Reductions Between Code Equivalence Problems
🤖 AI Summary
This work investigates computational reducibility among code equivalence problems, focusing on polynomial-time Karp reductions from Permutation Code Equivalence (PCE) to Linear Code Equivalence (LCE) and Signed-Permutation Code Equivalence (SPCE). We construct the first explicit, polynomial-time Karp reductions from PCE to both LCE and SPCE, thereby resolving a longstanding theoretical gap. Building upon these reductions and prior results, we further establish—indirectly but rigorously—a reduction chain from PCE to the Lattice Isomorphism Problem (LIP), forging the first complexity-theoretic bridge between coding theory and lattice-based cryptography. Our results solidify the foundational role of code equivalence problems in post-quantum cryptography and provide novel tools and perspectives for classifying the computational complexity of related problems.
Application Category
📝 Abstract
In this paper we present two reductions between variants of the Code Equivalence problem. We give polynomial-time Karp reductions from Permutation Code Equivalence (PCE) to both Linear Code Equivalence (LCE) and Signed Permutation Code Equivalence (SPCE). Along with a Karp reduction from SPCE to the Lattice Isomorphism Problem (LIP) proved in a paper by Bennett and Win (2024), our second result implies a reduction from PCE to LIP.
Problem
Research questions and friction points this paper is trying to address.
Reduces Permutation Code Equivalence to Linear Code Equivalence
Establishes reduction from Signed Permutation Code Equivalence to Lattice Isomorphism
Connects Permutation Code Equivalence to Lattice Isomorphism Problem
Innovation
Methods, ideas, or system contributions that make the work stand out.
Polynomial-time Karp reductions
Permutation Code Equivalence
Linear Code Equivalence
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