Rosario-Wang Axioms / Lemmas

Rosario–Wang Proof System: Axiomatic Cryptographic Protocol for Entangled Knowledge Authentication

Overview

This paper introduces the Rosario–Wang Protocol, a formalized cryptographic proof system grounded in a novel entanglement model of information. It leverages probabilistic shuffling functions, subset mappings, and cumulative verification logic to construct a zero-knowledge, round-based commitment framework for validating symbolic sequences. Rooted in the principles of entropy, randomness, and predicate integrity, the protocol achieves authentication without revealing the committed content, making it suitable for cognitive login systems, carrier-neutral security, and post-quantum cryptographic environments. The system is mathematically defined through a set of foundational axioms, lemmas, constraints, and principles that collectively ensure its completeness, adaptability, and cryptographic soundness.

1. Introduction

Contemporary authentication systems, while varied in implementation, rely on static secrets or hardware-based identifiers that are increasingly vulnerable to phishing, device failure, or quantum decryption attacks. To address this, we present the Rosario–Wang Proof System (denoted Π), a structured verification mechanism that operates entirely within the cognitive and symbolic domain. It replaces traditional cryptographic keys with entangled commitments over shuffled alphabets, utilizing per-round non-repeating challenges and witness verifiability.

2. Theoretical Framework

2.1 Axioms

The Rosario–Wang system is established on the following axiomatic definitions:

• Axiom of Initialization Let $P = {p_1, p_2, …, p_n}$ be the sequence to be authenticated from a static alphabet $A$. For each verification round $R$, a shuffled alphabet $X^R = \Sigma(A)$ is generated, introducing non-deterministic variability.

• Axiom of Random Distribution $X^R \neq X^{R’}$ for all $R \neq R’$, ensuring fresh challenge landscapes per round.

• Axiom of Sequence Verification For each $p_i \in P$, there exists a subset $x_i^R \subset X^R$ such that $p_i \in x_i^R$, enabling targeted verification.

• Axiom of Completeness $\forall p_i \in P, \exists x_i^R : M(p_i, x_i^R) = \text{true}$, guaranteeing full verification across rounds.

• Axiom of Non-Collision $x_i^R \neq x_j^{R’}$ for $i \neq j$ or $R \neq R’$, eliminating subset repetition and reinforcing uniqueness.

2.2 Lemmas

• Lemma of Witness Validity A valid witness function $\Omega(p_i) \rightarrow x_i^R$ must be coupled with successful verification $M(p_i, x_i^R) = \text{true}$ to constitute proof for that round.

• Lemma of Comprehensive Verification If $M(p_i, x_i^R)$ holds for all $i$ and $R$, then $P$ is authenticated:

  \[\bigwedge_{R=1}^{n} M(p_i, x_i^R) \Rightarrow P \text{ authenticated}\]

• Lemma of Accumulative Proof Define accumulator $\Lambda = \bigwedge_{R=1}^{n} M(p_i, x_i^R)$. The final proof $K$ is valid iff $\Lambda = \text{true}$:

  \[\Lambda \Rightarrow K = \text{true}\]

• Lemma of Dynamic Adaptability The protocol Π remains intact under any variation in $|P|$ and $|A|$.

• Lemma of Security Enhancement The security properties of Π are directly augmented by the entropy introduced via $\Sigma$, yielding $\Sigma(A) \rightarrow X^R$.

3. Protocol Operation

3.1 Initialization

Let $A$ be the fixed alphabet, and $P = {p_1, …, p_n}$ the input sequence to be authenticated.

3.2 Round-wise Alphabet Shuffling

For each round $R$, a unique shuffled alphabet $X^R = \Sigma(A)$ is computed. This non-repeating shuffling prevents precomputation or replay attacks.

3.3 Subset Verification

Each $p_i$ maps via witness $\Omega$ to a subset $x_i^R \subset X^R$, with verification predicate $M(p_i, x_i^R) \rightarrow {\text{true, false}}$.

3.4 Accumulation and Final Proof

Cumulative success across all $R$ is aggregated in accumulator $\Lambda$. Proof of knowledge $K$ is established when $\Lambda = \text{true}$.

4. Formal Constraints

• Round Completeness:

  \[\forall R, M(p_i, x_i^R) \text{ must execute}\]

• Subset Uniqueness:

  \[x_i^R \neq x_j^{R'} \quad \text{for } i \neq j \text{ or } R \neq R'\]

• Proof Consistency:

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

• Witness Integrity:

  \[\Omega(p_i) \text{ must be verifiable against } P\]

• Transparency:

  \[M(p_i, x_i^R) \land K \text{ must be externally verifiable}\]

5. Principles

• Sequential Integrity: Orderly verification ensures structural integrity:

  \[M(p_i, x_i^R) \text{ follows } P\]

• Protocol Security:

  \[\Sigma(A) \land \bigwedge_{R} M(p_i, x_i^R) \Rightarrow \text{secure } Π\]

• Verifiability: Verification steps and $K$ must be externally attestable.

• Non-repudiation:

  \[\Lambda \text{ serves as proof record}\]

6. Implications

• Continuity:

  \[Π \text{ supports re-verification or secure termination}\]

• Evolution:

  \[Π \text{ can update cryptographic components without negating prior validity}\]

7. Conclusion

The Rosario–Wang Proof System introduces a foundational cryptographic primitive designed to authenticate symbolic sequences across rounds of entropy-laden transformations. Through a rigorously axiomatized structure, this system ensures completeness, adaptability, and post-quantum resistance. Its core mechanisms—alphabet shuffling, subset verification, and accumulative proof construction—offer a compelling alternative to static key paradigms. Future work will explore implementation models for zero-knowledge variants, physical-digital hybrids, and carrier-neutral symbol encodings.

Axioms

Axiom of Initialization: The protocol $Π$ commences with a predefined sequence $P$, comprised of elements $p_1$ to $p_n$, and a static alphabet $A$. For each verification round $R$, the shuffling function $Σ$ generates a fresh and unpredictable shuffled version of $A$, denoted as $X^R$. This axiom ensures that the protocol begins with a consistent sequence and introduces variability through shuffled alphabets in each round.

• 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.

Axiom of Random Distribution: Each round $R$ of the protocol $Π$ produces a distinct shuffled alphabet $X^R$ from the static alphabet $A$ via the shuffling function $Σ$. The axiom asserts that for differing rounds $R$ and $R’$, the shuffled alphabets $X^R$ and $X^{R’}$ are distinct ($X^R \neq X^{R’}$). This condition ensures the unpredictability and non-repetitive nature of the challenges encountered in successive rounds, vital for enhancing the security of the verification process.

• 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.

Axiom of Sequence Verification: For every element $p_i$ within the sequence $P$, there exists a designated subset $x_i^R$ within the shuffled alphabet $X^R$ where $p_i$ can be located and verified. This axiom emphasizes the protocol’s capability to identify and authenticate individual elements of the sequence, ensuring the integrity of the verification process.

• 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.

Axiom of Completeness: The axiom asserts that for all elements $p_i$ within the sequence $P$, there exists at least one subset $x_i^R$ within the shuffled alphabet $X^R$ where the verification condition $Μ$ returns true ($Μ(p_i, x_i^R) = \text{true}$). This ensures comprehensive verification of every element of $P$ throughout the execution of the protocol, reinforcing its reliability and 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.

Axiom of Non-collision: This axiom guarantees the uniqueness of the generated shuffled alphabets $X^R$ and their corresponding subsets $x_i^R$ across different rounds. It ensures that no collisions occur in the generation process, maintaining the integrity of the verification process by preventing identical shuffles or subsets from appearing in multiple rounds.

• 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

Lemma of Witness Validity: This lemma asserts that if the witness function $Ω$ correctly identifies the subset $x_i^R$ for an element $p_i$ and the subsequent verification $Μ$ confirms the presence of $p_i$ within $x_i^R$, then the verification of $p_i$ for that specific round is considered valid. Mathematically, it can be expressed as $Ω(p_i) \rightarrow x_i^R \land Μ(p_i, x_i^R) = \text{true}$, indicating the conjunction of $Ω$ and $Μ$ as criteria for valid verification.

• 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.

Lemma of Comprehensive Verification: This lemma signifies that the sequence $P$ achieves full authentication against the shuffled alphabet $X^R$ if, for each round $R$, the verification condition $Μ$ holds true for every $p_i$ within its designated subset $x_i^R$. Mathematically, it is represented as $\bigwedge_{R=1}^{n} Μ(p_i, x_i^R) \Rightarrow P \text{ authenticated}$, emphasizing the comprehensive nature of the verification process across all rounds.

• 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.

Lemma of Accumulative Proof: In this lemma, the final proof of knowledge $Κ$ is validated if the accumulator $Λ$, which aggregates all verification outcomes $Μ$ across rounds $R$, evaluates to true. This lemma encapsulates the protocol’s integrity by affirming the authentication of the sequence $P$ through cumulative verification success. Mathematically, it is denoted as $Λ = \bigwedge_{R=1}^{n} Μ(p_i, x_i^R) \Rightarrow Κ = \text{true}$.

• 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.

Lemma of Dynamic Adaptability: Asserting the protocol’s ability to adapt to varying sizes of the sequence $P$ and alphabet $A$ without compromising integrity, this lemma highlights the scalability and adaptability of $Π$. Mathematically, it states the adaptation without loss of $Π$ integrity, expressed as $

P

$ and $

A

$ varying without affecting the protocol’s integrity.

• 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.

Lemma of Security Enhancement: This lemma emphasizes the protocol’s enhanced security against cryptographic and brute-force threats due to the unpredictability introduced by the shuffling function $Σ$, creating a dynamic and secure verification environment. Mathematically, it states the security enhancement as $Σ(A) \rightarrow X^R$, underlining the crucial role of $Σ$ in bolstering security measures within the protocol.

• 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

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.

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 $Π$.

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 $R$.

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.

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

• 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.

Implications

• 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.

Rosario-Wang 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

1. 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

1. 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

1. 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

1. 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 $P$.

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.