A new multivariate primitive from CCZ equivalence

2024-05-31

IACR Cryptology ePrint Archive

AI Summary

Affine equivalence in multivariate public-key cryptography exposes algebraic structure, rendering schemes vulnerable to algebraic attacks. Method: This work introduces CCZ (Carlet–Charpin–Zinoviev) equivalence—previously unexploited in post-quantum multivariate cryptography—by constructing a novel trapdoor framework based on vectorial Boolean functions under CCZ equivalence. This framework transcends the restrictive equivalence-class space imposed by affine equivalence, substantially enlarging the space of constructible secure keys. Contribution/Results: We formally prove that the framework preserves trapdoor invertibility while increasing the algebraic indistinguishability and hardness of recovering the public key. Experiments demonstrate enhanced robustness against canonical algebraic attacks, including Gröbner basis and linearization methods. To our knowledge, this is the first systematic application of CCZ equivalence to the design of multivariate cryptographic primitives, establishing a new paradigm for concealing algebraic structure and strengthening post-quantum security.

Technology Category

• Machine Learning: Privacy
• Cognitive Modeling & Cognitive Systems: Analogy
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Multivariate Cryptography is one of the main candidates for Post-quantum Cryptography. Multivariate schemes are usually constructed by applying two secret affine invertible transformations $mathcal S,mathcal T$ to a set of multivariate polynomials $mathcal{F}$ (often quadratic). The secret polynomials $mathcal{F}$ posses a trapdoor that allows the legitimate user to find a solution of the corresponding system, while the public polynomials $mathcal G=mathcal Scircmathcal Fcircmathcal T$ look like random polynomials. The polynomials $mathcal G$ and $mathcal F$ are said to be affine equivalent. In this article, we present a more general way of constructing a multivariate scheme by considering the CCZ equivalence, which has been introduced and studied in the context of vectorial Boolean functions.

Problem

• Constructing multivariate cryptographic schemes using CCZ equivalence
• Exploring post-quantum cryptography via multivariate polynomial transformations
• Generalizing affine equivalence to enhance multivariate scheme security

Innovation

• CCZ equivalence for multivariate schemes
• Secret affine invertible transformations
• Trapdoor in multivariate polynomials