Character Sums and Generating Sets Ming-Deh A. Huang, Lian Liu - - PowerPoint PPT Presentation

Character Sums and Generating Sets

Ming-Deh A. Huang, Lian Liu

University of Southern California

July 14, 2015

Introduction

Let p be a prime number, f ∈ Fp[x] be an irreducible polynomial of degree d ≥ 2 and q = pd be a prime power.

Theorem (Chung)

Given Fq ∼ = Fp[x]/f , if √p > d − 1, then Fp + x is a generating set for F⇥

q .

Fp + x := {a + x|a ∈ Fp}

Today’s topic

Today, we will discuss more on the relationship between character sums and group generating sets. To illustrate, we will take a detailed look the multiplicative group of the algebra A⇥, where A is of the form: A := Fp[x] /f e where e ≥ 1 is an integer.

Outline

Question

I Given S ⊆ A⇥ a subset of elements, what are the sufficient or

necessary conditions for S to generate A⇥?

I How to construct a small generating set for A⇥? I How strong are the above sufficient conditions for generating sets?

Can they be substantially weakened in practice?

Difference graphs

Given G, a nontrivial finite abelian group and S ⊆ G a subset of elements, the difference graph G defined by the pair (G, S) is constructed as follows:

Algorithm

• 1. For each element g ∈ G, create a vertex g in G;
• 2. Create an arc g → h in G if and only if gs = h for

some s ∈ S. E.g., in Chung’s situation, G = F⇥

q ∼

= (Fp[x]/f )⇥ and S = x + Fp.

Lemma

If G has a finite diameter, then S is a generating set for G.

Diameters and eigenvalues

Theorem (Chung)

Suppose a k-regular directed graph G which has out-degree k for every vertex, and the eigenvectors of its adjacency matrix form an orthogonal

• basis. Then

diam(G) ≤ & log(n − 1) log( k

λ)

' where n is the number of vertices and λ is the second largest eigenvalue (in absolute value) of the adjacency matrix.

Adjacency matrices defined on general finite abelian groups

Assume that G is any nontrivial finite abelian group, and assume the adjacency matrix, M, of G := (G, S) has rows and columns indexed by g1, . . . , gn ∈ G: B B B B B B B @ g1 . . . gj . . . gn g1 . . . . . . . . . gi . . . . . . I[∃s ∈ S : gj = sgi] . . . . . . . . . . . . gn . . . 1 C C C C C C C A

Dirichlet character sums

Let G be any nontrivial finite abelian group. Then G ∼ = Zd1 ⊕ . . . ⊕ Zdk for some integers di > 1. Consider Dirichlet characters χ : G → C⇥ of the following form: g ∼ = (g1, . . . , gk) → Y

i

ωgi

di

for every g ∈ G, where ωdi is a dth

i

root of unity.

A generalization of Chung’s results

The adjacency matrix M has the following properties:

Lemma

The eigenvectors of M are [χ(g1), . . . , χ(gn)]>, and the corresponding eigenvalutes are P

s2S χ(s).

Lemma

The set of eigenvectors [χ(g1), . . . , χ(gn)]> form an orthogonal basis for Cn.

A generalization of Chung’s results

Following the diameter theorem for directed graphs, we may generalize Chung’s results to obtain

Theorem (Main)

If

• X

s2S

χ(s)

• < |S|

for every nontrivial Dirichlet character χ of G, then S is a generating set for G.

The structure of A×

Now let us consider groups of the form A := Fp[x]/f e. Recall that f ∈ Fp[x] is a monic irreducible polynomial of degree d ≥ 2 and e ≥ 1 is an integer.

Lemma (Decomposition)

If p ≥ e, then A⇥ ∼ = Zpd1 ⊕ @ M

d(e1)

Zp 1 A

Theorem

If p ≥ e, then any generating set of A⇥ contains at least d(e − 1) + 1 elements.

The structure of A×

This isomorphism allows us to define a Dirichlet character from A⇥ to the unit circle. For every α ∈ A⇥, χ : α → ω

d(e1)

Y

i=1

θi where ω is a (pd − 1)th root of unity and each θi is a pth root of unity. χ is trivial if ω and every θi equals 1.

A as an Fp-algebra

Let us first consider if the set of linear elements S = Fp −x generates A⇥.

Theorem (Katz, Lenstra)

Given Fq a finite filed and B an arbitrary finite n-dimensional commutative Fq-algebra. For any nontrivial complex-valued multiplicative character χ on B⇥, extended by zero all of B,

• X

a2Fq

χ(a − x)

• ≤ (n − 1)√q

A as an Fp-algebra

Since A can be naturally regarded as an Fp-algebra of dimension de, by the Main theorem we get

Theorem

If √p > de − 1, then Fp − x is a generating set for A⇥. Furthermore, every element α ∈ A⇥ can be written as Qm

i=1 (ai − x) where ai ∈ Fq and

m < 2de + 1 + 4de log(de − 1) log p − 2 log(de − 1)

More on the structure of A×

The constraint √p > de − 1 might be critical on the size of the base field Fp, and hence we wonder whether we can use other base fields of A to build generating sets in a similar way. One candidate base field is Fq := Fp[x]/f , and we proved that A is indeed an Fq-algebra:

Lemma

A is an Fq-algebra of dimension e, and there exists a embedding π : Fq → A such that Fq ∼ = π(Fq) as rings.

The embedding

Given an element a ∈ F⇥

q , the image π(a) is uniquely determined by the

following constraints:

I π(a) ≡ a (mod f ); I (π(a))q1 ≡ 1 (mod f e).

We extend the embedding to all of Fq by enforcing π(0) = 0. Each image can be computed with O(de log p) group operations in (Fp[x]/f i)⇥ where 1 ≤ i ≤ e.

A as an Fq-algebra

Knowing that A as an Fq-algebra of dimension e, we may similarly consider whether or not the set π(Fq) − x generates A⇥. Again, by Katz and Lenstra’s character sum theorem, we have

Theorem

If p ≥ e, then π(Fq) − x is a generating set for A⇥. Furthermore, every element α ∈ A⇥ can be written as Qm

i=1 (π(ai) − x) where ai ∈ Fq and

m < 2e + 1 + 4e log(e − 1) d log p − 2 log(e − 1)

Constructing a small generating set

Based on previous discussions we observe that

I Fp − x generates A⇥ if √p > de − 1, but requires p to be large; I π(Fq) − x generates A⇥ if p ≥ e, but might be over-killing; I Next step: take a nice subfield K ⊂ Fq and build a generating set

from π(K) − x.

Constructing a small generating set

Let K ⊂ Fq be a subfield of size pc where c|d. Then Fp[x]/f can be considered as an K-algebra of dimension de/c. Based on our previous discussion we can similarly show that

Theorem

If pc/2 > de/c − 1 and p ≥ e, then π(K) − x is a generating set for A⇥. Furthermore, every element α ∈ A⇥ can be written as Qm

i=1 (π(ai) − x)

where ai ∈ K and m < 2de c + 1 + 4 de

c log( de c − 1) d c log p − 2 log( de c − 1)

Constructing a small generating set

Now we conclude the algorithm for constructing the smallest generating set for A⇥ in the situation that p ≥ e:

Algorithm

• 1. Find the smallest c such that c|d which satisfies

pc/2 > de/c − 1;

• 2. Take the subfield K ⊂ Fq of size pc and return π(K) − x

as a generating set for A⇥.

Theorem

Given fixed p and e with p ≥ e, if d is a perfect power, then there is (constructively) a generating set for A⇥ of size pO(log d).

Experiments

In the following experiments, we compare the size of the following three types of generating sets for A⇥:

I S := π(Fq) − x, the size is equal to pd; I S⇤ := π(K) − x, the size is equal to pc; I ˜

S⇤, the set generated by adding elements in S⇤ one-by-one to ∅, until it generates the whole group. We denote its size as pb for some real number b. Obviously, we have b ≤ c ≤ d. Also note that ˜ S⇤ might still be much bigger than the real smallest generating set.

The relationship between c and d

Experiment setting:

I p = 7, e = 5; I d = 21, 22, 23, . . ..

2 4 6 8 10 200 600 1000 log2 (d) 1 2 3 4 5 6 7 8 9 10 d c

(a) Comparison between c and d

200 400 600 800 1000 100 300 500 log2 (d) c fit(c)

(b) The logarithmic growth of c

The relationship between c and b

I d = 21, 22, 23, . . .; I fix e = 4 and increase the value of p.

1 2 3 4 5 6 7 2 4 6 8 10 log2 (d) c b fit(c) fit(b)

(c) p = 5, e = 4

1 2 3 4 5 6 7 2 4 6 8 10 log2 (d) c b fit(c) fit(b)

(d) p = 11, e = 4

The relationship between c and b

I d = 21, 22, 23, . . .; I fix p = 7 and increase the value of e.

1 2 3 4 5 6 7 2 4 6 8 10 log2 (d) c b fit(c) fit(b)

(e) p = 7, e = 3

1 2 3 4 5 6 7 2 4 6 8 10 log2 (d) c b fit(c) fit(b)

(f) p = 7, e = 5

Remarks and future work

We observe that both b and c grows linearly with log(d), and they may differ only by a constant ratio, i.e. ˜ S⇤ is still of size pO(log d) given d being a perfect power.

Problem

Given p ≥ e > 1 and f ∈ Fp[x] an irreducible polynomial of degree d, a perfect power, how to construct a generating set of size po(log d) for the group A⇥?

Remarks and future work

A big assumption we made in our work is that p ≥ e, which helps guarantee the decomposition of the group. It is therefore an important question to ask what if p < e?

Problem

If p < e, can we get similar results for the group A⇥?

Thanks! ,Y