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Using cyclic $(f,σ)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices

Susanne Pumpluen

TL;DR

The paper extends Construction A to nonassociative settings by using cyclic (f,σ)-codes over finite chain rings and Petit algebras to construct new Z_p- and F_q[[t]]-lattices. It develops natural orders in nonassociative algebras, analyzes their quotients, and shows how principal left ideals correspond to cyclic codes that lift to lattice codes via Construction A, with MRD codes obtained through left multiplication maps. It generalizes to generalized nonassociative cyclic algebras, enabling lattice-encoding of more complex codes and providing MRD-code constructions over p-adic and t-adic fields. The results point to potential post-quantum cryptographic applications, including LWE-like schemes and p-adic coset/wiretap coding, while also suggesting directions to explore alternative orders and broader algebraic frameworks for lattice design.

Abstract

We construct $\mathbb{Z}_p$-lattices and $\mathbb{F}_q[\![t]\!]$-lattices from cyclic $(f,σ)$-codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains $\mathbb{Z}$-lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use proper nonassociative cyclic algebras that are defined over field extensions of $p$-adic fields. This means we focus on $σ$-constacyclic codes to obtain $\mathbb{Z}_p$-lattices, hence $\mathbb{Z}_p$-lattice codes. We construct linear maximum rank distance (MRD) codes that are $\mathbb{Z}_p$-lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring. Possible applications of our constructions include post-quantum cryptography involving $p$-adic lattices, e.g. learning with errors, building rank-metric codes like MRD-codes, or $p$-adic coset coding, in particular wire-tap coding.

Using cyclic $(f,σ)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices

TL;DR

The paper extends Construction A to nonassociative settings by using cyclic (f,σ)-codes over finite chain rings and Petit algebras to construct new Z_p- and F_q[[t]]-lattices. It develops natural orders in nonassociative algebras, analyzes their quotients, and shows how principal left ideals correspond to cyclic codes that lift to lattice codes via Construction A, with MRD codes obtained through left multiplication maps. It generalizes to generalized nonassociative cyclic algebras, enabling lattice-encoding of more complex codes and providing MRD-code constructions over p-adic and t-adic fields. The results point to potential post-quantum cryptographic applications, including LWE-like schemes and p-adic coset/wiretap coding, while also suggesting directions to explore alternative orders and broader algebraic frameworks for lattice design.

Abstract

We construct -lattices and -lattices from cyclic -codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains -lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use proper nonassociative cyclic algebras that are defined over field extensions of -adic fields. This means we focus on -constacyclic codes to obtain -lattices, hence -lattice codes. We construct linear maximum rank distance (MRD) codes that are -lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring. Possible applications of our constructions include post-quantum cryptography involving -adic lattices, e.g. learning with errors, building rank-metric codes like MRD-codes, or -adic coset coding, in particular wire-tap coding.
Paper Structure (24 sections, 3 theorems, 62 equations)

This paper contains 24 sections, 3 theorems, 62 equations.

Table of Contents

  1. Preliminaries
  2. Nonassociative algebras
  3. Skew polynomial rings
  4. Petit's algebras
  5. Cyclic $(f,\sigma,\delta)$-codes
  6. Local nonarchimedean fields
  7. Natural orders in $S_f$ and their quotients by a prime ideal
  8. Quotients of rings on integers
  9. Natural orders
  10. How to obtain $\mathbb{Z}_p$-lattice and codes from cyclic $(f,\sigma)$-codes over $\mathcal{O}_K/\mathfrak{p}\mathcal{O}_K$ using Construction A
  11. Principal left ideals in $S_{\overline{f}}$ and $(f,\sigma)$-codes
  12. Construction A
  13. Examples of lifting $\overline{\sigma}$-constacyclic codes
  14. How to obtain lattice codes from cyclic $(\overline{f},\overline{\sigma})$-codes over $\mathcal{O}_K/\mathfrak{p}\mathcal{O}_K$ using Construction A and generalized nonassociative cyclic algebras
  15. Natural orders in generalized nonassociative cyclic algebras and their quotients by a prime ideal
  16. ...and 9 more sections

Key Result

Lemma 1.3

Eisenbud1995 Let $f(x)\in \mathcal{O}_F [x]$ be a polynomial and $f'(x)$ its formal derivative. If $a\in \mathcal{O}_F$ satisfies then there exists a unique $b\in \mathcal{O}_F$ such that

Theorems & Definitions (11)

  • Definition 1.1
  • Example 1.2
  • Lemma 1.3: Hensel's Lemma
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Example 3.1
  • Example 3.2
  • Remark 5.1
  • Theorem 6.1
  • ...and 1 more