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 2 Fp[x ] be an irreducible polynomial ofdegree d � 2 and q = pd be a prime power.

Theorem (Chung)Given Fq

⇠= Fp[x ]/f , ifpp > d � 1, then Fp + x is a generating set for

F⇥q .

Fp + x := {a+ x |a 2 Fp}

Today’s topic

Today, we will discuss more on the relationship between character sumsand group generating sets. To illustrate, we will take a detailed look themultiplicative 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 su�cient ornecessary conditions for S to generate A⇥?

I How to construct a small generating set for A⇥?

I How strong are the above su�cient conditions for generating sets?Can they be substantially weakened in practice?

Di↵erence graphs

Given G , a nontrivial finite abelian group and S ✓ G a subset ofelements, the di↵erence graph G defined by the pair (G , S) is constructedas follows:

Algorithm

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

some s 2 S.

E.g., in Chung’s situation, G = F⇥q⇠= (Fp[x ]/f )⇥ and S = x + Fp.

LemmaIf 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 everyvertex, and the eigenvectors of its adjacency matrix form an orthogonalbasis. Then

diam(G ) &log(n � 1)

log( k� )

'

where n is the number of vertices and � 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 theadjacency matrix, M, of G := (G , S) has rows and columns indexed byg1

, . . . , gn 2 G :

0

BBBBBBB@

g1

. . . gj . . . gn

g1

......

...gi . . . . . . I[9s 2 S : gj = sgi ] . . . . . ....

...

gn...

1

CCCCCCCA

Dirichlet character sums

Let G be any nontrivial finite abelian group. Then

G ⇠= Zd1

� . . .� Zdk

for some integers di > 1.Consider Dirichlet characters � : G ! C⇥ of the following form:

g ⇠= (g1

, . . . , gk) !Y

i

!gidi

for every g 2 G , where !di is a d th

i root of unity.

A generalization of Chung’s results

The adjacency matrix M has the following properties:

LemmaThe eigenvectors of M are [�(g

1

), . . . ,�(gn)]>, and the correspondingeigenvalutes are

Ps2S �(s).

LemmaThe set of eigenvectors [�(g

1

), . . . ,�(gn)]> form an orthogonal basis forCn.

A generalization of Chung’s results

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

Theorem (Main)If �����

X

s2S

�(s)

����� < |S |

for every nontrivial Dirichlet character � of G , then S is a generating setfor G.

The structure of A⇥

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

Lemma (Decomposition)If p � e, then

A⇥ ⇠= Zpd�1

�

0

@M

d(e�1)

Zp

1

A

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

The structure of A⇥

This isomorphism allows us to define a Dirichlet character from A⇥ tothe unit circle. For every ↵ 2 A⇥,

� : ↵ ! !

d(e�1)Y

i=1

✓i

where ! is a (pd � 1)th root of unity and each ✓i is a pth root of unity. �is trivial if ! and every ✓i 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-dimensionalcommutative Fq-algebra. For any nontrivial complex-valuedmultiplicative character � on B⇥, extended by zero all of B ,

������

X

a2Fq

�(a� x)

������ (n � 1)

pq

A as an Fp-algebra

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

TheoremIfpp > de � 1, then Fp � x is a generating set for A⇥. Furthermore,

every element ↵ 2 A⇥ can be written asQm

i=1

(ai � x) where ai 2 Fq and

m < 2de + 1 +4de log(de � 1)

log p � 2 log(de � 1)

More on the structure of A⇥

The constraintpp > 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 tobuild generating sets in a similar way.One candidate base field is Fq := Fp[x ]/f , and we proved that A isindeed an Fq-algebra:

LemmaA is an Fq-algebra of dimension e, and there exists a embedding⇡ : Fq ! A such that Fq

⇠= ⇡(Fq) as rings.

The embedding

Given an element a 2 F⇥q , the image ⇡(a) is uniquely determined by the

following constraints:

I ⇡(a) ⌘ a (mod f );

I (⇡(a))q�1 ⌘ 1 (mod f e).

We extend the embedding to all of Fq by enforcing ⇡(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 similarlyconsider whether or not the set ⇡(Fq)� x generates A⇥. Again, by Katzand Lenstra’s character sum theorem, we have

TheoremIf p � e, then ⇡(Fq)� x is a generating set for A⇥. Furthermore, everyelement ↵ 2 A⇥ can be written as

Qmi=1

(⇡(ai )� x) where ai 2 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⇥ ifpp > de � 1, but requires p to be large;

I ⇡(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 setfrom ⇡(K )� x .

Constructing a small generating set

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

TheoremIf pc/2 > de/c � 1 and p � e, then ⇡(K )� x is a generating set for A⇥.Furthermore, every element ↵ 2 A⇥ can be written as

Qmi=1

(⇡(ai )� x)where ai 2 K and

m < 2de

c+ 1 +

4 dec log( dec � 1)

dc log p � 2 log( dec � 1)

Constructing a small generating set

Now we conclude the algorithm for constructing the smallest generatingset 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 ⇡(K )� xas a generating set for A⇥

.

TheoremGiven 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 threetypes of generating sets for A⇥:

I S := ⇡(Fq)� x , the size is equal to pd ;

I S⇤ := ⇡(K )� x , the size is equal to pc ;

I S̃⇤, 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 somereal number b.

Obviously, we have b c d . Also note that S̃⇤ might still be muchbigger 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

0200

600

1000

log2 (d)

1 2 3 4 5 6 7 8 9 10

dc

(a) Comparison between c and d

0 200 400 600 800 1000

0100

300

500

log2 (d)

cfit(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

02

46

810

log2 (d)

cbfit(c)fit(b)

(c) p = 5, e = 4

1 2 3 4 5 6 70

24

68

10

log2 (d)

cbfit(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

02

46

810

log2 (d)

cbfit(c)fit(b)

(e) p = 7, e = 3

1 2 3 4 5 6 70

24

68

10

log2 (d)

cbfit(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 maydi↵er only by a constant ratio, i.e. S̃⇤ is still of size pO(log d) given dbeing a perfect power.

ProblemGiven p � e > 1 and f 2 Fp[x ] an irreducible polynomial of degree d , aperfect power, how to construct a generating set of size po(log d) for thegroup A⇥?

Remarks and future work

A big assumption we made in our work is that p � e, which helpsguarantee the decomposition of the group. It is therefore an importantquestion to ask what if p < e?

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

Thanks! ,Y