Permutation Polynomials and Polynomial Generators of a General

Permutation Polynomials and GL( 2 ) Fp Permutation Polynomials

Introduction and Polynomial Generators Permutation Polynomials of a General Linear Group Representing Groups

Equivalence of Groups of Polynomials Chris Castillo An Loyola Blakeﬁeld Unexpected Result about GL( ) Fp2 MD-DC-VA Section of the MAA Fall 2016 Meeting Johns Hopkins University November 5, 2016 Theorem (Lagrange Interpolation)

Any function ϕ: Fpn → Fpn may be represented as a polynomial f (X ) ∈ Fpn [X ] according to the formula:

X n f (X ) = 1 − (X − x)p −1 ϕ(x).

x∈Fpn

Deﬁnition (Permutation Polynomial)

A polynomial f (X ) ∈ Fpn [X ] is a permutation polynomial if it induces a bijection of Fpn under evaluation.

Finite Fields and Permutation Polynomials

Permutation Polynomials and GL( ) Fp2 Let p be prime and consider the ﬁnite ﬁeld Fpn

Introduction

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Deﬁnition (Permutation Polynomial)

A polynomial f (X ) ∈ Fpn [X ] is a permutation polynomial if it induces a bijection of Fpn under evaluation.

Finite Fields and Permutation Polynomials

Permutation Polynomials and GL( ) Fp2 Let p be prime and consider the ﬁnite ﬁeld Fpn Theorem (Lagrange Interpolation) Introduction

Permutation Any function ϕ: n → n may be represented as a Polynomials Fp Fp Representing polynomial f (X ) ∈ n [X ] according to the formula: Groups Fp

Equivalence of X n Groups of f (X ) = 1 − (X − x)p −1 ϕ(x). Polynomials x∈ n An Fp Unexpected Result about GL(Fp2 ) Finite Fields and Permutation Polynomials

Permutation Polynomials and GL( ) Fp2 Let p be prime and consider the ﬁnite ﬁeld Fpn Theorem (Lagrange Interpolation) Introduction

Permutation Any function ϕ: n → n may be represented as a Polynomials Fp Fp Representing polynomial f (X ) ∈ n [X ] according to the formula: Groups Fp

Equivalence of X n Groups of f (X ) = 1 − (X − x)p −1 ϕ(x). Polynomials x∈ n An Fp Unexpected Result about GL( ) Fp2 Deﬁnition (Permutation Polynomial)

A polynomial f (X ) ∈ Fpn [X ] is a permutation polynomial if it induces a bijection of Fpn under evaluation. Example (Monomials) X m if and only if (m, pn − 1) = 1

Example (All-ones polynomials) 1 + X + X 2 + ··· + X k if and only if k ≡ 1 (mod p(pn − 1))

Example (Dickson polynomials of the ﬁrst kind) k b 2 c X k k − j g (X , a) := (−a)j X k−2j for a ∈ ∗ k k − j j Fq j=0 if and only if (k, (pn)2 − 1) = 1

Examples of Permutation Polynomials

Permutation Polynomials Example (Linear polynomials) and GL(Fp2 ) ∗ aX + b for any a ∈ Fpn and any b ∈ Fpn

Introduction

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Example (All-ones polynomials) 1 + X + X 2 + ··· + X k if and only if k ≡ 1 (mod p(pn − 1))

Example (Dickson polynomials of the ﬁrst kind) k b 2 c X k k − j g (X , a) := (−a)j X k−2j for a ∈ ∗ k k − j j Fq j=0 if and only if (k, (pn)2 − 1) = 1

Examples of Permutation Polynomials

Permutation Polynomials Example (Linear polynomials) and GL(Fp2 ) ∗ aX + b for any a ∈ Fpn and any b ∈ Fpn

Introduction Permutation Example (Monomials) Polynomials Representing m n Groups X if and only if (m, p − 1) = 1

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Example (Dickson polynomials of the ﬁrst kind) k b 2 c X k k − j g (X , a) := (−a)j X k−2j for a ∈ ∗ k k − j j Fq j=0 if and only if (k, (pn)2 − 1) = 1

Examples of Permutation Polynomials

Permutation Polynomials Example (Linear polynomials) and GL(Fp2 ) ∗ aX + b for any a ∈ Fpn and any b ∈ Fpn

Introduction Permutation Example (Monomials) Polynomials Representing m n Groups X if and only if (m, p − 1) = 1

Equivalence of Groups of Polynomials Example (All-ones polynomials) An 2 k n Unexpected 1 + X + X + ··· + X if and only if k ≡ 1 (mod p(p − 1)) Result about GL(Fp2 ) Examples of Permutation Polynomials

Permutation Polynomials Example (Linear polynomials) and GL(Fp2 ) ∗ aX + b for any a ∈ Fpn and any b ∈ Fpn

Introduction Permutation Example (Monomials) Polynomials Representing m n Groups X if and only if (m, p − 1) = 1

Equivalence of Groups of Polynomials Example (All-ones polynomials) An 2 k n Unexpected 1 + X + X + ··· + X if and only if k ≡ 1 (mod p(p − 1)) Result about GL(Fp2 ) Example (Dickson polynomials of the ﬁrst kind) k b 2 c X k k − j g (X , a) := (−a)j X k−2j for a ∈ ∗ k k − j j Fq j=0 if and only if (k, (pn)2 − 1) = 1 Example (Linear binomials) ∼ ∗ hX + bi = Cp for a ﬁxed b ∈ Fpn

Example (Linearized polynomials) ∼ Pn−1 pi hL(X )i = GL(Fpn ) where L(X ) = i=0 `i X such that the unique zero of L(X ) of 0

Example (Dickson polynomials of the ﬁrst kind)

gk (X , a) is an abelian group if and only if a ∈ {−1, 0, 1}

Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) Example (Linear monomials) ∼ ∗ haX i = Cpn−1 for a ﬁxed primitive a ∈ Fpn Introduction

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Example (Linearized polynomials) ∼ Pn−1 pi hL(X )i = GL(Fpn ) where L(X ) = i=0 `i X such that the unique zero of L(X ) of 0

Example (Dickson polynomials of the ﬁrst kind)

gk (X , a) is an abelian group if and only if a ∈ {−1, 0, 1}

Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) Example (Linear monomials) ∼ ∗ haX i = Cpn−1 for a ﬁxed primitive a ∈ Fpn Introduction

Permutation Polynomials Example (Linear binomials) Representing Groups ∼ ∗ hX + bi = Cp for a ﬁxed b ∈ Fpn Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Example (Dickson polynomials of the ﬁrst kind)

gk (X , a) is an abelian group if and only if a ∈ {−1, 0, 1}

Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) Example (Linear monomials) ∼ ∗ haX i = Cpn−1 for a ﬁxed primitive a ∈ Fpn Introduction

Permutation Polynomials Example (Linear binomials) Representing Groups ∼ ∗ hX + bi = Cp for a ﬁxed b ∈ Fpn Equivalence of Groups of Polynomials Example (Linearized polynomials) An Unexpected ∼ Pn−1 pi Result about hL(X )i = GL( pn ) where L(X ) = `i X such that the GL( ) F i=0 Fp2 unique zero of L(X ) of 0 Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) Example (Linear monomials) ∼ ∗ haX i = Cpn−1 for a ﬁxed primitive a ∈ Fpn Introduction

Example (Dickson polynomials of the ﬁrst kind)

gk (X , a) is an abelian group if and only if a ∈ {−1, 0, 1} 1 Injection: σ : G ,→ Fpn 2 Action of G on Fpn : ( σ g · σ−1(x) , x ∈ σ(G) g ∗ x := x, x ∈/ σ(G)

3 Interpolation:

X q−1 fg (X ) = 1 − (X − x) (g ∗ x)

x∈Fpn

n 4 Operation: composition and reduction modulo X p − X

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL( 2 ) n Fp Let G be a group of order at most p

Introduction

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) 2 Action of G on Fpn : ( σ g · σ−1(x) , x ∈ σ(G) g ∗ x := x, x ∈/ σ(G)

3 Interpolation:

X q−1 fg (X ) = 1 − (X − x) (g ∗ x)

x∈Fpn

n 4 Operation: composition and reduction modulo X p − X

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL( 2 ) n Fp Let G be a group of order at most p

1 Injection: σ : G ,→ n Introduction Fp

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) 3 Interpolation:

X q−1 fg (X ) = 1 − (X − x) (g ∗ x)

x∈Fpn

n 4 Operation: composition and reduction modulo X p − X

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL( 2 ) n Fp Let G be a group of order at most p

An Unexpected Result about GL(Fp2 ) n 4 Operation: composition and reduction modulo X p − X

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL( 2 ) n Fp Let G be a group of order at most p

1 Injection: σ : G ,→ n Introduction Fp Permutation 2 Action of G on Fpn : Polynomials Representing ( Groups σ g · σ−1(x) , x ∈ σ(G) Equivalence of g ∗ x := Groups of x, x ∈/ σ(G) Polynomials An 3 Interpolation: Unexpected Result about GL(F 2 ) X q−1 p fg (X ) = 1 − (X − x) (g ∗ x)

x∈Fpn Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL( 2 ) n Fp Let G be a group of order at most p

x∈Fpn

n 4 Operation: composition and reduction modulo X p − X fe (X ) = X (identity) [−1] fg (X ) = fg −1 (X ) (inverse)

Theorem The representation polynomials form a group under composition modulo X pn − X which is isomorphic to G: ∼ G = {fg (X ): g ∈ G} .

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) fg fh(X ) = fgh(X ) (closure) Introduction

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) [−1] fg (X ) = fg −1 (X ) (inverse)

Theorem The representation polynomials form a group under composition modulo X pn − X which is isomorphic to G: ∼ G = {fg (X ): g ∈ G} .

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) fg fh(X ) = fgh(X ) (closure) Introduction f (X ) = X (identity) Permutation e Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Theorem The representation polynomials form a group under composition modulo X pn − X which is isomorphic to G: ∼ G = {fg (X ): g ∈ G} .

Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) fg fh(X ) = fgh(X ) (closure) Introduction f (X ) = X (identity) Permutation e Polynomials [−1] Representing fg (X ) = fg −1 (X ) (inverse) Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Constructing Groups of Permutation Polynomials

Permutation Polynomials and GL(Fp2 ) fg fh(X ) = fgh(X ) (closure) Introduction f (X ) = X (identity) Permutation e Polynomials [−1] Representing fg (X ) = fg −1 (X ) (inverse) Groups

Equivalence of Groups of Theorem Polynomials

An The representation polynomials form a group under n Unexpected composition modulo X p − X which is isomorphic to G: Result about GL(Fp2 ) ∼ G = {fg (X ): g ∈ G} . Example (Cyclic group of order p2) The polynomials

2 1 ± X + X p−1 + X 2(p−1) + ··· + X p −p

are permutation polynomials over Fp2 .

Examples

Permutation Polynomials Example (Cyclic group of order pn) and GL(Fp2 ) For any z ∈ {1, 2,..., pn − 1} and any primitive element Introduction ξ ∈ Fpn , the polynomials Permutation Polynomials z 1−z 1−z 2 1−z pn−2 Representing ξX + ξ 1 + ξ X + (ξ X ) + ··· + (ξ X ) Groups

Equivalence of Groups of are permutation polynomials over pn . Polynomials F

An Unexpected Result about GL(Fp2 ) Examples

Equivalence of Groups of are permutation polynomials over pn . Polynomials F

An Unexpected 2 Result about Example (Cyclic group of order p ) GL(F 2 ) p The polynomials

2 1 ± X + X p−1 + X 2(p−1) + ··· + X p −p

are permutation polynomials over Fp2 . Fix a basis [β] = [β0, β1] of (Fp2 , +) over Fp Write the p-adic expansion of k as k = κ0 + κ1p

Write x ∈ Fp2 and x = λ0β0 + λ1β1

The Example of Interest: Cp2

Permutation Polynomials and GL(Fp2 )

Introduction

Permutation Polynomials Let C 2 = hgi Representing p Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Write the p-adic expansion of k as k = κ0 + κ1p

Write x ∈ Fp2 and x = λ0β0 + λ1β1

The Example of Interest: Cp2

Permutation Polynomials and GL(Fp2 )

Introduction

Permutation Polynomials Let C 2 = hgi Representing p Groups Fix a basis [β] = [β0, β1] of (Fp2 , +) over Fp Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Write x ∈ Fp2 and x = λ0β0 + λ1β1

The Example of Interest: Cp2

Permutation Polynomials and GL(Fp2 )

Introduction

An Unexpected Result about GL(Fp2 ) The Example of Interest: Cp2

Permutation Polynomials and GL(Fp2 )

Introduction

Permutation Polynomials Let C 2 = hgi Representing p Groups Fix a basis [β] = [β0, β1] of (Fp2 , +) over Fp Equivalence of Groups of Write the p-adic expansion of k as k = κ + κ p Polynomials 0 1 An Write x ∈ Fp2 and x = λ0β0 + λ1β1 Unexpected Result about GL(Fp2 ) Action: k k −1 g ∗ x = σ g · σ (λ0β0 + λ1β1) = σ g κ0+κ1p · g λ0+λ1p = σ g (κ0+λ0)+(κ1+λ1)p ( (κ + λ )β + (κ + λ )β , κ + λ < p = 0 0 0 1 1 1 0 0 (κ0 + λ0)β0 + (κ1 + λ1 + 1)β1, κ0 + λ0 ≥ p ( x + (κ β + κ β ), λ < p − κ = 0 0 1 1 0 0 x + (κ0β0 + κ1β1) + β1, λ0 ≥ p − κ0

The Example of Interest: Cp2

Permutation Polynomials Injection: and GL(Fp2 ) k κ0+κ1p σ(g ) = σ g = κ0β0 + κ1β1

Introduction

Permutation Polynomials Representing Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) The Example of Interest: Cp2

Permutation Polynomials Injection: and GL(Fp2 ) k κ0+κ1p σ(g ) = σ g = κ0β0 + κ1β1 Introduction Action: Permutation Polynomials k k −1 Representing g ∗ x = σ g · σ (λ0β0 + λ1β1) Groups κ +κ p λ +λ p Equivalence of = σ g 0 1 · g 0 1 Groups of Polynomials (κ0+λ0)+(κ1+λ1)p An = σ g Unexpected Result about ( GL( 2 ) (κ + λ )β + (κ + λ )β , κ + λ < p Fp = 0 0 0 1 1 1 0 0 (κ0 + λ0)β0 + (κ1 + λ1 + 1)β1, κ0 + λ0 ≥ p ( x + (κ β + κ β ), λ < p − κ = 0 0 1 1 0 0 x + (κ0β0 + κ1β1) + β1, λ0 ≥ p − κ0 The Example of Interest: Cp2

Permutation Polynomials and GL(Fp2 ) Cp2 = hgi

Introduction Let [β] = [β0, β1] be a basis of (Fp2 , +) over Fp Permutation Write k = κ + κ p Polynomials 0 1 Representing Groups 2 Equivalence of Example (The “Additive Representation” of Cp ) Groups of Polynomials [β ,β ] An f 0 1 (X ) = X + κ β + κ β Unexpected g κ0+κ1p 0 0 1 1 Result about 2 GL(Fp2 ) p−1 p−1 p −2 X X X −1 ` − β1 (λ0β0 + λ1β1) X

λ0=p−κ0 λ1=0 `=0 Equivalence of Representations

Permutation Polynomials and GL(Fp2 )

Introduction

Permutation Deﬁnition (Equivalence) Polynomials Representing The polynomial representations f , f 0 generated by Groups 0 σ, σ : G ,→ n , respectively, are equivalent if there exists a Equivalence of Fp Groups of group automorphism α:( n , +) → ( n , +) such that for Polynomials Fp Fp

An all g ∈ G, Unexpected f (X ) = α−1 ◦ f 0 ◦ α(X ). Result about g g GL(Fp2 ) Moreover,

[γ] [−1] [β] fg (X ) = L(X ) ◦ fg (X ) ◦ L(X ),

where L(X ) is a polynomial of the form

p L(X ) = `1X + `0X

that represents the change of basis of (Fp2 , +) from [γ] to [β].

Equivalence of Polynomials Representing Cp2

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Equivalence of Polynomials Representing Cp2

Permutation Polynomials and GL(Fp2 ) Theorem Introduction The “additive representations” of Cp2 in any two bases [β] Permutation Polynomials and [γ] of (Fp2 , +) over Fp are equivalent. Moreover, Representing Groups [γ] [−1] [β] Equivalence of f (X ) = L(X ) ◦ f (X ) ◦ L(X ), Groups of g g Polynomials

An where L(X ) is a polynomial of the form Unexpected Result about GL( ) p Fp2 L(X ) = `1X + `0X

that represents the change of basis of (Fp2 , +) from [γ] to [β]. Equivalence of Polynomials Representing Cpn

Permutation Polynomials and GL(F 2 ) p Theorem

n Introduction The “additive representations” of Cp in any two bases [β] Permutation and [γ] of (Fpn , +) over Fp are equivalent. Moreover, Polynomials Representing Groups [γ] [−1] [β] fg (X ) = L(X ) ◦ fg (X ) ◦ L(X ), Equivalence of Groups of Polynomials where L(X ) is a polynomial of the form An Unexpected Result about n−1 i GL( ) X p Fp2 L(X ) = `i X i=0

that represents the change of basis of (Fpn , +) from [γ] to [β]. Equivalence of Polynomials Representing Cp2

Permutation Polynomials Theorem and GL(Fp2 ) Let [β0, β1] and [γ0, γ1] be two bases of (Fp2 , +) over Fp . Then there exist unique r ∈ ∗ , s ∈ , and t ∈ ∗ such that Introduction Fp2 Fp Fp

Permutation Polynomials [γ0,γ1] [−1] [β0,β1] Representing fg (X ) = Nt (Ms (rX ) ◦ fg (X ) ◦ Nt (Ms (rX )), Groups

Equivalence of Groups of where Polynomials

An 1 2 p p p p+1 Unexpected Ms (X ) = p p sβ1 X + (β0 β1 − β0β1 − sβ1 )X Result about β0 β1 − β0β1 GL(Fp2 ) and

1 p p−1 p−1 Nt (X ) = p−1 p−1 (t − 1)X + (β0 − tβ1 )X . β0 − β1 The Unexpected Result

Permutation Polynomials and GL( ) Fp2 Theorem (Generators of GL(Fp2 )) ∗ Let β0, β1 ∈ 2 be linearly independent over p , let ρ ∈ 2 Introduction Fp F Fp be primitive, and let ψ, τ ∈ with τ nonzero. Then the Permutation Fp Polynomials polynomials ρX, Representing Groups

Equivalence of 1 2 p p p p+1 Groups of Mψ(X ) = p p ψβ1 X + (β0 β1 − β0β1 − ψβ1 )X Polynomials β0 β1 − β0β1 An and Unexpected Result about 1 p−1 p−1 GL(F 2 ) p p Nτ (X ) = p−1 p−1 (τ − 1)X + (β0 − τβ1 )X β0 − β1 generate a group of permutation polynomials isomorphic to the general linear group GL(Fp2 ). Are there any questions?

Permutation Polynomials and GL(Fp2 )

Introduction

Permutation Polynomials Representing Thank you for attending! Groups

Equivalence of Groups of Polynomials

An Unexpected Result about GL(Fp2 ) Permutation Polynomials and GL(Fp2 )

Introduction

Permutation Polynomials Representing Thank you for attending! Groups

Equivalence of Groups of Polynomials Are there any questions? An Unexpected Result about GL(Fp2 )