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Outline Preliminaries Part 1 Part 2 Q

Non-extendable Fq-quadratic Perfect Nonlinear Maps

Ferruh ¨ Ozbudak (joint work with Alexander Pott)

Department of Mathematics and Institute of Applied Mathematics, METU, Ankara, Turkey. (joint work with Alexander Pott.) Special Semester on Applications of Algebra and Number Theory, Emerging Applications of Finite Fields, RICAM, Linz, AUSTRIA.

December 09-13, 2013

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 1/57

Outline Preliminaries Part 1 Part 2 Q

Outline

Part 1 (uniqueness)

special case general case

Part 2 (existence)

notations and some results examples

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 2/57

Outline Preliminaries Part 1 Part 2 Q

Notations

Fq: Finite Field, where q is a power of an odd prime.

n, m: positive integers with m | n.

TrFqn/Fqm (x) = x + xqm + · · · + xq( n

m −1)m

NormFqn/Fqm (x) = x · xqm · · · · · xq( n

m −1)m

Tr = TrFq3/Fq and Norm = NormFq3/Fq

K ∗ = K\ {0}

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 3/57

Outline Preliminaries Part 1 Part 2 Q

Preliminaries

arbitrary Fq-quadratic form f on Fq3 f :Fq3 → Fq x −→ Tr(ax2 + bxq+1) a, b ∈ Fq3. arbitrary Fq-quadratic map from Fq3 to F3

q.

F :Fq3 → F3

q

x −→

         

f1(x) f2(x) f3(x)

          ,

where f1, f2, f3 are Fq-quadratic forms on F3

q.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 4/57

Outline Preliminaries Part 1 Part 2 Q

Preliminaries

arbitrary Fq-quadratic map F from Fq3 to F2

q

F :Fq3 → F2

q

x −→

• f1(x)

f2(x)

• ,

where f1, f2 are Fq-quadratic forms on Fq3.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 5/57

Outline Preliminaries Part 1 Part 2 Q

Equivalence

Definition 1 Let

• f1

f2

• ,
• g1

g2

• be arbitrary Fq-quadratic maps from Fq3 to Fq2.

We call that they are equivalent if there exists an Fq-lineralized permutation polynomial L(x) ∈ Fq3 [x] and an invertible 2 × 2 matrix [ai,j] with entries from Fq such that

[ai,j] ·

• f1(x)

f2(x)

• =
• g1(L(x))

g2(L(x))

• for all x ∈ Fq3.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 6/57

Outline Preliminaries Part 1 Part 2 Q

Remark

Remark 1 Here we give notations and definitions for Fq-quadratic maps from

Fq3 to Fq2. It is simple to generalize these to Fq-quadratic maps

from Fqn to Fqr with 1 ≤ r ≤ n. Remark 2 The equivalence above would seem rather restricted at first. However we show that our results also hold if we change the equivalence above with Extended Affine (EA) Equivalence or with Carlet-Charpin-Zinoviev (CCZ) Equivalence as well.

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Outline Preliminaries Part 1 Part 2 Q

Definition

Definition 2 Let 1 ≤ r ≤ 3, a ∈ F∗

q3 be.

F : Fq-quadratic maps from Fq3 to Fqr DF,a: the difference map from F3

q to Fr q.

DF,a :Fq3 → Fqr x −→ F(x + a) − F(x) − F(a) We call F perfect nonlinear or (q3, qr)-bent if the cardinality of the set

• x ∈ F3

q : DF,a(x) = b

• is the same and is equal to q3−r for all

a ∈ F∗

q3 and b ∈ Fqr.

If r = 3, F : Planar, If r = 1, F : Bent.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 8/57

Outline Preliminaries Part 1 Part 2 Q

Problem

Problem 1 Classification of Fq-quadratic perfect nonlinear maps from Fqn to

Fqr, where 1 ≤ r ≤ n.

Classified completely if n ≥ 1, r = 1 (non-degenerate Fq-quadratic form) n = 2, r = 2 (Dickson, finite field) n = 3, r = 3 (Menichetti, finite field or twisted finite field) Our results We give a complete classification of the case n = 3, r = 2. Namely, we prove that all Fq-quadratic perfect nonlinear maps from

Fq3 to Fq2 are equivalent. Also we give a geometric method to find

an equivalence between two given Fq-quadratic (q3, q2)-bent maps explicitly.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 9/57

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Content

Part1 First consider some special cases using elementary techniques. The general case is more difficult, we use some results from Algebraic Geometry (in particular, Bezout’s Theorem). These results do not generalize to other (n, r) with 1 ≤ r ≤ n in general. Part2: a corollary: no non-extendable Fq-quadratic (3, 2) perfect nonlinear (PN) map. a proposition: no non-extendable Fq-quadratic (n, 1) PN map. existence of non-extendable Fq-quadratic (4, 3) PN map (in a sense an “atomic” structure).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 10/57

Outline Preliminaries Part 1 Part 2 Q

Part 1: Some Special and Elementary Cases

Lemma 1 Let {w1, w2} ⊆ Fq3 be a subset.

Tr(w1x2) Tr(w2x2)

• is (q3, q2)-bent ⇐⇒ {w1, w2} is linear

independent over Fq.

Tr(w1xq+1) Tr(w2xq+1)

• is (q3, q2)-bent ⇐⇒ {w1, w2} is linear

independent over Fq

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Outline Preliminaries Part 1 Part 2 Q

Proposition 1 Let {w1, w2} ⊆ Fq3 be an Fq-linearly independent subset and put w = w2

w1 . Then

Tr(w1x2) Tr(w2x2)

• ∼

Tr(x2) Tr(wx2)

• Tr(w1xq+1)

Tr(w2xq+1)

• ∼

Tr(xq+1) Tr(wxq+1)

• Ferruh ¨

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Outline Preliminaries Part 1 Part 2 Q

Proposition 1 Let {w1, w2} ⊆ Fq3 be an Fq-linearly independent subset and put w = w2

w1 . Then

Tr(w1x2) Tr(w2x2)

• ∼

Tr(x2) Tr(wx2)

• Tr(w1xq+1)

Tr(w2xq+1)

• ∼

Tr(xq+1) Tr(wxq+1)

• Proposition 2

Let w1, w2 ∈ Fq3\Fq. Then

• Tr(x2)

Tr(w1x2)

• ∼
• Tr(x2)

Tr(w2x2)

• and both are (q3, q2)bent.
• Tr(xq+1)

Tr(w1xq+1)

• ∼
• Tr(xq+1)

Tr(w2xq+1)

• and both are (q3, q2) -bent.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 12/57

Outline Preliminaries Part 1 Part 2 Q

Sketch of the proof of Prop. 2:

Determine a0, a1, a2 ∈ Fq such that

1 w2 = a0 + a1w1 + a2w2

• 1. Here

(a1, a2) (0, 0) as w2 Fq. There exist a, b, c, d ∈ Fq with (c, d) (0, 0) such that

a + bw1 c + dw1

= 1

w2

.

Consider H(x) =

• Tr((a + bw1)x2)

Tr((c + dw1)x2)

• .

H(x) is (q3, q2)-bent. Using Proposition 1, H(x) ∼

• Tr(x2)

Tr( c+dw1

a+bw1 x2)

• =
• Tr(x2)

Tr(w2x2)

• .

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Outline Preliminaries Part 1 Part 2 Q

Theorem 1 Let {w1, w2} , {w3, w4} ⊆ Fq3 be linearly independent over Fq. Then

Tr(w1x2) Tr(w2x2)

• ∼

Tr(w3xq+1) Tr(w4xq+1)

• and they are (q3, q2)-bent.

Sketch of the Proof: Note that x + xq − xq2 is a linearized permutation polynomial over Fq3 as gcd(1 + t − t2, t3 − 1) = 1. Hence L(x) = b(x + xq − xq2) ∈ Fq3 [x] is a linearized permutation polynomial for all b ∈ F∗

q3.

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Outline Preliminaries Part 1 Part 2 Q

Sketch of the Proof (Cont.)

Claim: There exists b ∈ F∗

q3 and α1, α2 ∈ Fq3 such that

Tr(L(x)2) = Tr(α1xq+1) and Tr(wL(x)2) = Tr(α2xq+1)

for all x ∈ Fq3. In the proof of the claim we use existence of a basis which is trace orthogonal to the basis

• 1, w, w2

. Then

Tr(α1xq+1) Tr(α2xq+1)

• ∼

Tr(w1x2) Tr(w2x2)

• .

The equivalence of

Tr(α1xq+1) Tr(α2xq+1)

• ∼

Tr(w3xq+1) Tr(w4xq+1)

• follows from Proposition 2.

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The General Case

Let h(x) be non-degenerate Fq-quadratic form on Fq3 and η ∈ F∗

q

be non-square element in Fq . Then it is well-known there exists a linearized permutation polynomial L(x) ∈ Fq3 [x] such that either h(x) = Tr(x2) for all x ∈ Fq3 or h(x) = ηTr(x2) for allx ∈ Fq3. Hence, any Fq-quadratic (q3, q2)-bent is of the form F(x) =

• Tr(x2)

Tr(θx2 + wxq+1)

• without lost of generality. We fix such F(x).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 16/57

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The General Case

Proposition 3 F is (q3, q2)-bent if and only if the polynomial T3 + A2T2 + A1T + A0 ∈ Fq[T], is irreducible over Fq, where A2 =

Tr(2θ),

A1 =

Tr

• (2θ)q+1 − w2

,

A0 =

Norm(2θ) − Tr

• 2θw2q

+ 2Norm(w).

.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 17/57

Outline Preliminaries Part 1 Part 2 Q

The General Case

There are many such (q3, q2)-bent maps. The number of choices

(θ, w) ∈ Fq3 × F∗

q3 giving (q3, q2)-bent.

168 for q = 3 4680 for q = 5 37296 for q = 7 172080 for q = 9 579480 for q = 11 1587768 for q = 13 Put w0 ∈ Fq3\Fq and fix G(x) =

• Tr(xq+1)

Tr(w0xq+1)

• .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 18/57

Outline Preliminaries Part 1 Part 2 Q

The General Case

Lemma 2 Assume that F is (q3, q2)-bent and there exists a linearized permutation polynomial L(x) = a0x + aq

1xq + aq2 2 xq2 ∈ Fq3[x]

such that

          

a2

0 + a2 1 + a2 2 = 0, and

(a2

0θ + a2 1θq2 + a2 2θq) + (a0a1wq2 + a0a2w + a1a2wq) = 0.

Then F ∼ G.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 19/57

Outline Preliminaries Part 1 Part 2 Q

Sketch of the Proof:

There exist certain δ1, δ2 ∈ Fq3 such that

Tr(L(x)2) = Tr

• x2

a2

0 + a2 1 + a2 2

• + δ1xq+1

and

Tr

• θL(x)2 + wL(x)q+1

= Tr

• x2

θa2

0 + θq2a2 1 + θqa2 2

• +
• wa0a2 + wq2a1a0 + wqa1a2
• + δ2xq+1

,

for all x ∈ Fq3. This follows from an arithmetic manipulation.

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Outline Preliminaries Part 1 Part 2 Q

Some nice properties and observations:

Lemma 3 Assume that F is (q3, q2)-bent and let S be the set consisting of the triples (a0, a1, a2) ∈ F3

q3 such that

L(x) = a0x + aq

1xq + aq2 2 xq2 ∈ Fq3[x] is a permutation and the

condition of Lemma 2 are satisfied. If S is not empty, then we have the following properties on S:

1

For c ∈ F∗

q3, (a0, a1, a2) ∈ S ⇒ (ca0, ca1, ca2) ∈ S

2

(a0, a1, a2) ∈ S ⇒ (aq

1, aq 2, aq 0) ∈ S.

Moreover (a0, a1, a2) ∈ S ⇒ (a0, a1, a2) (0, 0, 0).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 21/57

Outline Preliminaries Part 1 Part 2 Q

Definition

PG(2, K): the projective plane over K, which is a field. (a0 : a1 : a2): an arbitrary element of PG(2, K). (a0 : a1 : a2) = (ta0 : ta1 : ta2) for any t ∈ K∗.

Using Lemma 3 we define ˜

S as follows.

Definition 3

˜ S: the set consisting of (a0 : a1 : a2) ∈ PG(2, Fq3) such that (a0, a1, a2) ∈ S. We have ˜ S ∅ ⇐⇒ S ∅. ι : PG(2, Fq3) → PG(2, Fq3) (a0 : a1 : a2) → ι(a0 : a1 : a2) := (a0 : a1 : a2).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 22/57

Outline Preliminaries Part 1 Part 2 Q

Definition (Cont.)

(b0 : b1 : b2) ∈ ι(PG(2, Fq3)) ⇐⇒ ∃c ∈ F

∗ q3 and a0, a1, a2 ∈ Fq3 with

(a0, a1, a2) (0, 0, 0) such that b0 = ca0, b1 = ca1 and b2 = ca2.

E: the action of PG(2, Fq3) E : PG(2, Fq3)

→ PG(2, Fq3) (a0 : a1 : a2) →

E(a0 : a1 : a2) := (aq

1 : aq 2 : aq 0).

We denote this action over PG(2, Fq3) by E as well.

(E ◦ E ◦ E): the identity action on PG(2, Fq3) and on ι

• PG(2, Fq3)
• but not identity on PG(2, Fq3).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 23/57

Outline Preliminaries Part 1 Part 2 Q

Definition (Cont.)

T1: the set consisting of (a0 : a1 : a2) ∈ PG(2, Fq3) such that

a2

0 + a2 1 + a2 2 = 0.

T2: the set consisting of (a0 : a1 : a2) ∈ PG(2, Fq3) such that (a2

0θ + a2 1θq2 + a2 2θq) + (a0a1wq2 + a0a2w + a1a2wq) = 0.

T = T1 ∩ T2.

Similarly over PG(2, Fq3) define T 1, T 2 and put T = T 1 ∩ T 2.

˜ S, T1, T2 and T are closed under the action E. ι(T1), ι(T2), ι(T ), T 1, T2 and T are closed under the action E.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 24/57

Outline Preliminaries Part 1 Part 2 Q

A simple lemma is given now, to be used later. Lemma 4 For P = (a0 : a1 : a2) ∈ PG(2, Fq3) P ∈ ι(PG(2, Fq3)) ⇐⇒ (a0 : a1 : a2) = (aq3

0 : aq3 1 : aq3 2 ).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 25/57

Outline Preliminaries Part 1 Part 2 Q

Consider fixed points of the action E. Lemma 5 P ∈ PG(2, Fq3) E(P) = P ⇐⇒ there exists a ∈ F∗

q3such that P = (a : aq2 : aq).

The number of P ∈ T such that E(P) = P is N − 1 q − 1 , where N: the number of a ∈ Fq3 satisfying the system

           Tr(a2) = 0 and Tr(θa2 + waq+1) = 0.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 26/57

Outline Preliminaries Part 1 Part 2 Q

Lemma 6 F : (q3, q2)-bent. The number N of solutions a ∈ Fq3 of the system

           Tr(a2) = 0 and Tr(θa2 + waq+1) = 0

is q. We prove it using exponential sums. Corollary 1 Let a ∈ F∗

q3 be a nonzero solution of the system in Lemma 6. Put

Q = (a : aq2 : aq) ∈ PG(2, Fq3). Then E(Q) = Q. Q ∈ T . There is no P ∈ T such that E(P) = P and Q P.

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Outline Preliminaries Part 1 Part 2 Q

Lemma 7 For P ∈ PG(2, Fq3).

(E ◦ E)(P) = P ⇐⇒ ∃a ∈ F∗

q6such that P = (a : aq2 : aq4).

The number P ∈ T such that (E ◦ E)(P) = P is M − 1 q2 − 1, where M: the number of a ∈ Fq6 satisfying the system

           TrFq6/Fq2(a2) = 0 and TrFq6/Fq2(θa2 + wq2aq2+1) = 0.

(1)

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Outline Preliminaries Part 1 Part 2 Q

Lemma 8 F is (q3, q2)-bent. The number M of solutions a ∈ Fq6 of the system

Trq6/q2(a2) = 0 and Trq6/q2(θa2 + wq2aq2+1) = 0

is q2.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 29/57

Outline Preliminaries Part 1 Part 2 Q

Corollary 2 Let a ∈ F∗

q3 be a nonzero solution of the system in Lemma 8. Let

Q = (a : aq2 : aq) ∈ PG(2, Fq3) be the point in Corollary 1. Then

(E ◦ E)(Q) = Q.

There is no P ∈ T such that (E ◦ E)(P) = P and Q P. Sketch of the proof: E(P) = P =⇒ (E ◦ E)(P) = P.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 30/57

Outline Preliminaries Part 1 Part 2 Q

From now on: Assume that F is (q3, q2)-bent Q ∈ T is the point defined in Corollary 2 so that E(Q) = Q. Another simple but useful Lemma. Lemma 9 If P ∈ PG(2, Fq3) and E(P) = Q, then P = Q.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 31/57

Outline Preliminaries Part 1 Part 2 Q

Recall and observe that:

T 1 : x2

0 + x2 1 + x2 2 = 0 in PG(2, Fq3) .

T 1 is absolutely irreducible. T 2 :

• θx2

0 + θq2x2 1 + θqx2 2

• +
• wq2x0x1 + wx0x2 + wqx1x2
• = 0

in PG(2, Fq3) .

T 1 and T 2 have no common component.

Note that T = T 1 ∩ T 2, deg(T 1) = deg(T 2) = 2. By Bezout’s Theorem, 4 =

• P∈T

I(P; T 1 ∩ T 2).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 32/57

Outline Preliminaries Part 1 Part 2 Q

Recall also that Q ∈ T . Lemma 10 I(Q; T 1 ∩ T 2) = 1. Sketch of the proof: T 1 is simple at Q easily. We prove the following claim. Claim: For c ∈ Fq3 (c + 2θ)x + wxq + wq2xq2 ∈ Fq3[x] is a permutation polynomial. Using the claim above and some algebraic manipulations we prove that the tangents of T 1 and T 2 are distinct at Q, and also T 2 is simple at Q.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 33/57

Outline Preliminaries Part 1 Part 2 Q

Corollary 3

T \{Q} is nonempty.

Proof. 4 =

• P∈T

I(P; T 1 ∩ T 2) and I(Q; T 1 ∩ T 2) = 1

• Ferruh ¨

Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 34/57

Outline Preliminaries Part 1 Part 2 Q

Choose and fix a point P1 in T \{Q}. Moreover put P2 = E(P1), P3 = E(P2). Proposition 4 P2 Q P2 P1 P3 Q P3 P1 P3 P2 E(P3) = P1 Moreover T = {Q, P1, P2, P3} and ι(T ) = T .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 35/57

Outline Preliminaries Part 1 Part 2 Q

Sketch of the proof of Prop. 4:

If P2 = Q, then E(P1) = Q and hence P1 = Q by Lemma 9, which is a contradiction. If P2 = P1, then E(P1) = P1 and hence P1 = Q by Corollary 1, which is a contradiction. Hence, |{Q, P1, P2}| = 3 E(P3) ∈ {Q, P1, P2, P3} as |T | ≤ 4 by Bezout’s Theorem. E(P3) Q as P3 = E(P2) and P2 Q. E(P3) P3 as P3 Q E(P3) P2 as E(P3) = (E ◦ E)(P2) = P2 is impossible using P2 Q. Hence, E(P3) = P1. Then, (E ◦ E ◦ E)(P1) = P1. By Lemma 4, P1 ∈ ι(T ).

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Outline Preliminaries Part 1 Part 2 Q

Another technical lemma. Lemma 11 b0, b1, b2 ∈ Fq3 with (b0, b1, b2) (0, 0, 0) B(x) = b0x + bq

1xq + bq2 2 xq2 ∈ Fq3[x]

P = (b0 : b1 : b2) ∈ PG(2, Fq3) W: the set of the roots of B(x) in Fq3. Then W is an Fq-linear subspace of Fq3. The following three statements are equivalent: i) dimFq W = 2 ii) E(P) = P iii) There exist α, c ∈ F∗

q3 such that

B(x) = α

• cx + cqxq + cq2xq2

∈ Fq3[x].

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 37/57

Outline Preliminaries Part 1 Part 2 Q

Recall we fixed Q ∈ T and we choosed P1 ∈ T \Q. Put P1 = (a0 : a1 : a2), P2 = (aq

1 : aq 2 : aq 0) and P3 = (aq2 2 : aq2 0 : aq2 1 ).

Let L(x) = a0x + aq

1xq + aq2 2 xq2 and W: the roots of L(x) in Fq3.

Lemma 12 dimFq W {2, 3}. Proof. dimFq W 3 is clear. dimFq W 2 by Lemma 11 directly.

• Ferruh ¨

Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 38/57

Outline Preliminaries Part 1 Part 2 Q

Another tricky result using Bezout’s Theorem: Lemma 13 dimFq W 1. Sketch of the Proof: Assume dimFq W = 1. Put

ϕ : Fq3 → Fq3

x

→

a0x + aq

1xq + aq2 2 xq2.

Let U = Image(ϕ). Then dimFq U = 2. Using Lemma 11 we obtain c ∈ F∗

q3 such that

cy + cqyq + cq2yq2 = 0 for all y ∈ U.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 39/57

Outline Preliminaries Part 1 Part 2 Q

Sketch of the Proof: (Cont.)

Then we conclude that P1, P2 and P3 are on

H : cx0 + cq2x1 + cqx2 = 0.

Hence we have

• P∈H∩T 1

I(P; H ∩ T 1) ≥ 3, But deg(H) = 1, deg(T 1) = 2 and by Bezout’s Theorem 2 =

• P∈H∩T 1

I(P; H ∩ T 1), which is a contradiction.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 40/57

Outline Preliminaries Part 1 Part 2 Q

Theorem 2 F : (q3, q2) − bent, F =

• Tr(x2)

Tr(θx2 + wxq+1)

• , w 0.

T1 : a2

0 + a2 1 + a2 2 = 0, and

T2 :

• a2

0θ + a2 1θq2 + a2 2θq

+

• a0a1wq2 + a0a2w + a1a2wq

= 0

There exists Q, P ∈ PG(2, Fq3) such that

T1 ∩ T2 = {Q, P, E(P), (E ◦ E)(P)}. Put P = (a0 : a1 : a2) and

L(x) = a0x + aq

1xq + aq2 2 xq2.

Then L(x) is permutation and F ∼

• Tr(xq+1)

Tr(w0xq+1)

• for any w0 ∈ Fq3\Fq using L(x).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 41/57

Outline Preliminaries Part 1 Part 2 Q

An Algorithm:

Step 1(Case w 0): Solve x2

0 + x2 1 + x2 2 = 0, and

• x2

0θ + x2 1θq2 + x2 2θq

+

• x0x1wq2 + x0x2w + x1x2wq

= 0

in PG(2, Fq3) and choose a solution P1 with E(P1) P1. Put P1 = (a0 : a1 : a2) and L(x) = a0x + aq

1xq + aq2 2 xq2.

Put

δ1 = 2a0aq

1 + 2aq 0a2 + 2a1aq 2,

δ2 =

• 2θa0aq

1 + 2θqaq 0a2 + 2θq2a1aq 2

• + waq+1

+

wq2a0aq

2 + wqaq 0a1 + wq2aq+1 1

+ waq

1a2 + wqaq+1 2

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 42/57

Outline Preliminaries Part 1 Part 2 Q

An Algorithm:

Step 1(Case w = 0): Choose b ∈ F∗

q3 such that

b2 + b2q + b2q2 = 0 and θb2 + (θb2)q + (θb2)q2 = 0 Put L(x) = b(x + xq − xq2),

δ1 = 2b2 − 2b2q − 2b2q2 and δ2 = 2θb2 − 2(θb2)q − 2(θb2)q2.

Step 2:Put δ = δ2

δ1 . If δ1 is a (q + 1)-power inF∗ q3, then choose

c1 ∈ F∗

q3 such that cq+1 1

= δ1 and also put µ1 = 1. If δ1 is not

a(q + 1)-power in F∗

q3, then choose c1 ∈ F∗ q3 and µ1 ∈ F∗ q such

that cq+1

1

= µ1δ1. Put L1(x) = x

c1 . We have

µ1 µ1        Tr

• δ1L1(x)q+1

Tr

• δ2L1(x)q+1

       =        Tr

• xq+1

Tr

• δxq+1

       .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 43/57

Outline Preliminaries Part 1 Part 2 Q

An Algorithm:

Step 3: G(x) =

• Tr(xq+1)

Tr(w0xq+1)

• andw0 ∈ Fq3\Fq.

Choose a1,1, a1,2, a2,1, a2,2 ∈ Fq such that a1,1 + a1,2δ a2,1 + a2,2δ = 1 w0

.

Put δ(2)

1

= a1,1 + a1,2δ and δ(2)

2

= a2,1 + a2,2δ.

Then we have

• a1,1

a1,2 a2,1 a2,2

       Tr

• xq+1

Tr

• δxq+1

       =            Tr

• δ(2)

1 xq+1

Tr

• δ(2)

2 xq+1

           .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 44/57

Outline Preliminaries Part 1 Part 2 Q

An Algorithm:

Step 4: Choose c2 ∈ F∗

q3 and µ2 ∈ F∗ q such that cq+1 2

= µ2δ(2)

1 .

Put L2(x) = x

c2 . Then we have

µ2 µ2            Tr

• δ(2)

1 L2(x)q+1

Tr

• δ(2)

2 L2(x)q+1

           = G(x).

Combining we get

µ2 µ2

a1,1 a1,2 a2,1 a2,2

µ1 µ1

• F ((L ◦ L1 ◦ L2)(x)) = G(x).

Corollary 4 There is only one Fq-quadratic (q3, q2)-bent map up to

• equivalence. Finding this equivalence is the same as finding the

intersection of two degree 2 curves T1 and T2 in principle.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 45/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

The results above do not generalize to (qn, qr)-bent maps with 1 ≤ r ≤ n in general. Example 1 Put q = 3. As {w1, w2} runs through all 2-dimensional Fq-linear subspaces of Fq4, there are exactly 5 equivalence classes of

(q4, q2)-bent maps of the form Tr(w1x2) Tr(w2x2)

• .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 46/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

Example 2 Put q = 3. Let ξ be a primitive element of F35 such that

ξ5 + 2ξ + 1 = 0. Then

F(x) =

Tr(x2) Tr(ξx2 + ξxq+1 + xq2+1)

• is (q5, q2) − bent.

Moreover F(x) is not extendable to x2 or any Albert’s twisted semifield planar map. Namely there is no Fq-linearly independent set {w1, w2} ⊆ Fq5 such that F(x) is equivalent to

Tr(w1f(x)) Tr(w2f(x))

• for f(x) ∈ {x2, xq+1}.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 47/57

Outline Preliminaries Part 1 Part 2 Q

Example 3 Put q = 3. As {w1, w2} runs through all 2-dimensional Fq-linear subspaces of F4, there are exactly 7 equivalence classes of

(q4, q2)-bent maps of the form Tr(w1f(x)) Tr(w2f(x))

• , where f(x) = x4 + x10 − x36

corresponds to the Dickson’s semifield on F34. Corollary 5 F(x) : any Fq-quadratic (q3, q2)-bent map. There exists

Fq-quadratic forms f1(x), f2(x) on Fq3 such that

• F(x)

f1(x)

• ∼ x2 and
• F(x)

f2(x)

• ∼ xq+1

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 48/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

Definition 4 q : any power of an odd prime. F(x) : any Fq-quadratic

(qn, qr)-bent map such that 1 ≤ r ≤ n − 1. We call that F is

extendable if there exists on Fq-quadratic form f(x) on Fqn such that

• F(x)

f(x)

• is(qn, qr+1) − bent.

Otherwise we call that F(x) is non-extendable. Non-extendable Fq-quadratic (qn, qr)-bent maps could be considered as “essential” or “atomic” bent maps.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 49/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

Proposition 5 n ≥ 2. q : any power of an odd prime. F(x) : Fq-quadratic

(qn, q)-bent map (or just bent map). Then F(x) is extendable.

Fact: q = 3 There are exactly 7 non-equivalent Fq-quadratic

(q5, q5)-bent (or just planar) maps.

1

x2

2

xq+1

3

xq2+1

4

x10 + x6 − x2

5

x10 − x6 − x2

6

x90 + x2

7

−(xq − x)2 + D(xq − x) + 1

2x2 with

D(x) = −x36 + x28 + x12 + x4.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 50/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

Example 4 q = 3. There is no non-extendable Fq-quadratic (q5, q2)-bent map. In particular the (q5, q2)-bent map in Example 2 above is extendable to one of the last 4 planar maps among 7 planar maps

• f semifields of order 243 = 35 given above.

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 51/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

The lattice of Fq-quadratic (q4, qr)- bent maps for 3 ≤ r ≤ 4 and q = 3. w : a primitive element of F34 such that w4 + 2w3 + 2 = 0. E1 :

          Tr(x2) Tr(wx10) Tr(wx4)           , E2 :           Tr(x2) Tr(wx10) Tr(w2x2)           , E3 :           Tr(x2) Tr(wx10) Tr(w8x2)           ,

E4 :

          Tr(x2) Tr(w13x10) Tr(w5x4)           , E5 :           Tr(x2) Tr(wx4) Tr(w2x4)           .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 52/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

NE1 :

          Tr(x2) Tr(wx10) Tr(wx4)           , NE2 :           Tr(x2) Tr(wx10) Tr(w5x4)           , NE3 :           Tr(x2) Tr(wx10) Tr(w7x4)           ,

NE4 :

          Tr(x2) Tr(wx10) Tr(w3x10 + w8x4 + wx2)           ,

NE5 :

          Tr(x2) Tr(wx10) Tr(w3x10 + w25x4 + wx2)           .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 53/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

NE6 :

          Tr(x2) Tr(wx10) Tr(w3x10 + wx4 + w2x2)           ,

NE7 :

          Tr(x2) Tr(wx10) Tr(w30x4 + w6x2)           , NE8 :           Tr(x2) Tr(wx10) Tr(w15x2)           ,

NE9 :

          Tr(x2) Tr(wx10) Tr(w4x4 + w15x2)           , NE10 :           Tr(x2) Tr(w13x10) Tr(wx4)           ,

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 54/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

NE11 :

          Tr(x2) Tr(wx4) Tr(w2x10 + wx2)           ,

NE12 :

          Tr(x2) Tr(wx4) Tr(w14x10 + w8x4 + wx2)           ,

NE13 :

          Tr(x2) Tr(wx4) Tr(w4x10 + w8x4 + w2x2)           .

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 55/57

Outline Preliminaries Part 1 Part 2 Q

Part 2

The lattice for F3- quadratic (q4, qr) maps for 3 ≤ r ≤ 4. NE1, NE2, · · · , NE13 are non-extendable. All Fq-quadratic (q4, q2)-maps are extendable ( there are exactly 7 non-equivalent ones). All Fq-quadratic (q4, q)-maps are extendable ( and equivalent to each other).

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 56/57

Outline Preliminaries Part 1 Part 2 Q

Any Question ? Thank you !

Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable Fq-quadratic Perfect Nonlinear Maps 57/57