Converted to Latex (about Symmetric polynomials,Galois theory and primality testing)

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Enhancing the Frobenius Primality Test Using Cubic Recurrence Relations

Introduction

In the pursuit of developing robust primality tests, the integration of symmetric polynomials, linear recurrence relations, and the Frobenius automorphism within the framework of Galois Theory offers a profound and efficient methodology. Building upon the foundational understanding of how the Frobenius automorphism cyclically permutes the roots of an irreducible cubic polynomial, we now explore the extension of this approach by incorporating additional sequence terms $V_{p+2}$ and $V_{p+3}$. This extension aims to fortify the primality test, thereby enhancing its reliability and reducing the likelihood of pseudoprimes passing the test.

Recurrence Relation for Cubic Polynomials

Consider the irreducible cubic polynomial over the finite field $\mathbb{F}p$: $$ f(x) = x^3 + Sx^2 + Px + Q = 0 $$ with distinct roots $\alpha$, $\beta$, $\gamma$ in the extension field $\mathbb{F}{p^3}$.

Defining the Sequence $V_k$

We define the sequence $V_k$ as the sum of the $k$-th powers of the roots:
$$
V_k = \alpha^k + \beta^k + \gamma^k \quad \text{for } k = 0, 1, 2, \dots
$$
From Vieta’s formulas, we have:
$$
\alpha + \beta + \gamma = -S, \quad \alpha\beta + \alpha\gamma + \beta\gamma = P, \quad \alpha\beta\gamma = -Q
$$

Establishing the Correct Recurrence Relation

Given the minimal polynomial $f(x) = 0$, each root satisfies:
$$
\alpha^3 = -S\alpha^2 – P\alpha – Q
$$
$$
\beta^3 = -S\beta^2 – P\beta – Q
$$
$$
\gamma^3 = -S\gamma^2 – P\gamma – Q
$$

Multiplying each equation by $\alpha^{k-3}$, $\beta^{k-3}$, $\gamma^{k-3}$ respectively and summing, we obtain the linear recurrence relation:
$$
V_k = -S V_{k-1} – P V_{k-2} – Q V_{k-3} \quad \text{for } k \geq 3
$$

Frobenius Automorphism and Root Permutation

Cyclic Permutation of Roots

In the context of finite fields, the Frobenius automorphism $\sigma$ maps each root to its $p$-th power:
$$
\sigma(\alpha) = \alpha^p = \beta
$$
$$
\sigma(\beta) = \beta^p = \gamma
$$
$$
\sigma(\gamma) = \gamma^p = \alpha
$$
This cyclic permutation arises because $f(x)$ is irreducible over $\mathbb{F}_p$, ensuring that the Frobenius automorphism generates the entire Galois group, which is cyclic of order 3 in this cubic case.

Determining Special Values $V_p$, $V_{p+1}$, and $V_{p+2}$

Value of $V_p$

$$
V_p = \alpha^p + \beta^p + \gamma^p = \beta + \gamma + \alpha = \alpha + \beta + \gamma = -S
$$
Result:
$$
V_p = -S
$$

Value of $V_{p+1}$

$$
V_{p+1} = \alpha^{p+1} + \beta^{p+1} + \gamma^{p+1} = \alpha\beta + \beta\gamma + \gamma\alpha = P
$$
Result:
$$
V_{p+1} = P
$$

Value of $V_{p+2}$

Using the recurrence relation:
$$
V_{p+2} = -S V_{p+1} – P V_p – Q V_{p-1}
$$
Given that $V_p = -S$ and considering the congruence class of $p$ modulo 3, we analyze three cases:

1. Case 1: $p \equiv 0 \pmod{3}$

• Implications: $p$ is a multiple of 3.
• Sequence Behavior:
  $$
  V_{p-1} = V_{3k} = V_0 = 3 \quad (\text{since } \alpha^0 + \beta^0 + \gamma^0 = 3)
  $$
• Computation:
  $$
  V_{p+2} = -S \cdot P – P \cdot (-S) – Q \cdot 3 = -SP + SP – 3Q = -3Q
  $$
• Result:
  $$
  V_{p+2} = -3Q
  $$

1. Case 2: $p \equiv 1 \pmod{3}$

• Implications: $p = 3k + 1$
• Sequence Behavior:
  $$
  V_{p-1} = V_{3k+1 -1} = V_{3k} = V_0 = 3
  $$
• Computation:
  $$
  V_{p+2} = -S \cdot P – P \cdot (-S) – Q \cdot 3 = -SP + SP – 3Q = -3Q
  $$
• Result:
  $$
  V_{p+2} = -3Q
  $$

1. Case 3: $p \equiv 2 \pmod{3}$

• Implications: $p = 3k + 2$
• Sequence Behavior:
  $$
  V_{p-1} = V_{3k+2 -1} = V_{3k+1} = V_1 = -S
  $$
• Computation:
  $$
  V_{p+2} = -S \cdot P – P \cdot (-S) – Q \cdot (-S) = -SP + SP + QS = QS
  $$
• Result:
  $$
  V_{p+2} = QS
  $$

Summary of Special Values:

Congruence Class	$V_p$	$V_{p+1}$	$V_{p+2}$
$p \equiv 0 \pmod{3}$	$-S$	$P$	$-3Q$
$p \equiv 1 \pmod{3}$	$-S$	$P$	$-3Q$
$p \equiv 2 \pmod{3}$	$-S$	$P$	$QS$

Primality Testing Conditions

For a candidate number $n = p$ to be prime, the following congruence conditions must simultaneously hold:
$$
V_p \equiv -S \pmod{p}
$$
$$
V_{p+1} \equiv P \pmod{p}
$$
$$
V_{p+2} \equiv
\begin{cases}
-3Q \pmod{p} & \text{if } p \equiv 0, 1 \pmod{3} \
QS \pmod{p} & \text{if } p \equiv 2 \pmod{3}
\end{cases}
$$

These conditions are derived from the properties of the Frobenius automorphism and the established recurrence relations.

Matrix Representation of the Recurrence Relation

To facilitate efficient computation and analysis, the cubic recurrence relation can be elegantly represented in matrix form. Define the state vector as:
$$
\mathbf{S}k = \begin{bmatrix} V{k} \ V_{k+1} \ V_{k+2} \end{bmatrix}
$$
The transition matrix $\mathbf{T}$ is then:
$$
\mathbf{T} = \begin{bmatrix}
0 & 1 & 0 \
0 & 0 & 1 \
-Q & -P & -S \
\end{bmatrix}
$$
This matrix encapsulates the recurrence relation:
$$
V_k = -S V_{k-1} – P V_{k-2} – Q V_{k-3}
$$
Ensuring that $\mathbf{T}$ is invertible is crucial for the test’s strength, as it guarantees that the sequence $V_k$ evolves deterministically without degeneracies.

Conclusion

By meticulously deriving the special values of $V_p$, $V_{p+1}$, and $V_{p+2}$ in terms of the polynomial coefficients $S$, $P$, $Q$ and considering the prime $p$’s congruence class modulo 3, we establish a robust framework for primality testing. This approach, grounded in Galois Theory and symmetric polynomial principles, leverages the Frobenius automorphism to impose stringent congruence conditions that primes must satisfy, thereby enhancing the reliability and strength of the primality test.

Key Takeaways

• Invertible Transition Matrix: Incorporating the $x^2$ term ensures that the transition matrix $\mathbf{T}$ is invertible, which is essential for maintaining the sequence’s integrity.
• Multiple Congruence Conditions: Verifying several sequence terms across different congruence classes reduces the likelihood of pseudoprimes passing the test.
• Matrix Representation Benefits: Facilitates efficient computation and provides a foundation for deeper analytical insights into the sequence’s behavior.

Further Reading

1. Abstract Algebra by David S. Dummit and Richard M. Foote
2. Galois Theory by Ian Stewart
3. Finite Fields by Rudolf Lidl and Harald Niederreiter

Stay Engaged and Continue Exploring the Elegant Synergies of Algebra and Number Theory! 😊

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