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Key exchange protocol based on circulant matrix action over congruence-simple semiring

Alvaro Otero Sanchez

TL;DR

The paper proposes a post-quantum key-exchange protocol based on the action of circulant matrices from $Circ_n(\mathbb{N})$ on vectors in $C_R[M]^n$ over a congruence-simple semiring $R$, with public data $(M,v)$ and private circulant selections leading to a shared key via $ Apk_2 = (AB)v$. It furnishes an explicit construction of large-order matrices $M$ through a partition-based block-structure guided by least common multiples and demonstrates that many distinct powers can be generated (e.g., a $20\times20$ example yielding at least 280 powers). It analyzes setup and execution costs and argues resistance to brute-force, Pohlig–Hellman-type, O–L, and quantum attacks by bounding search spaces and favoring polynomials in $M$ over monoid powers. The work delivers a concrete, potentially practical post-quantum key-exchange primitive in a semiring setting, with explicit construction rules for $M$ and $v$ and a thorough security perspective against known cryptanalytic classes.

Abstract

We present a new key exchange protocol based on circulant matrices acting on matrices over a congruence-simple semiring. We describe how to compute matrices with the necessary properties for the implementation of the protocol. Additionally, we provide an analysis of its computational cost and its security against known attacks.

Key exchange protocol based on circulant matrix action over congruence-simple semiring

TL;DR

The paper proposes a post-quantum key-exchange protocol based on the action of circulant matrices from on vectors in over a congruence-simple semiring , with public data and private circulant selections leading to a shared key via . It furnishes an explicit construction of large-order matrices through a partition-based block-structure guided by least common multiples and demonstrates that many distinct powers can be generated (e.g., a example yielding at least 280 powers). It analyzes setup and execution costs and argues resistance to brute-force, Pohlig–Hellman-type, O–L, and quantum attacks by bounding search spaces and favoring polynomials in over monoid powers. The work delivers a concrete, potentially practical post-quantum key-exchange primitive in a semiring setting, with explicit construction rules for and and a thorough security perspective against known cryptanalytic classes.

Abstract

We present a new key exchange protocol based on circulant matrices acting on matrices over a congruence-simple semiring. We describe how to compute matrices with the necessary properties for the implementation of the protocol. Additionally, we provide an analysis of its computational cost and its security against known attacks.
Paper Structure (9 sections, 16 theorems, 29 equations)

This paper contains 9 sections, 16 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. Example
  4. Key Exchange Protocol and Its Security
  5. Brute force and random attack
  6. Pohlig-Hellman type attacks
  7. O-L attack
  8. Quantum attack
  9. Conclusion

Key Result

Theorem 2.3

The set $Circ_n(R)$ of ciruclar matrix of $n\times n$ over $R$ form a commutative subring of $\rm{Mat}_n(R)$.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 24 more