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Engel p-adic Isogeny-based Cryptography over Laurent Series: Foundations, Security, and an ESP32 Implementation

Ilias Cherkaoui, Indrakshi Dey

TL;DR

This work introduces a novel post-quantum cryptography framework that fuses Engel expansions with $p$-adic Laurent-series arithmetic to parameterize supersingular isogenies. Encoding torsion data via Engel coefficients yields compact public keys while enabling fixed-precision, branch-regular arithmetic suitable for embedded devices, demonstrated on an ESP32 with constant-time kernels. The security model rests on SIDP and a new EEIP hardness assumption for the Engel encoding, offering IND‑CPA security under these conjectures. Empirical evaluation shows linear-time cryptographic computation with respect to message size, with network latency dominating end-to-end performance in multi-node IoT deployments, and energy profiles indicating predictable, hardware-friendly operation. The results highlight a viable path to quantum-resistant, lightweight cryptography on resource-constrained devices, while outlining directions for deeper cryptanalytic validation and hardware co-design.

Abstract

Securing the Internet of Things (IoT) against quantum attacks requires public-key cryptography that (i) remains compact and (ii) runs efficiently on microcontrollers, capabilities many post-quantum (PQ) schemes lack due to large keys and heavy arithmetic. We address both constraints simultaneously with, to our knowledge, the first-ever isogeny framework that encodes super-singular elliptic-curve isogeny data via novel Engel expansions over the p-adic Laurent series. Engel coefficients compress torsion information, thereby addressing the compactness constraint, yielding public keys of ~1.1 - 16.9 kbits preserving the hallmark small sizes of isogeny systems. Engel arithmetic is local and admits fixed-precision p-adic operations, enabling micro-controller efficiency with low-memory, branch-regular kernels suitable for embedded targets.

Engel p-adic Isogeny-based Cryptography over Laurent Series: Foundations, Security, and an ESP32 Implementation

TL;DR

This work introduces a novel post-quantum cryptography framework that fuses Engel expansions with -adic Laurent-series arithmetic to parameterize supersingular isogenies. Encoding torsion data via Engel coefficients yields compact public keys while enabling fixed-precision, branch-regular arithmetic suitable for embedded devices, demonstrated on an ESP32 with constant-time kernels. The security model rests on SIDP and a new EEIP hardness assumption for the Engel encoding, offering IND‑CPA security under these conjectures. Empirical evaluation shows linear-time cryptographic computation with respect to message size, with network latency dominating end-to-end performance in multi-node IoT deployments, and energy profiles indicating predictable, hardware-friendly operation. The results highlight a viable path to quantum-resistant, lightweight cryptography on resource-constrained devices, while outlining directions for deeper cryptanalytic validation and hardware co-design.

Abstract

Securing the Internet of Things (IoT) against quantum attacks requires public-key cryptography that (i) remains compact and (ii) runs efficiently on microcontrollers, capabilities many post-quantum (PQ) schemes lack due to large keys and heavy arithmetic. We address both constraints simultaneously with, to our knowledge, the first-ever isogeny framework that encodes super-singular elliptic-curve isogeny data via novel Engel expansions over the p-adic Laurent series. Engel coefficients compress torsion information, thereby addressing the compactness constraint, yielding public keys of ~1.1 - 16.9 kbits preserving the hallmark small sizes of isogeny systems. Engel arithmetic is local and admits fixed-precision p-adic operations, enabling micro-controller efficiency with low-memory, branch-regular kernels suitable for embedded targets.

Paper Structure

This paper contains 62 sections, 13 theorems, 31 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Works
  4. Field Construction
  5. Mathematical Preliminaries
  6. Preliminaries on $p$-adics and Valuation
  7. Laurent Series over $\mathbb{Q}_p$
  8. Engel Expansions on $\mathbb{Q}_p((t))$
  9. Engel Field Isomorphism and 2‑Isogeny via Vélu
  10. The Engel--Field Correspondence
  11. Elliptic Curves over $\mathbb{Q}_p((t))$ and Reduction
  12. Hensel Lifts of $2$-Torsion and Engel Encoding
  13. Explicit $2$-Isogeny via Vélu
  14. Reduction and Super-singularity Preservation
  15. Algorithm Design
  16. ...and 47 more sections

Key Result

Theorem 1

The map $\phi:\mathbb{Q}_p((t))\to\mathcal{E}$ sending $f(t)$ to its Engel expansion is bijective and preserves addition, multiplication, and inversion (on $\mathbb{Q}_p((t))^\times$). Consequently $(\mathcal{E},+,\cdot)$ is a field isomorphic to $\mathbb{Q}_p((t))$.

Figures (10)

  • Figure 1: Single-Board Cryptographic Process Flow: Sequential encryption–decryption and metric collection process on an ESP32 node.
  • Figure 2: Four-Board Chain Implementation Flow: Sequential communication and acknowledgement flow among ESP32 nodes.
  • Figure 3: Timing Analysis: Encryption and decryption execution times versus message size at 80, 160, and 240 MHz. Red and blue markers denote encryption and decryption, respectively.
  • Figure 4: Frequency Scaling Efficiency: Relative execution time improvements between CPU frequencies for encryption and decryption.
  • Figure 5: Latency and computation time of the cryptographic system across three frequencies: $80$, $160$, and $240$ MHz for data sizes up to $600$ bytes. Latency dominates the total time, while computation remains nearly constant with message size, indicating that communication overhead is the primary bottleneck.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Theorem 1: Engel--Field Isomorphism
  • Theorem 2: Supersingularity via $2$-Isogeny
  • Lemma 3
  • proof
  • Lemma 4: Convergence
  • proof
  • Lemma 5: Existence
  • proof
  • Lemma 6: Uniqueness
  • proof
  • ...and 15 more