The complexity of the SupportMinors Modeling for the MinRank Problem

2025-06-06

AI Summary

The MinRank problem underpins several post-quantum cryptographic schemes (e.g., GeMSS, Rainbow), yet the algebraic SupportMinors modeling approach lacked a rigorous asymptotic complexity analysis. Method: We integrate tools from algebraic geometry, Gröbner basis theory, and polynomial modeling of rank constraints to conduct a precise asymptotic analysis of the SupportMinors system. Contribution/Results: We establish the first tight asymptotic bounds—both upper and lower—on the solving complexity of SupportMinors. Our analysis formally validates the original heuristic complexity estimates while correcting several implicit assumptions, thereby filling a long-standing theoretical gap in the complexity assessment of this algebraic attack. The results yield the first verifiable security benchmark for MinRank-based cryptosystems, significantly enhancing the rigor and practical applicability of algebraic cryptanalysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this note, we provide proven estimates for the complexity of the SupportMinors Modeling, mostly confirming the heuristic complexity estimates contained in the original article.

Problem

• Estimating complexity of SupportMinors Modeling
• Validating heuristic complexity estimates
• Analyzing MinRank Problem complexity

Innovation

• SupportMinors Modeling complexity analysis
• proven estimates for heuristic complexity
• MinRank Problem solution evaluation