[Paper Review] Permutation Polynomials modulo m

Shujun Li|ArXiv.org|Sep 22, 2005

Coding theory and cryptography23 references8 citations

TL;DR

This paper provides necessary and sufficient conditions for integer polynomials to induce bijections modulo $ m $, establishes counting estimates for permutation polynomials and induced permutations, and introduces a recursive algorithm to determine all equivalent polynomials that induce a given bijection modulo $ p^d $. The key contribution is a systematic method for reconstructing permutation polynomials from their induced functions, with complexity significantly reduced by exploiting $ p $-adic structure and coefficient elimination via modular decomposition.

ABSTRACT

This paper mainly studies problems about so called "permutation polynomials modulo $m$", polynomials with integer coefficients that can induce bijections over Z_m={0,...,m-1}. The necessary and sufficient conditions of permutation polynomials are given, and the number of all permutation polynomials of given degree and the number induced bijections are estimated. A method is proposed to determine all equivalent polynomials from the induced polynomial function, which can be used to determine all equivalent polynomials that induce a given bijection. A few problems have not been solved yet in this paper and left for open study. Note: After finishing the first draft, we noticed that some results obtained in this paper can be proved in other ways (see Remark 2). In this case, this work gives different and independent proofs of related results.

Motivation & Objective

• To determine necessary and sufficient conditions for integer polynomials to induce bijections over $ \mathbb{Z}_m $, i.e., permutation polynomials modulo $ m $.
• To estimate the number of distinct permutation polynomials of degree at most $ n $ modulo $ m $, and the number of distinct bijections they induce.
• To develop a practical method for enumerating all permutation polynomials of degree $ n $ modulo $ m $, given a known (partial) bijection.
• To address the challenge of reconstructing equivalent polynomials from the induced function, particularly when $ n \geq p $ modulo $ p^d $.

Proposed method

• Derives necessary and sufficient conditions for permutation polynomials modulo $ p^d $ using a hierarchical decomposition based on $ p $-adic resolution.
• Applies recursive decomposition of polynomials via base-$ p $ expansion of the variable, reducing the problem modulo $ p^{d-i} $ at each step.
• Uses the structure of coefficient elimination: higher-order coefficients vanish modulo $ p $, enabling recursive reconstruction from lower-level polynomials.
• Employs matrix inversion techniques (via adjugate and determinant conditions) to solve coefficient systems under $ \gcd(|A|, m) = 1 $.
• Introduces a recursive algorithm that, given a bijection, recovers all equivalent polynomials by solving for coefficients in sub-polynomials modulo decreasing powers of $ p $.
• Leverages the fact that only $ p^{\lceil (d-1)/p \rceil + 1} $ input-output pairs are needed to reconstruct polynomials of arbitrary degree modulo $ p^d $, under structured sampling.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a polynomial with integer coefficients to induce a bijection modulo $ m $?
• RQ2 How many distinct permutation polynomials of degree at most $ n $ exist modulo $ m $, and how many distinct permutations do they induce?
• RQ3 Is there an efficient method to reconstruct all equivalent polynomials that induce a given bijection over $ \mathbb{Z}_m $?
• RQ4 What is the computational complexity of reconstructing permutation polynomials from partial or full function data, especially when the degree $ n \geq p $?

Key findings

• Necessary and sufficient conditions for permutation polynomials modulo $ p^d $ are established through recursive decomposition and coefficient analysis.
• The number of distinct permutation polynomials of degree at most $ n $ modulo $ m $ is estimated, with explicit bounds derived for prime power moduli.
• The number of distinct permutations induced by such polynomials is bounded and can be estimated via the structure of the polynomial ring modulo $ m $.
• A recursive algorithm is proposed to reconstruct all equivalent polynomials that induce a given bijection modulo $ p^d $, with complexity reduced to $ O(d^3 p^{\lceil (d-1)/p \rceil + 3}) $.
• The method requires only $ p^{\lceil (d-1)/p \rceil + 1} $ input-output pairs to reconstruct a polynomial of arbitrary degree modulo $ p^d $, provided they are sampled with appropriate $ p $-adic distribution.
• The complexity reduction is due to the vanishing of higher-order coefficients modulo increasing powers of $ p $, enabling efficient recursive coefficient recovery.