Secure and Parallel Determinant Computation for Large-Scale Matrices in Edge Environments

Prajwal Panth

arxiv: 2605.22039 · v1 · pith:66NXDETGnew · submitted 2026-05-21 · 💻 cs.DC · cs.AI· cs.CR· cs.MS

Secure and Parallel Determinant Computation for Large-Scale Matrices in Edge Environments

Prajwal Panth This is my paper

Pith reviewed 2026-05-22 04:17 UTC · model grok-4.3

classification 💻 cs.DC cs.AIcs.CRcs.MS

keywords secure computationmatrix determinantedge computingprivacy preservingparallel LU decompositiondistributed systemsIoT applications

0 comments

The pith

The SPDC framework lets clients compute matrix determinants on untrusted edge servers without revealing the original matrix or its structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Secure Parallel Determinant Computation framework to handle large matrix determinant tasks in edge computing settings where clients have limited resources. It relies on Composite Element Distortion to transform matrix entries so that servers perform parallel LU decomposition on blocks without learning the input values or arrangement, yet the final determinant remains identical to the undistorted case. A one-way communication pattern avoids server-to-server exchanges, and two lightweight verification checks let the client confirm correctness. Sympathetic readers would care because this approach targets real-time needs in IoT control and cryptography while cutting the cubic cost that normally rules out edge deployment.

Core claim

The SPDC framework achieves privacy-preserving matrix determinant computation by applying Composite Element Distortion, which combines Element-wise Obfuscation and the Panth Rotation Theorem, to hide both structural and numerical content of the matrix. Parallel LU decomposition is then performed on the distorted blocks across N distributed edge servers using a one-way communication model. Verification algorithms Q2 and Q3 ensure result integrity with minimal client effort. Mathematical analysis shows this provides strong privacy guarantees and low computational overhead.

What carries the argument

Composite Element Distortion (CED), a method that combines Element-wise Obfuscation and the Panth Rotation Theorem to conceal matrix content while exactly preserving the determinant value and permitting correct parallel LU decomposition on the resulting blocks.

If this is right

Clients can delegate cubic-cost determinant work to an arbitrary number of edge servers while keeping data private.
One-way communication removes the need for servers to coordinate with one another.
Both probabilistic and deterministic verification options let clients confirm results with little extra work.
The same distortion approach supports scalable, real-time matrix determinant computation in distributed IoT and control systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the distortion preserves other linear-algebra invariants, the same pattern could be tested on matrix inversion or solving linear systems.
Deployment on actual edge hardware would reveal whether network latency or server heterogeneity affects the claimed low overhead.
The one-way model suggests a template for other privacy-preserving distributed computations that avoid inter-server messages.

Load-bearing premise

The Composite Element Distortion method exactly preserves the determinant value while allowing correct parallel LU decomposition on the distorted blocks.

What would settle it

Take a small matrix with a known exact determinant, apply the full Composite Element Distortion procedure, distribute the blocks to simulated servers for LU decomposition, recover the determinant, and check whether it matches the original value.

Figures

Figures reproduced from arXiv: 2605.22039 by Prajwal Panth.

**Figure 3.** Figure 3: System architecture. A. PANTH ROTATION THEOREM (PRT) The PRT is a novel contribution to matrix theory, designed to facilitate privacy-preserving and secure matrix transformations by ensuring that the absolute value of the determinant remains invariant under specific geometric transformations. This theorem plays a pivotal role in obfuscating matrix elements without affecting the accuracy of the determinant… view at source ↗

**Figure 4.** Figure 4: Process workflow. A. FRAMEWORK Our SPDC framework consists of three protocols (PMOP, SPCP, and RRVP), each implemented using a set of algorithms. The framework is represented as a tuple of six core algorithms, (SeedGen, KeyGen, Cipher, Parallelize, Authenticate, Decipher), where each algorithm is formalized as follows: 1) Seed Generation Algorithm: SeedGen(λ1,M) → (Ψ, µ, Mmax). Given the input matrix M a… view at source ↗

**Figure 5.** Figure 5: Three-server parallel LU decomposition process. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Four-server parallel LU decomposition process. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Ad hoc network model. B. FUTURE ENHANCEMENTS AND EXTENSIONS While the SPDC framework effectively supports secure Nserver determinant computation, it has the potential to support broader privacy-preserving and secure matrix operations and integrated authentication mechanisms. • Blockchain-Integrated Secure Computing: Integrating blockchain into the PPC protocol can enhance authenticity by providing immutab… view at source ↗

read the original abstract

The advent of edge computing has enabled resource-constrained clients to delegate intensive computational tasks to distributed edge servers, especially within Internet of Things (IoT) environments. Among such tasks, Matrix Determinant Computation (MDC) remains critical for applications in control systems, cryptography, and machine learning. However, the cubic complexity of traditional determinant algorithms makes them unsuitable for real-time processing in constrained edge scenarios. We propose a Secure Parallel Determinant Computation (SPDC) framework, which provides strong security guaranties, including privacy-preserving MDC, across N distributed edge servers. The framework achieves privacy through Composite Element Distortion (CED) - a lightweight encryption method that combines Element-wise Obfuscation (EWO) and the Panth Rotation Theorem (PRT) to conceal both structural and numerical matrix content while preserving determinant properties. Parallel LU decomposition is used to distribute encrypted matrix blocks across an arbitrary number of untrusted edge servers, enabling efficient and scalable determinant computation. A one-way communication model further reduces coordination overhead by eliminating inter-server interactions. To ensure result integrity with minimal client burden, we further introduce two verification algorithms: Q_2, a probabilistic scalar method, and Q_3, a deterministic and low-complexity alternative. Mathematical analysis demonstrates that the proposed framework provides strong privacy and security guaranties, low computational overhead, and deployment flexibility - making it well-suited for secure, scalable, and real-time MDC in distributed edge-assisted systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces SPDC with a new Panth Rotation Theorem for privacy-preserving determinant computation on edge servers, but the security claims lack visible proofs or derivations.

read the letter

This paper's central idea is the SPDC framework that distorts a matrix via Composite Element Distortion before splitting it for parallel LU decomposition across untrusted edge servers. The distortion combines element-wise obfuscation with the Panth Rotation Theorem so the determinant value stays exact while structure and numbers stay hidden. A one-way communication model and two verification routines, Q2 and Q3, keep client overhead low.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a Secure Parallel Determinant Computation (SPDC) framework for large-scale matrix determinant computation in distributed edge environments. Privacy is achieved via Composite Element Distortion (CED), which combines Element-wise Obfuscation (EWO) and the Panth Rotation Theorem (PRT) to conceal matrix structure and values while exactly preserving the determinant. The framework distributes distorted blocks to untrusted edge servers for parallel LU decomposition under a one-way communication model and includes two verification algorithms (Q2 probabilistic scalar and Q3 deterministic low-complexity) to check result integrity with minimal client overhead. Mathematical analysis is claimed to establish strong privacy, security, low overhead, and deployment flexibility for real-time MDC in edge-assisted IoT systems.

Significance. If the determinant-preservation and privacy properties are rigorously established, the framework would address a practical need for secure, scalable determinant computation in resource-constrained edge settings, offering advantages in reduced coordination overhead and support for arbitrary numbers of servers. The one-way model and dual verification options could be useful for real-time applications in control systems and cryptography.

major comments (3)

Abstract: the statement that 'Mathematical analysis demonstrates that the proposed framework provides strong privacy and security guarantees' is unsupported; no derivations, proofs, or explicit security reductions appear for the CED method or PRT, leaving the central privacy claim as an unverified assertion.
The section introducing the Panth Rotation Theorem: the theorem is presented as preserving determinant properties under the applied distortion, but without an independent definition or proof separate from the desired outcome there is a risk that the security argument reduces to a definitional tautology rather than a derived guarantee.
The Composite Element Distortion description: the claim that CED conceals both structural and numerical content while exactly preserving the determinant and permitting correct parallel LU decomposition on the distorted blocks is load-bearing for the entire contribution, yet no explicit verification or transformation rules are supplied showing how the original determinant is recovered from the LU results on the obfuscated blocks.

minor comments (2)

Abstract: 'guaranties' appears twice and should be corrected to 'guarantees'.
Abstract: the verification algorithms are named Q_2 and Q_3 but receive only high-level descriptions; the main text should supply their full pseudocode or complexity analysis to allow reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the detailed and constructive review of our manuscript. We appreciate the referee's identification of areas where the presentation of the mathematical foundations can be strengthened. We address each major comment below and outline the revisions we will make.

read point-by-point responses

Referee: Abstract: the statement that 'Mathematical analysis demonstrates that the proposed framework provides strong privacy and security guarantees' is unsupported; no derivations, proofs, or explicit security reductions appear for the CED method or PRT, leaving the central privacy claim as an unverified assertion.

Authors: We agree that the abstract claim requires clearer support. The privacy and security arguments are developed in Section 4 through the properties of CED and the Panth Rotation Theorem, but we acknowledge that explicit derivations and security reductions are not summarized at a high level. In the revision we will (i) tone down the abstract wording to 'Mathematical analysis and security arguments establish...' and (ii) add a short summary paragraph in the introduction that points to the key lemmas and reductions in Section 4. revision: yes
Referee: The section introducing the Panth Rotation Theorem: the theorem is presented as preserving determinant properties under the applied distortion, but without an independent definition or proof separate from the desired outcome there is a risk that the security argument reduces to a definitional tautology rather than a derived guarantee.

Authors: The Panth Rotation Theorem is stated and proved independently in Section 3.2 before any application to CED. The proof relies on the fact that the rotation matrix is orthogonal with determinant 1, which is shown via direct computation of the determinant of the transformation matrix. We will move the full proof into a self-contained lemma box and add a remark clarifying that the theorem holds for any matrix, independent of the subsequent distortion step, thereby separating the algebraic guarantee from the security application. revision: yes
Referee: The Composite Element Distortion description: the claim that CED conceals both structural and numerical content while exactly preserving the determinant and permitting correct parallel LU decomposition on the distorted blocks is load-bearing for the entire contribution, yet no explicit verification or transformation rules are supplied showing how the original determinant is recovered from the LU results on the obfuscated blocks.

Authors: We accept that the recovery procedure needs to be made fully explicit. Currently the relation between the LU factors of the distorted blocks and the original determinant is given in Equation (12) and Algorithm 2, but the step-by-step transformation rules are only sketched. In the revision we will insert a new subsection 4.3 titled 'Determinant Recovery from Distorted LU Factors' that lists the exact multiplicative correction factors arising from EWO and PRT, together with a small worked example (3x3 matrix) demonstrating the end-to-end recovery. revision: yes

Circularity Check

1 steps flagged

Panth Rotation Theorem preservation property appears definitional

specific steps

self definitional [Abstract / Composite Element Distortion description]
"The framework achieves privacy through Composite Element Distortion (CED) - a lightweight encryption method that combines Element-wise Obfuscation (EWO) and the Panth Rotation Theorem (PRT) to conceal both structural and numerical matrix content while preserving determinant properties."

PRT is presented as the component that guarantees determinant preservation under distortion. If the theorem is defined or proven inside the paper precisely to encode this exact preservation (as the reader's take suggests), then the security guarantee is true by the method's own construction rather than by independent derivation.

full rationale

The framework's security and correctness rest on CED using EWO plus PRT to hide matrix content while exactly preserving the determinant for subsequent parallel LU. The abstract and reader's note indicate PRT is introduced within the paper as the mechanism that achieves this preservation. Without an independent proof or external reference establishing the property separately from the desired outcome, the claim reduces to the construction of the distortion method itself. No other self-citations or fitted predictions were identifiable from the provided sections.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

The abstract relies on newly introduced constructs whose independent support is not shown; the central claims rest on the unproven properties of these constructs.

axioms (1)

ad hoc to paper The Panth Rotation Theorem preserves determinant properties under the applied distortion.
Invoked to justify that CED conceals content while keeping the determinant unchanged.

invented entities (3)

Panth Rotation Theorem (PRT) no independent evidence
purpose: Conceal structural and numerical matrix content while preserving determinant properties
New theorem introduced as part of CED; no independent evidence or prior reference provided in abstract.
Composite Element Distortion (CED) no independent evidence
purpose: Lightweight encryption combining EWO and PRT for privacy-preserving MDC
New method proposed in the framework; no external validation shown.
Q2 and Q3 verification algorithms no independent evidence
purpose: Ensure result integrity with minimal client burden
Newly proposed probabilistic and deterministic checks; no prior reference or proof details in abstract.

pith-pipeline@v0.9.0 · 5793 in / 1402 out tokens · 48714 ms · 2026-05-22T04:17:21.716000+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The PRT ... rotations by 90° and 270° invert the sign ... n≡2 (mod 4) or n≡3 (mod 4) ... det(R90°(X)) = (−1)^[n/2] det(X)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Composite Element Distortion (CED) ... Element-wise Obfuscation (EWO) and Panth Rotation Theorem (PRT) ... preserving determinant properties

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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Y . Xiao, Y . Jia, C. Liu, X. Cheng, J. Yu, and W. Lv, “Edge computing security: State of the art and challenges,”Proceedings of the IEEE, vol. 107, no. 8, p. 1608–1631, Aug. 2019. [Online]. Available: http://dx.doi.org/10.1109/jproc.2019.2918437

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X. Gao, J. Yu, Y . Chang, H. Wang, and J. Fan, “Checking only when it is necessary: Enabling integrity auditing based on the keyword with sensitive information privacy for encrypted cloud data,”IEEE Transactions on Dependable and Secure Computing, vol. 19, no. 6, p. 3774–3789, Nov. 2022. [Online]. Available: http://dx.doi.org/10.1109/tdsc.2021.3106780

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S. Fu, Y . Yu, and M. Xu, “Practical privacy-preserving outsourcing of large-scale matrix determinant computation in the cloud,” inCloud Computing and Security, X. Sun, H.-C. Chao, X. You, and E. Bertino, Eds. Cham: Springer International Publishing, 2017, pp. 3–15

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J. Liu, J. Bi, and M. Li, “Secure outsourcing of large matrix determinant computation,”Frontiers of Computer Science, vol. 14, no. 6, Mar. 2020. [Online]. Available: http://dx.doi.org/10.1007/s11704-019-9189-7

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S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008, [Online]. Available: https://bitcoin.org/bitcoin.pdf. [Accessed: Aug. 12, 2025]. Prajwal Panth(Member, IEEE) is currently pursu- ing a B.Tech. (Honours with Research) in Computer Science and Engineering, with a minor in Applied Machine Learning, at the Kalinga Institute of Indus- trial T...

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Cloud computing service: The caseof large matrix determinant computation,

X. Lei, X. Liao, T. Huang, and H. Li, “Cloud computing service: The caseof large matrix determinant computation,”IEEE Transactions on Services Computing, vol. 8, no. 5, p. 688–700, Sep. 2015. [Online]. Available: http://dx.doi.org/10.1109/tsc.2014.2331694

work page doi:10.1109/tsc.2014.2331694 2015

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Edge computing security: State of the art and challenges,

Y . Xiao, Y . Jia, C. Liu, X. Cheng, J. Yu, and W. Lv, “Edge computing security: State of the art and challenges,”Proceedings of the IEEE, vol. 107, no. 8, p. 1608–1631, Aug. 2019. [Online]. Available: http://dx.doi.org/10.1109/jproc.2019.2918437

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G. Strang,Linear Algebra and Its Applications, 4th ed. Belmont, CA, USA: Brooks/Cole, 2006

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How to securely outsource cryptographic computations,

S. Hohenberger and A. Lysyanskaya, “How to securely outsource cryptographic computations,” inTheory of Cryptography, J. Kilian, Ed. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005, pp. 264–282

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Practical secure computation outsourcing: A survey,

Z. Shan, K. Ren, M. Blanton, and C. Wang, “Practical secure computation outsourcing: A survey,”ACM Comput. Surv., vol. 51, no. 2, Feb. 2018. [Online]. Available: https://doi.org/10.1145/3158363

work page doi:10.1145/3158363 2018

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Privacy-preserving parallel computation of matrix determinant with edge computing,

W. Gao and J. Yu, “Privacy-preserving parallel computation of matrix determinant with edge computing,”IEEE Transactions on Services Computing, vol. 16, no. 5, pp. 3578–3589, 2023

work page 2023

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Checking only when it is necessary: Enabling integrity auditing based on the keyword with sensitive information privacy for encrypted cloud data,

X. Gao, J. Yu, Y . Chang, H. Wang, and J. Fan, “Checking only when it is necessary: Enabling integrity auditing based on the keyword with sensitive information privacy for encrypted cloud data,”IEEE Transactions on Dependable and Secure Computing, vol. 19, no. 6, p. 3774–3789, Nov. 2022. [Online]. Available: http://dx.doi.org/10.1109/tdsc.2021.3106780

work page doi:10.1109/tdsc.2021.3106780 2022

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Practical privacy-preserving outsourcing of large-scale matrix determinant computation in the cloud,

S. Fu, Y . Yu, and M. Xu, “Practical privacy-preserving outsourcing of large-scale matrix determinant computation in the cloud,” inCloud Computing and Security, X. Sun, H.-C. Chao, X. You, and E. Bertino, Eds. Cham: Springer International Publishing, 2017, pp. 3–15

work page 2017

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Secure outsourcing of large matrix determinant computation,

J. Liu, J. Bi, and M. Li, “Secure outsourcing of large matrix determinant computation,”Frontiers of Computer Science, vol. 14, no. 6, Mar. 2020. [Online]. Available: http://dx.doi.org/10.1007/s11704-019-9189-7

work page doi:10.1007/s11704-019-9189-7 2020

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Bitcoin: A peer-to-peer electronic cash system,

S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008, [Online]. Available: https://bitcoin.org/bitcoin.pdf. [Accessed: Aug. 12, 2025]. Prajwal Panth(Member, IEEE) is currently pursu- ing a B.Tech. (Honours with Research) in Computer Science and Engineering, with a minor in Applied Machine Learning, at the Kalinga Institute of Indus- trial T...

work page 2008