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Block-MDS QC-LDPC Codes for Information Reconciliation in Key Distribution

Lev Tauz, Debarnab Mitra, Jayanth Shreekumar, Murat Can Sarihan, Chee Wei Wong, Lara Dolecek

TL;DR

The paper tackles improving QKD secret-key rate by relaxing the Information Reconciliation (IR) constraint through a sampling strategy that couples IR with Privacy Amplification (PA). It introduces Block-MDS QC-LDPC codes and a Multiple Subset Codeword (MSC) decoding framework that enables decoding only a subset of reconciled bits while preserving security, supported by entropy-based arguments. A key contribution is a sufficient condition for Block-MDS QC-LDPC codes, expressed via determinant and gcd criteria on associated polynomials, along with a Vandermonde-based decoupling that scales the field size with the code's row degree. Empirical results on high-girth QC-LDPC codes show that the MSC approach substantially lowers IR failure probability and improves SKR in high-noise scenarios, validating the viability of joint IR/PA and guiding further exploration of block-structured LDPC families for QKD and related applications.

Abstract

Quantum key distribution (QKD) is a popular protocol that provides information theoretically secure keys to multiple parties. Two important post-processing steps of QKD are 1) the information reconciliation (IR) step, where parties reconcile mismatches in generated keys through classical communication, and 2) the privacy amplification (PA) step, where parties distill their common key into a new secure key that the adversary has little to no information about. In general, these two steps have been abstracted as two distinct problems. In this work, we consider a new technique of performing the IR and PA steps jointly through sampling that relaxes the requirement on the IR step, allowing for more success in key creation. We provide a novel LDPC code construction known as Block-MDS QC-LDPC codes that can utilize the relaxed requirement by creating LDPC codes with pre-defined sub-matrices of full-rank. We demonstrate through simulations that our technique of sampling can provide notable gains in successfully creating secret keys.

Block-MDS QC-LDPC Codes for Information Reconciliation in Key Distribution

TL;DR

The paper tackles improving QKD secret-key rate by relaxing the Information Reconciliation (IR) constraint through a sampling strategy that couples IR with Privacy Amplification (PA). It introduces Block-MDS QC-LDPC codes and a Multiple Subset Codeword (MSC) decoding framework that enables decoding only a subset of reconciled bits while preserving security, supported by entropy-based arguments. A key contribution is a sufficient condition for Block-MDS QC-LDPC codes, expressed via determinant and gcd criteria on associated polynomials, along with a Vandermonde-based decoupling that scales the field size with the code's row degree. Empirical results on high-girth QC-LDPC codes show that the MSC approach substantially lowers IR failure probability and improves SKR in high-noise scenarios, validating the viability of joint IR/PA and guiding further exploration of block-structured LDPC families for QKD and related applications.

Abstract

Quantum key distribution (QKD) is a popular protocol that provides information theoretically secure keys to multiple parties. Two important post-processing steps of QKD are 1) the information reconciliation (IR) step, where parties reconcile mismatches in generated keys through classical communication, and 2) the privacy amplification (PA) step, where parties distill their common key into a new secure key that the adversary has little to no information about. In general, these two steps have been abstracted as two distinct problems. In this work, we consider a new technique of performing the IR and PA steps jointly through sampling that relaxes the requirement on the IR step, allowing for more success in key creation. We provide a novel LDPC code construction known as Block-MDS QC-LDPC codes that can utilize the relaxed requirement by creating LDPC codes with pre-defined sub-matrices of full-rank. We demonstrate through simulations that our technique of sampling can provide notable gains in successfully creating secret keys.
Paper Structure (8 sections, 8 theorems, 22 equations, 1 figure, 2 tables)

This paper contains 8 sections, 8 theorems, 22 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and Motivation
  2. Background and Model
  3. System Model
  4. LDPC code preliminaries
  5. Privacy Amplification with Sampling
  6. Block-MDS QC-LDPC Codes
  7. Simulations and Conclusion
  8. Code Parameters

Key Result

Lemma 1

Fossorier2004 A QC-LDPC code in the form of Eq.eq:qc_ldpc has girth at least $2(g+1)$ if and only if for all $m$, $2 \leq m \leq g$, all $i_k$, $i\in[\gamma]$, and all $j_k$, $j\in[\kappa]$ with $i_1 = i_{m}$, $i_k \neq i_{k+1}$, and $j_k\neq j_{k+1}$.

Figures (1)

  • Figure 1: Probability of IR failure for different transition probabilities for a 8-ary symmetric channel. Bold line indicates the FC decoder and dotted lines indicates the MSC decoder.

Theorems & Definitions (16)

  • Lemma 1
  • Definition 1
  • Theorem 1
  • proof
  • Definition 2
  • Corollary 2
  • Definition 3
  • Example 1
  • Lemma 2
  • Lemma 3
  • ...and 6 more