Mathematical Formulation

In the context of the multi-round proof of knowledge ceremony ($Π$), operating over a sequence $P$ derived from a static alphabet $A$, we formalize the operations, verification, and the final synthesis of proof through a detailed mathematical exposition, ensuring clarity and alignment with foundational principles.

Protocol Operation and Verification Rounds

1. Initialization and Shuffling:

   • Given: $P = {p_1, p_2, \ldots, p_n}$ and $A$ as inputs.

   • Operation: For each verification round $R$, apply $Σ$ to $A$ to yield $X^R$.

   Meaning: $Σ$ denotes the shuffling function, transforming $A$ into a uniquely shuffled set $X^R$ for each round, enhancing unpredictability and security.

2. Witness Function and Verification: - Process: $Ω$ determines a target subset $x_i^R \subseteq X^R$ for each $p_i$.

Verification: Assess $p_i$'s presence within $x_i^R$, denoted by $Μ$.

$$Μ(p_i, x_i^R) = \text{true} \iff p_i \in x_i^R$$

Implication: This step authenticates each $p_i$ against its assigned subset, validating authenticity per round.

Accumulation of Verification Results

3. Results Accumulation and Proof Validation:

   • Accumulation: Compile outcomes of $Μ$ across all rounds into $Λ$.

$$\Lambda = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R)$$

Final Proof: Validate $Κ$ based on the collective truth of $Λ$.

$$Κ \Leftrightarrow Λ$$

Interpretation: $Λ$ embodies the cumulative verification success, with $Κ$'s validity contingent upon unanimous positive verifications, affirming $P$'s authentication within the dynamic context of $Π$.

Proof of Accumulation Efficacy

Theorem: The effective aggregation of verification results ($Λ$) precisely reflects the comprehensive authentication of sequence $P$ across all rounds ($R$), encapsulated by $Π$.

Forward Assertion: If $Λ$ is true, implying the aggregate of $Μ$ over $R$ is uniformly positive, then each $p_i$ is verified within the correct $x_i^R$, thus:

$$Λ = \text{true} \implies \forall p_i \in P, \, Μ(p_i, x_i^R) = \text{true}, \, \forall R$$

Backward Assertion: Conversely, if each $p_i$ is successfully authenticated within its designated subset $x_i^R$ for all $R$, then $Λ$ must be true, encapsulating the protocol’s integrity:

$$\bigwedge_{R=1}^{n} Μ(p_i, x_i^R) = \text{true}, \forall i \implies Λ = \text{true}$$

Conclusion: This delineation affirms that $Λ$, as an accumulation of $Μ$ across rounds, serves as a robust metric for the authentication of $P$, with $Κ$ as the conclusive proof of knowledge, underscoring $Π$'s efficacy in sequence verification within a dynamically secure framework.

To ensure alignment with our established lemmas, axioms, constraints, and systemic framework, we refine the proof of accumulation efficacy to mirror the intricacies and specifications of our system $Π$. This revised proof elucidates the critical role of $Λ$ in confirming the authentication of the sequence $P$ throughout all verification rounds $R$, in accordance with the operational principles and verification logic of $Π$.

Extended Proof of Accumulation Efficacy

Theorem: The accumulator $Λ$, through the effective aggregation of verification results, accurately represents the thorough authentication of the sequence $P$ across every round $R$ within the protocol $Π$.

Refined Assertions

Forward Assertion: Assuming $Λ$ holds true, indicating a universal affirmation of the verification condition $Μ$ across all rounds $R$, it logically follows that every element $p_i$ of $P$ has been validated within its respective subset $x_i^R$. This assertion can be formally captured as:

$$Λ = \text{true} \implies \forall p_i \in P, \, \forall R, \, Μ(p_i, x_i^R) = \text{true}$$

This implies that the integrity of $Λ$ as true necessitates the successful verification of every $p_i$ within its designated $x_i^R$ across all rounds, ensuring the completeness and correctness of the sequence $P$ authentication.

Backward Assertion: If, for each round $R$, every $p_i$ is affirmatively verified within its intended subset $x_i^R$, thereby fulfilling the verification condition $Μ$, then the cumulative verification result $Λ$ must inherently be true. This logical proposition can be succinctly expressed as:

$$\forall i, \, \forall R, \, Μ(p_i, x_i^R) = \text{true} \implies Λ = \text{true}$$

The sufficiency condition mandates that the aggregate verification success of all $p_i$ in their corresponding $x_i^R$ for every $R$ compels the truth of $Λ$, encapsulating the protocol’s verification integrity and the sequential authentication's authenticity.

Conclusion

By analytically delineating both the forward and backward assertions, we solidify the theorem's validity, demonstrating that the truth value of $Λ$—as the logical conjunction of all individual verification outcomes $Μ(p_i, x_i^R)$—is both necessary and sufficient for affirming the comprehensive authentication of $P$ within the dynamic verification framework of $Π$. This refined proof underscores $Π$'s robust verification mechanism, ensuring $P$'s integrity and validating $Κ$ as the definitive proof of knowledge. Through this elaboration, $Π$’s efficacy in securely authenticating sequences within a dynamically secure and algorithmically precise environment is irrefutably established, adhering to the rigorous standards set forth by our system's lemmas, axioms, and constraints.

Table of Functions and Operators

Following key of operators

Function/Operator	Symbol/Notation	Description
Shuffled	$Σ(A) \rightarrow X^R$	Function to generate a shuffled alphabet $X^R$ from $A$.
Indicating	$Ω(p_i) \rightarrow x_i^R$	Function indicating the subset $x_i^R$ where $p_i$ is expected.
Verifying	$Μ(p_i, x_i^R)$	Verification condition for the presence of $p_i$ in $x_i^R$.
Authenticated	$Κ \Leftrightarrow Λ$	States $Κ$ is validated iff $Λ$ is true.
Expected	$Ω(p_i)$	Indicates the expected subset $x_i^R$ for $p_i$.
For Round	$X^R$, $x_i^R$	Specifies that $X^R$ and $x_i^R$ are for round $R$.
Present Within	$Μ(p_i, x_i^R)$	Checks if $p_i$ is present within subset $x_i^R$.
Accumulator	$Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R)$	Accumulates verification results across all rounds.
Across All Rounds	$\bigwedge_{R=1}^{n}$	Logical AND operation across all rounds $R$.
Is True If and Only If	$k\Leftrightarrow Λ$	Logical equivalence, used to relate $Κ$ and $Λ$.
Validated If	$Κ \Leftrightarrow Λ$	States the condition under which $Κ$ is considered validated.

List of Equations

1. Shuffling Function:

   • $Σ(A) \rightarrow X^R$

2. Indicating Function:

   • $Ω(p_i) \rightarrow x_i^R$

3. Verification Condition:

   • $Μ(p_i, x_i^R)$

4. Authentication Relation:

   • $Κ \Leftrightarrow Λ$

5. Accumulation of Results:

   • $Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R)$