[Paper Review] A general method for finding the compositional inverses of permutations from the AGW criterion.

Tailin Niu, Kangquan Li+2 more|arXiv (Cornell University)|Apr 27, 2020

Coding theory and cryptography28 references4 citations

TL;DR

This paper presents a unified method for computing the compositional inverses of permutation polynomials constructed via the AGW criterion. By leveraging algebraic structures and transformation rules derived from the AGW framework, the approach systematically determines inverses for various polynomial forms, offering a general solution to a longstanding challenge in finite field cryptography and coding theory.

ABSTRACT

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a unified approach to find these inverses of permutation polynomials constructed by the AGW criterion.

Motivation & Objective

• Address the lack of a general method for computing compositional inverses of permutation polynomials constructed using the AGW criterion.
• Unify disparate results on inverses of specific permutation polynomial forms into a single, systematic framework.
• Provide a method applicable across diverse polynomial families while preserving computational efficiency.
• Enable broader application of permutation polynomials in cryptographic and coding-theoretic constructions by ensuring inverse computation is tractable.

Proposed method

• Utilize the AGW criterion as a foundational framework to characterize permutation polynomials over finite fields.
• Establish transformation rules that map a permutation polynomial to its compositional inverse using algebraic identities.
• Apply the AGW criterion’s equivalence conditions to derive explicit inverse expressions in terms of the original polynomial’s components.
• Generalize known inverse-finding techniques by embedding them within a unified algebraic structure.
• Verify the method’s consistency across multiple polynomial families by applying it to representative cases.
• Ensure the method remains computationally feasible by relying on field operations and known polynomial identities.

Experimental results

Research questions

• RQ1 How can a general method be developed to compute compositional inverses of permutation polynomials constructed via the AGW criterion?
• RQ2 What algebraic transformations underlie the inverse computation across different polynomial forms under the AGW framework?
• RQ3 To what extent can existing inverse-finding results be unified under a single theoretical framework?
• RQ4 Can the proposed method be applied systematically and efficiently to various classes of permutation polynomials?
• RQ5 What structural properties of the AGW criterion enable the derivation of compositional inverses in a general setting?

Key findings

• The proposed method provides a systematic and general procedure to compute the compositional inverse of any permutation polynomial constructed using the AGW criterion.
• The method unifies previously isolated results on inverse computation into a single coherent framework.
• The approach is applicable to a wide range of permutation polynomial families, including those of binomial, trinomial, and other structured forms.
• The derivation of inverses relies on algebraic identities derived from the AGW criterion, ensuring correctness and consistency.
• The method preserves computational efficiency, making it suitable for practical applications in cryptography and coding theory.
• The framework enables the inverse of a permutation polynomial to be expressed explicitly in terms of its defining components and field operations.