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Public-Key Encryption from the MinRank Problem

Rohit Chatterjee, Changrui Mu, Prashant Nalini Vasudevan

TL;DR

This work constructs a public-key encryption scheme whose security rests on the average-case hardness of the MinRank problem on uniformly random instances, reframing MinRank as decoding random rank-metric codes. The authors introduce a novel block-wise, matrix-valued inner product to define a dual MinRank problem and prove a duality connecting the primal and dual hardness, enabling a search-to-decision reduction. The resulting PKE, with KeyGen, Enc, and Dec, achieves semantic security and correctness under parameter regimes that ensure the duality and hardness assumptions hold, while avoiding reliance on structured codes. The contribution includes a comprehensive analysis of MinRank attacks (combinatorial, algebraic, and quantum) and parameter trade-offs, highlighting potential post-quantum security and practical simplicity of the construction.

Abstract

We construct a public-key encryption scheme from the hardness of the (planted) MinRank problem over uniformly random instances. This corresponds to the hardness of decoding random linear rank-metric codes. Existing constructions of public-key encryption from such problems require hardness for structured instances arising from the masking of efficiently decodable codes. Central to our construction is the development of a new notion of duality for rank-metric codes.

Public-Key Encryption from the MinRank Problem

TL;DR

This work constructs a public-key encryption scheme whose security rests on the average-case hardness of the MinRank problem on uniformly random instances, reframing MinRank as decoding random rank-metric codes. The authors introduce a novel block-wise, matrix-valued inner product to define a dual MinRank problem and prove a duality connecting the primal and dual hardness, enabling a search-to-decision reduction. The resulting PKE, with KeyGen, Enc, and Dec, achieves semantic security and correctness under parameter regimes that ensure the duality and hardness assumptions hold, while avoiding reliance on structured codes. The contribution includes a comprehensive analysis of MinRank attacks (combinatorial, algebraic, and quantum) and parameter trade-offs, highlighting potential post-quantum security and practical simplicity of the construction.

Abstract

We construct a public-key encryption scheme from the hardness of the (planted) MinRank problem over uniformly random instances. This corresponds to the hardness of decoding random linear rank-metric codes. Existing constructions of public-key encryption from such problems require hardness for structured instances arising from the masking of efficiently decodable codes. Central to our construction is the development of a new notion of duality for rank-metric codes.

Paper Structure

This paper contains 40 sections, 7 theorems, 41 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Cryptography from MinRank.
  3. Public-Key Encryption from MinRank.
  4. Our Results
  5. Non-Scalar Inner Products.
  6. Duality.
  7. Significance.
  8. Preliminaries
  9. Notation.
  10. Public key encryption
  11. Coding Theory
  12. MinRank Problems
  13. The MinRank Problems
  14. Dual MinRank Problem
  15. Properties of the Block-Wise Inner Product
  16. ...and 25 more sections

Key Result

Lemma 3.10

Consider any polynomially bounded functions $r=r(n)$, $k=k(n)$, $l=l(n)$, and $t=t(n)$ that have the following properties: If there exists a distinguisher for ${\rm dualMinRk}({n,l,r})$ that runs in time $T(n)$ and has advantage $\epsilon(n)$, then there exists a distinguisher for ${\rm MinRk}({n,k,r})$ that runs in time $T(n)+\mathrm{poly}(n)$ and has advantage $(\epsilon(n)-\mathrm{negl}(n))$.

Figures (1)

  • Figure 1: PKE from MinRank

Theorems & Definitions (37)

  • Definition 2.1: Public Key Encryption Scheme
  • Definition 2.2: Rank Distance
  • Definition 2.3: Matrix Code Hua51
  • Definition 2.4: Matrix-Valued Inner Product
  • Definition 2.5: Dual Matrix Code
  • Definition 2.6: Square Matrix Trace
  • Definition 2.7: Frobenius Inner Product
  • Definition 2.9: Total Variation Distance
  • Definition 3.1: Decision MinRank Problem
  • Definition 3.2: Search MinRank Problem
  • ...and 27 more