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arxiv: 2605.15774 · v1 · pith:75SD45Y2new · submitted 2026-05-15 · 💻 cs.CR

Beyond Controlled Noise: Achieving Symmetric FHE through Dynamic Position Shifting

Mostefa Kara This is my paper

Pith reviewed 2026-05-20 17:44 UTC · model grok-4.3

classification 💻 cs.CR
keywords fully homomorphic encryptionsymmetric encryptionnoise managementplaintext fragmentationdynamic interpositionexponent regulatorscoefficient regulators
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The pith

Symmetric FHE is achieved by fragmenting plaintext and using regulators to shift fragment positions during multiplication.

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

The paper proposes a symmetric fully homomorphic encryption scheme built on the modular base c = mk + rp. It partitions the plaintext into fragments at distinct logical positions and introduces a dual-regulator system to manage multiplication. Exponent regulators redirect fragment products to new target positions while coefficient regulators normalize scalars, avoiding exponential noise growth. Security rests on a binding mechanism that makes exponents and coefficients mutually dependent, shielding the secret key from algebraic manipulation. A reader would care because this approach aims to simplify noise control, a central obstacle to practical FHE deployment.

Core claim

The central claim is that plaintext fragmentation across logical positions, governed by a dual-regulator system, extends the naturally additive property of the base encryption to multiplication. Exponent regulators t_i redirect the product of fragments to a new target position, preventing accumulation of secret key exponents, while coefficient regulators d_i normalize the resulting scalars. Security follows from the mutual dependence of exponents and coefficients, which blocks substitution attacks on the secret key k.

What carries the argument

The interposition framework's dual-regulator system, where exponent regulators t_i redirect fragment products to new positions and coefficient regulators d_i normalize scalars.

If this is right

  • Multiplication operations preserve the additive character of the base scheme without exponential noise accumulation.
  • The secret key remains protected from direct substitution or algebraic attacks through the exponent-coefficient binding.
  • Encryption stays modular while extending homomorphic multiplication via dynamic position redirection.
  • Overall scheme complexity may decrease by replacing traditional noise management with position shifting.

Where Pith is reading between the lines

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

  • If regulators can be computed efficiently, the scheme could support longer sequences of multiplications than current noise-limited methods.
  • The position-shifting idea might combine with existing lattice-based techniques to address other attack vectors.
  • Practical testing on standard benchmark circuits would reveal whether overhead stays below that of noise flooding approaches.

Load-bearing premise

The dual-regulator system can be implemented without introducing new algebraic vulnerabilities or prohibitive overhead.

What would settle it

A concrete algebraic attack that substitutes or extracts the secret key k by manipulating fragment positions and regulators despite their claimed mutual binding, or a multiplication sequence in which noise still grows exponentially.

Figures

Figures reproduced from arXiv: 2605.15774 by Mostefa Kara.

Figure 1
Figure 1. Figure 1: A global overview of the interposition mechanism. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

Traditional Fully Homomorphic Encryption (FHE) schemes often suffer from prohibitive computational overhead and complex noise management. In this paper, we propose a novel symmetric FHE through a mechanism of plaintext fragmentation and dynamic interposition. Our approach is built upon a modular encryption foundation, c = mk + rp, which is naturally additive but typically limited by exponential noise growth during multiplication. To resolve this, we introduce an interposition framework where the plaintext is partitioned into multiple fragments across distinct logical positions. We introduce a dual-regulator system to govern the multiplication process; exponent regulators (t_i) redirect the product of fragments to a new target position, preventing the accumulation of secret key exponents, while coefficient regulators (d_i) normalize the resulting scalars. Security is established through a binding mechanism where exponents and coefficients are mutually dependent, shielding the secret key k from algebraic manipulation and substitution attacks.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a symmetric FHE construction built on the modular base c = mk + rp. Plaintext is fragmented across logical positions; a dual-regulator system (exponent regulators t_i that redirect fragment products to new target positions and coefficient regulators d_i that normalize scalars) is introduced to prevent exponential noise growth under multiplication. Security is claimed to follow from a binding in which exponents and coefficients become mutually dependent, thereby shielding the secret key k from algebraic manipulation and substitution attacks.

Significance. If the regulators can be shown to bound noise while preserving semantic security without prohibitive overhead or new attack surfaces, the position-shifting approach would constitute a meaningful alternative to conventional noise-control techniques in symmetric FHE. The absence of any concrete algebraic definitions, update rules, noise bounds, or security reductions currently prevents evaluation of whether this potential is realized.

major comments (2)
  1. Abstract and interposition-framework description: the central security claim rests on the assertion that t_i and d_i create mutual dependence between exponents and coefficients, yet no functional definitions, recurrence relations, or update rules across multiplications are supplied. Without these, it is impossible to verify that the mechanism actually prevents an adversary from solving the resulting system for k after one or more homomorphic multiplications.
  2. Abstract: no noise-growth analysis, ciphertext-size bounds, or reduction to a hard problem is provided to support the claim that position shifting eliminates the exponential noise growth that normally accompanies multiplication in the base scheme c = mk + rp.
minor comments (1)
  1. Abstract: the variables m, k, r, p in the base equation c = mk + rp are introduced without a short definitional sentence; adding one would improve readability for readers outside the immediate sub-area.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We agree that the current manuscript requires additional formalization to allow verification of the claims. We will revise the paper to supply the missing definitions, relations, and analyses while preserving the core contribution of the position-shifting approach.

read point-by-point responses

  1. Referee: Abstract and interposition-framework description: the central security claim rests on the assertion that t_i and d_i create mutual dependence between exponents and coefficients, yet no functional definitions, recurrence relations, or update rules across multiplications are supplied. Without these, it is impossible to verify that the mechanism actually prevents an adversary from solving the resulting system for k after one or more homomorphic multiplications.

    Authors: We accept this criticism. The abstract provides only a high-level overview, and the full text does not yet contain explicit functional definitions or recurrence relations for the regulators t_i and d_i. In the revised manuscript we will add a new subsection that defines t_i as a position-mapping function t_i(j) = (j + f(i)) mod m where f encodes fragment index, and d_i as a coefficient normalizer d_i = 1 / (product of selected scalars). We will also supply the update rules that apply these regulators after each multiplication and include a small worked example demonstrating that the resulting system remains underdetermined for k. revision: yes

  2. Referee: Abstract: no noise-growth analysis, ciphertext-size bounds, or reduction to a hard problem is provided to support the claim that position shifting eliminates the exponential noise growth that normally accompanies multiplication in the base scheme c = mk + rp.

    Authors: The referee correctly identifies the absence of quantitative analysis. We will add a dedicated section deriving noise bounds: after each multiplication the noise term is redirected to a fresh position and scaled by a bounded d_i, yielding linear rather than exponential growth in the number of multiplications. Ciphertext-size growth will be expressed as O(number of fragments + regulator bits). A full reduction to a standard hard problem is not currently available; we will instead provide an informal argument that recovering k requires solving a system whose binding is at least as hard as the original modular equation, and we will explicitly note the lack of a formal reduction as a limitation to be addressed in future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper begins from the modular base scheme c = mk + rp and describes an interposition framework that partitions plaintext into fragments governed by a dual-regulator system (exponent regulators t_i and coefficient regulators d_i). The security claim is presented as following from the resulting mutual dependence in the binding mechanism. No load-bearing step reduces a prediction, uniqueness result, or security property back to its own inputs by construction; the regulators are introduced as new components without self-referential definitions, fitted-parameter renaming, or self-citation chains that carry the central argument. The derivation therefore remains self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claim depends on the unproven effectiveness of the dual-regulator system and the binding mechanism; these are introduced without independent evidence or reduction to prior results in the provided abstract.

free parameters (2)
  • exponent regulators t_i
    Values chosen to redirect fragment products to target positions; no specific selection method or bounds given in abstract.
  • coefficient regulators d_i
    Scalars introduced to normalize results after position shifting; selection criteria not specified.
axioms (1)
  • domain assumption The base encryption c = mk + rp is naturally additive.
    Invoked as the modular foundation that the interposition framework extends.
invented entities (1)
  • dual-regulator system no independent evidence
    purpose: Govern multiplication by redirecting fragments and normalizing scalars
    New framework postulated to solve noise accumulation; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5667 in / 1401 out tokens · 98827 ms · 2026-05-20T17:44:15.350663+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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