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A Survey on Code-Based Cryptography

Violetta Weger, Niklas Gassner, Joachim Rosenthal

TL;DR

This survey analyzes code-based cryptography as a leading post-quantum candidate, emphasizing the decoding-hardness of random linear codes and the standardization landscape for public-key encryption and signatures. It systematically reviews core frameworks (McEliece, Niederreiter, Alekhnovich, QC, GPT, and related rank-metric variants), along with key code families (GRS, Goppa, cyclic, LDPC/MDPC, RM, Gabidulin, and LRPC) and their cryptanalytic considerations. It then covers code-based signature schemes, notably hash-and-sign approaches and ZK-based constructions via Fiat-Shamir, including the CFS scheme, and discusses practical considerations such as key sizes, decoding attacks, and MPC-in-the-head techniques. The chapter aims to equip researchers with a structured, background-rich understanding of code-based PKEs and signatures, highlighting security assumptions, standardization progress, and open research directions in the quantum era.

Abstract

The improvements on quantum technology are threatening our daily cybersecurity, as a capable quantum computer can break all currently employed asymmetric cryptosystems. In preparation for the quantum-era the National Institute of Standards and Technology (NIST) has initiated in 2016 a standardization process for public-key encryption (PKE) schemes, key-encapsulation mechanisms (KEM) and digital signature schemes. In 2023, NIST made an additional call for post-quantum signatures. With this chapter we aim at providing a survey on code-based cryptography, focusing on PKEs and signature schemes. We cover the main frameworks introduced in code-based cryptography and analyze their security assumptions. We provide the mathematical background in a lecture notes style, with the intention of reaching a wider audience.

A Survey on Code-Based Cryptography

TL;DR

This survey analyzes code-based cryptography as a leading post-quantum candidate, emphasizing the decoding-hardness of random linear codes and the standardization landscape for public-key encryption and signatures. It systematically reviews core frameworks (McEliece, Niederreiter, Alekhnovich, QC, GPT, and related rank-metric variants), along with key code families (GRS, Goppa, cyclic, LDPC/MDPC, RM, Gabidulin, and LRPC) and their cryptanalytic considerations. It then covers code-based signature schemes, notably hash-and-sign approaches and ZK-based constructions via Fiat-Shamir, including the CFS scheme, and discusses practical considerations such as key sizes, decoding attacks, and MPC-in-the-head techniques. The chapter aims to equip researchers with a structured, background-rich understanding of code-based PKEs and signatures, highlighting security assumptions, standardization progress, and open research directions in the quantum era.

Abstract

The improvements on quantum technology are threatening our daily cybersecurity, as a capable quantum computer can break all currently employed asymmetric cryptosystems. In preparation for the quantum-era the National Institute of Standards and Technology (NIST) has initiated in 2016 a standardization process for public-key encryption (PKE) schemes, key-encapsulation mechanisms (KEM) and digital signature schemes. In 2023, NIST made an additional call for post-quantum signatures. With this chapter we aim at providing a survey on code-based cryptography, focusing on PKEs and signature schemes. We cover the main frameworks introduced in code-based cryptography and analyze their security assumptions. We provide the mathematical background in a lecture notes style, with the intention of reaching a wider audience.
Paper Structure (81 sections, 66 theorems, 441 equations, 3 figures, 33 tables, 5 algorithms)

This paper contains 81 sections, 66 theorems, 441 equations, 3 figures, 33 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Organization of the Chapter
  3. Preliminaries
  4. Notation
  5. Algebraic Coding Theory
  6. Basics on Hamming-Metric Codes
  7. Matrix Codes
  8. Generalized Reed-Solomon Codes
  9. Goppa Codes
  10. Cyclic Codes
  11. LDPC Codes
  12. Reed-Muller Codes
  13. Concatenated Codes
  14. $(U,U+V)$-Codes
  15. Product Codes
  16. ...and 66 more sections

Key Result

Theorem 8

Let $k \leq n$ be positive integers and let $\mathcal{C}$ be an $[n,k]$ linear code over $\mathbb{F}_q$. Then,

Figures (3)

  • Figure 1: Compression Technique for $N$ Rounds
  • Figure 2: Overview of algorithms following the splitting of Lee-Brickell, adapted from ballcoll.
  • Figure 3: Overview of the weight splitting in the different algorithms.

Theorems & Definitions (190)

  • Definition 1: Linear Code
  • Definition 2: Hamming Metric
  • Definition 3: Minimum Distance
  • Theorem 8: Singleton Bound sing
  • Definition 10: Generator Matrix
  • Definition 11: Parity-Check Matrix
  • Definition 13: Dual Code
  • Definition 16: Information Set
  • Definition 20: Systematic Form
  • Theorem 21: Gilbert-Varshamov bound
  • ...and 180 more