Character Sums and Generating Setslianliu/files/pres-fq12.pdf · Introduction Let p be a prime...
Transcript of Character Sums and Generating Setslianliu/files/pres-fq12.pdf · Introduction Let p be a prime...
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