SCR-Based Representation in Finite Fields

• SCR-Based Representation is a framework that uses self-conjugate-reciprocal symmetry to define permutation polynomials over quadratic finite fields like F(q²) with rigorous bijectivity criteria.
• The methodology leverages discriminant, trace, and Möbius transformation techniques to reduce complex root testing in extended fields to manageable checks in the subfield F(q).
• This representation facilitates the construction of few-term permutation polynomials, unifying diverse approaches in finite field theory with practical applications in cryptography.

Self-Conjugate-Reciprocal (SCR)–Based Representation

The SCR-based representation is a mathematical and algebraic framework for constructing and characterizing permutation polynomials over quadratic finite fields, specifically $\mathbb{F}_{q^2}$, using a class of polynomials termed self-conjugate-reciprocal (SCR) polynomials. An SCR polynomial $C(x)\in \mathbb{F}_{q^2}[x]$ is defined by symmetry constraints linking its reciprocal and Frobenius-conjugated coefficient forms, and is leveraged to produce new permutation polynomials with provable necessary and sufficient criteria for bijectivity on the field extension. The approach gains prominence due to its ability to systematically yield few-term permutation polynomials and reduce root-testing over extension fields to manageable lower-degree instances over the smaller subfield $\mathbb{F}_q$ (Sharma et al., 2024).

1. Definition and Algebraic Properties of SCR Polynomials

For $q=p^n$ and $\mathbb{F}=\mathbb{F}_{q^2}$, a polynomial $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$ ($c_n \neq 0$) is called self-conjugate-reciprocal (SCR) if there exists a $\beta \in \mathbb{F}^*$ with $\beta^{q+1}=1$ so that

$x^n C^{(q)}(1/x) = \beta C(x)$

where $C(x)\in \mathbb{F}_{q^2}[x]$0 denotes application of the Frobenius automorphism on the coefficients. This is equivalent to the coefficient relation $C(x)\in \mathbb{F}_{q^2}[x]$1 for $C(x)\in \mathbb{F}_{q^2}[x]$2, with $C(x)\in \mathbb{F}_{q^2}[x]$3 constrained to the $C(x)\in \mathbb{F}_{q^2}[x]$4-th roots of unity in $C(x)\in \mathbb{F}_{q^2}[x]$5. Thus, SCR polynomials exhibit reciprocal symmetry under $C(x)\in \mathbb{F}_{q^2}[x]$6 and conjugation symmetry under $C(x)\in \mathbb{F}_{q^2}[x]$7; their inherent structure facilitates root-count and factorization properties crucial for field permutation analysis.

2. Explicit Forms and Root Criteria: Degrees 2 and 3

A. Degree-2 SCR polynomials have canonical form

$C(x)\in \mathbb{F}_{q^2}[x]$8

with $C(x)\in \mathbb{F}_{q^2}[x]$9 and $\mathbb{F}_q$0. The root exclusion criterion in $\mathbb{F}_q$1 (the $\mathbb{F}_q$2-th roots of unity) depends on the discriminant:

• If $\mathbb{F}_q$3 is odd, $\mathbb{F}_q$4 has no root in $\mathbb{F}_q$5 iff $\mathbb{F}_q$6 is a nonzero nonsquare in $\mathbb{F}_q$7.
• If $\mathbb{F}_q$8 is even, $\mathbb{F}_q$9 is root-free in $q=p^n$0 precisely when $q=p^n$1.

B. Degree-3 SCR polynomials take

$q=p^n$2

with appropriate conjugation-constrained coefficients. For monomial-plus-constant forms $q=p^n$3, bijectivity conditions on $q=p^n$4 reduce to the non-root criterion $q=p^n$5, and more complex forms require paired trace and cubic-residue checks over $q=p^n$6 (Sharma et al., 2024).

3. Construction of Permutation Polynomials from SCR Factors

The SCR paradigm applies these polynomials as building blocks for permutation polynomials over $q=p^n$7. The central recipe is:

• Choose an SCR polynomial $q=p^n$8 and a base function $q=p^n$9 (often binomial, satisfying $\mathbb{F}=\mathbb{F}_{q^2}$0 permutes $\mathbb{F}=\mathbb{F}_{q^2}$1 and is non-vanishing there).
• Form $\mathbb{F}=\mathbb{F}_{q^2}$2, with $\mathbb{F}=\mathbb{F}_{q^2}$3 coprime to $\mathbb{F}=\mathbb{F}_{q^2}$4.
• $\mathbb{F}=\mathbb{F}_{q^2}$5 permutes $\mathbb{F}=\mathbb{F}_{q^2}$6 iff $\mathbb{F}=\mathbb{F}_{q^2}$7 has no $\mathbb{F}=\mathbb{F}_{q^2}$8-root (and base conditions hold for $\mathbb{F}=\mathbb{F}_{q^2}$9, $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$0).

This reduces challenging root existence or permutation checks in high-degree field extensions to manageable discriminant or trace checks, parameterizable by the degree and form of $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$1. In many cases, higher-degree SCR polynomials can be reduced (via Möbius transform bijections) to root analysis within $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$2 for lower-degree factor polynomials.

4. Specialized Families and Practical Construction Templates

A notable family, $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$3, appears ubiquitously. This is re-expressed as $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$4 with $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$5 (degree 2 SCR). The permutation property is fully characterized by simple arithmetic:

• $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$6: in odd $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$7, $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$8 is a nonzero nonsquare; in even $C(x)=\sum_{i=0}^n c_i x^i \in \mathbb{F}[x]$9, $c_n \neq 0$0.
• $c_n \neq 0$1: revert to discriminant-based or trace-based root exclusion per Section 2.

Consequently, a broad spectrum of binomial, trinomial, and quadrinomial permutation polynomials derive from SCR-based factorization templates—parametrically adjustable for desired degree and field size, with diagnostics for bijectivity sourced from SCR structural algebra.

5. Reduction of Higher-Degree SCR Root Testing

The Möbius transformation $c_n \neq 0$2, with $c_n \neq 0$3, $c_n \neq 0$4, acts as a bijection $c_n \neq 0$5. When composed with a degree-$c_n \neq 0$6 SCR polynomial, $c_n \neq 0$7 admits a numerator in $c_n \neq 0$8; determining if $c_n \neq 0$9 has a root in $\beta \in \mathbb{F}^*$0 reduces further to a solvability query of a degree-$\beta \in \mathbb{F}^*$1 polynomial in $\beta \in \mathbb{F}^*$2. This algebraic reduction operationalizes the SCR method for higher-degree constructions and ensures that root detection—and thus permutation property checking—remains computationally tractable for all SCR-based polynomials.

6. Significance and Applications in Finite Field Theory

The SCR-based representation synthesizes permutation polynomial theory for quadratic extensions by shifting root characterization and permutation conditions from the large field $\beta \in \mathbb{F}^*$3 to arithmetic in the subfield $\beta \in \mathbb{F}^*$4. This has direct algebraic and cryptographic consequence, underpinning efficient classification and generation of polynomial permutations with prescribed structural properties. The method subsumes several previously scattered construction paradigms, yielding a unified and theoretically grounded framework for polynomial permutation generation, root-detection reductions, and the systematic expansion of known classes (Sharma et al., 2024).

7. Summary Table: SCR Polynomial Types and Permutation Criteria

Degree	Canonical SCR Form	No $\beta \in \mathbb{F}^*$5-root Criterion	Specific Application
2	$\beta \in \mathbb{F}^*$6	Odd $\beta \in \mathbb{F}^*$7: $\beta \in \mathbb{F}^*$8 nonsquare; Even $\beta \in \mathbb{F}^*$9: trace	Families $\beta^{q+1}=1$0
3	$\beta^{q+1}=1$1	Odd $\beta^{q+1}=1$2: $\beta^{q+1}=1$3 not root; Even $\beta^{q+1}=1$4: complex trace/cube tests	Monomial-plus-constant, trinomial case
$\beta^{q+1}=1$5	By $\beta^{q+1}=1$6 composition, numerator $\beta^{q+1}=1$7 in $\beta^{q+1}=1$8	Solvability of $\beta^{q+1}=1$9 in $x^n C^{(q)}(1/x) = \beta C(x)$0	Higher-degree PPs, quadrinomials

This table encapsulates the algebraic hierarchy and necessary conditions for SCR-based permutation polynomial construction. The approach fundamentally enhances the systematic modeling, generation, and structural analysis of polynomial permutations over quadratic extension fields.