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 ∈ F p [ x ] be an irreducible polynomial of degree d ≥ 2 and q = p d be a prime power. Theorem (Chung) = F p [ x ] / f , if √ p > d − 1 , then F p + x is a generating set for Given F q ∼ F ⇥ q . F p + x := { a + x | a ∈ F p }
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 := F p [ x ] / f e where e ≥ 1 is an integer.
Outline Question I Given S ⊆ A ⇥ a subset of elements, what are the su ffi cient or necessary conditions for S to generate A ⇥ ? I How to construct a small generating set for A ⇥ ? I How strong are the above su ffi cient conditions for generating sets? Can they be substantially weakened in practice?
Di ff erence graphs Given G , a nontrivial finite abelian group and S ⊆ G a subset of elements, the di ff erence 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 . = ( F p [ x ] / f ) ⇥ and S = x + F p . q ∼ E.g., in Chung’s situation, G = F ⇥ 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 & ' log( n − 1) diam ( G ) ≤ 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 g 1 , . . . , g n ∈ G : . . . . . . g 1 g j g n . 0 1 . g 1 . . . B C . . . B . C B C I [ ∃ s ∈ S : g j = sg i ] g i B . . . . . . . . . . . . C B C . . B C . . . B . C @ A . . g n .
Dirichlet character sums Let G be any nontrivial finite abelian group. Then G ∼ = Z d 1 ⊕ . . . ⊕ Z d k for some integers d i > 1. Consider Dirichlet characters χ : G → C ⇥ of the following form: g ∼ Y ω g i = ( g 1 , . . . , g k ) → d i i for every g ∈ G , where ω d i is a d th root of unity. i
A generalization of Chung’s results The adjacency matrix M has the following properties: Lemma The eigenvectors of M are [ χ ( g 1 ) , . . . , χ ( g n )] > , and the corresponding eigenvalutes are P s 2 S χ ( s ) . Lemma The set of eigenvectors [ χ ( g 1 ) , . . . , χ ( g n )] > form an orthogonal basis for C n .
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 χ ( s ) � < | S | � � � � � s 2 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 := F p [ x ] / f e . Recall that f ∈ F p [ x ] is a monic irreducible polynomial of degree d ≥ 2 and e ≥ 1 is an integer. Lemma (Decomposition) If p ≥ e , then 0 1 A ⇥ ∼ @ M = Z p d � 1 ⊕ Z p A d ( e � 1) 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 ⇥ , d ( e � 1) Y χ : α → ω θ i i =1 where ω is a ( p d − 1) th root of unity and each θ i is a p th root of unity. χ is trivial if ω and every θ i equals 1.
A as an F p -algebra Let us first consider if the set of linear elements S = F p − x generates A ⇥ . Theorem (Katz, Lenstra) Given F q a finite filed and B an arbitrary finite n -dimensional commutative F q -algebra. For any nontrivial complex-valued multiplicative character χ on B ⇥ , extended by zero all of B , � � � � ≤ ( n − 1) √ q X � � χ ( a − x ) � � � � a 2 F q � �
A as an F p -algebra Since A can be naturally regarded as an F p -algebra of dimension de , by the Main theorem we get Theorem If √ p > de − 1 , then F p − x is a generating set for A ⇥ . Furthermore, every element α ∈ A ⇥ can be written as Q m i =1 ( a i − x ) where a i ∈ F q and 4 de log( de − 1) m < 2 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 F p , 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 F q := F p [ x ] / f , and we proved that A is indeed an F q -algebra: Lemma A is an F q -algebra of dimension e , and there exists a embedding π : F q → A such that F q ∼ = π ( F q ) 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 )) q � 1 ≡ 1 (mod f e ). We extend the embedding to all of F q by enforcing π (0) = 0. Each image can be computed with O ( de log p ) group operations in ( F p [ x ] / f i ) ⇥ where 1 ≤ i ≤ e .
A as an F q -algebra Knowing that A as an F q -algebra of dimension e , we may similarly consider whether or not the set π ( F q ) − x generates A ⇥ . Again, by Katz and Lenstra’s character sum theorem, we have Theorem If p ≥ e , then π ( F q ) − x is a generating set for A ⇥ . Furthermore, every element α ∈ A ⇥ can be written as Q m i =1 ( π ( a i ) − x ) where a i ∈ F q and 4 e log( e − 1) m < 2 e + 1 + d log p − 2 log( e − 1)
Constructing a small generating set Based on previous discussions we observe that I F p − x generates A ⇥ if √ p > de − 1, but requires p to be large; I π ( F q ) − x generates A ⇥ if p ≥ e , but might be over-killing; I Next step: take a nice subfield K ⊂ F q and build a generating set from π ( K ) − x .
Constructing a small generating set Let K ⊂ F q be a subfield of size p c where c | d . Then F p [ 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 p c / 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 Q m i =1 ( π ( a i ) − x ) where a i ∈ K and 4 de c log( de c − 1) m < 2 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 p c / 2 > de / c − 1 ; 2. Take the subfield K ⊂ F q of size p c 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 p O (log d ) .
Experiments In the following experiments, we compare the size of the following three types of generating sets for A ⇥ : I S := π ( F q ) − x , the size is equal to p d ; I S ⇤ := π ( K ) − x , the size is equal to p c ; S ⇤ , the set generated by adding elements in S ⇤ one-by-one to ∅ , I ˜ until it generates the whole group. We denote its size as p b for some real number b . S ⇤ might still be much Obviously, we have b ≤ c ≤ d . Also note that ˜ bigger than the real smallest generating set.
The relationship between c and d Experiment setting: I p = 7 , e = 5; I d = 2 1 , 2 2 , 2 3 , . . . . 1000 500 d c c fit(c) 600 300 200 100 0 0 1 2 2 3 4 4 5 6 6 7 8 8 9 10 10 0 200 400 600 800 1000 log 2 (d) log 2 (d) (a) Comparison between c and d (b) The logarithmic growth of c
The relationship between c and b I d = 2 1 , 2 2 , 2 3 , . . . ; I fix e = 4 and increase the value of p . 10 10 c c b b 8 8 fit(c) fit(c) fit(b) fit(b) 6 6 4 4 2 2 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 log 2 (d) log 2 (d) (c) p = 5 , e = 4 (d) p = 11 , e = 4
The relationship between c and b I d = 2 1 , 2 2 , 2 3 , . . . ; I fix p = 7 and increase the value of e . 10 10 c c b b 8 8 fit(c) fit(c) fit(b) fit(b) 6 6 4 4 2 2 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 log 2 (d) log 2 (d) (e) p = 7 , e = 3 (f) p = 7 , e = 5
Remarks and future work We observe that both b and c grows linearly with log( d ), and they may S ⇤ is still of size p O (log d ) given d di ff er only by a constant ratio, i.e. ˜ being a perfect power. Problem Given p ≥ e > 1 and f ∈ F p [ x ] an irreducible polynomial of degree d , a perfect power, how to construct a generating set of size p o (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
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