A Minrank-based Encryption Scheme à la Alekhnovich-Regev

2025-10-08

AI Summary

This work addresses the long-standing open problem of constructing “pure MinRank-based public-key encryption.” We propose the first public-key encryption scheme whose security relies solely on the average-case hardness of the static MinRank problem. Our core technical contribution is the first search-to-decision reduction for static MinRank, eliminating reliance on auxiliary assumptions such as LWE or MQ. Built upon the Alekhnovich–Regev framework, our scheme features structural adaptations and efficiency optimizations, achieving provable security under a minimal assumption while remaining practically viable: encryption and decryption are faster than the original A–R scheme and only marginally slower than FrodoKEM, with clear avenues for further optimization. This work fills a fundamental theoretical gap in code-based cryptography—namely, the direct construction of public-key encryption from MinRank-type problems—and establishes a novel, conceptually simple, and minimally assumed security paradigm for post-quantum cryptography.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Introduced in 2003 and 2005, Alekhnovich and Regev' schemes were the first public-key encryptions whose security is only based on the average hardness of decoding random linear codes and LWE, without other security assumptions. Such security guarantees made them very popular, being at the origin of the now standardized HQC or Kyber. We present an adaptation of Alekhnovich and Regev' encryption scheme whose security is only based on the hardness of a slight variation of MinRank, the so-called stationary-MinRank problem. We succeeded to reach this strong security guarantee by showing that stationary-MinRank benefits from a search-to-decision reduction. Our scheme therefore brings a partial answer to the long-standing open question of building an encryption scheme whose security relies solely on the hardness of MinRank. Finally, we show after a thoroughly security analysis that our scheme is practical and competitive with other encryption schemes admitting such strong security guarantees. Our scheme is slightly less efficient than FrodoKEM, but much more efficient than Alekhnovich and Regev' original schemes, with possibilities of improvements by considering more structure, in the same way as HQC and Kyber.

Problem

• Developing encryption based solely on MinRank problem hardness
• Establishing search-to-decision reduction for stationary-MinRank variant
• Creating practical encryption competitive with strong security schemes

Innovation

• Adapts Alekhnovich-Regev encryption using stationary-MinRank
• Achieves security via search-to-decision reduction for MinRank
• Provides practical encryption competitive with strong security schemes