Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields

Abstract In 2006, Hubert, Mauduit and Sárközy extended the notion of binary sequences to n-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. In 2011, Gyarmati, Mauduit and Sárközy extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices. In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in finite fields and study the pseudorandom measure of order k, family complexity, collision and avalanche effect. Results indicate that such binary lattices are “good,” and their families possess a nice structure in terms of family complexity, collision and avalanche effect.


Introduction
The need for pseudorandom binary lattices arises in many applications, so numerous papers have been written on this subject. In these papers, some measures are introduced and studied. For example, Hubert, Mauduit and Sárközy [1] extended the notion of binary sequences to n-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. For details, let I N n denote the set of n-dimensional vectors all whose coordinates are in { … − } N 0, 1, , 1 . That is, : , , 0, 1, , 1 . An n-dimensional binary N-lattice η is considered as a "good" pseudorandom binary lattice if ( ) η k is "small" in terms of N for small k. This terminology is justified since Hubert, Mauduit and Sárközy [1] proved that for a fixed ∈ k and for a truly random n-dimensional binary N-lattice η we have with probability greater than − ε 1 , while the trivial upper bound for ( ) η k is N n . In 2011, Gyarmati, Mauduit and Sárközy [2] extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices.
Assume that ∈ n N , , and is a family of n-dimensional binary N-lattices The family complexity or f-complexity of the family , denoted by ( ) Γ , is defined as the greatest integer j such that for any specification of length j, there is at least one ∈ η satisfying , , . Obviously, we have the trivial bound , is a given finite set (e.g., a set of certain polynomials), to each ∈ s we assign a unique n-dimensional binary N-lattice = →{ − +} η η I : 1 , 1 s N n , and let = ( ) denote the family of the binary lattices obtained in this way: s , then this is said to be a collision in = ( ). If there is no collision in = ( ), then is said to be collision free. , then is said to possess the strict avalanche property.
, between η and ′ η is defined by is collision free if ( ) > m 0, and possesses the strict avalanche property if Many pseudorandom binary lattices have been obtained and studied by using the subsets in finite fields (see [1][2][3][4][5][6][7][8][9][10][11]). Suppose that = q p n is an odd prime power and q is a finite field with q elements. Let … v v , , n 1 be linearly independent elements of q over p and let α be a primitive element of the finite field q , and > d 1 be a divisor of − q 1. The dth cyclotomic classes ( ) In this paper, we shall give large families of binary lattices by using the cyclotomic classes in finite fields and study their properties. Our results are the following.
has no multiple zero in ¯q , where ¯q is an algebraic closure of q . Let > d 1 be a divisor of − q 1 and d be even. Define Assume that one of the following conditions holds f K of which the multiplicity of each zero in ¯q is less than d. Let > d 1 be a divisor of − q 1 and d be even. Define

Estimates for character sums of polynomials
We need the following lemmas to prove the theorems. Proof. This is Lemma 4A of [12].  Proof. This is Theorem 2C′ of [12]. □ Lemma 2.3. Suppose that = q p n is an odd prime power and q is a finite field. Let … ∈ z z , , k q 1 has no multiple zero in ¯q , and assume that one of the following conditions holds is not a constant times of a dth power of a polynomial.
Proof. This is Lemma 5 of [11]. Proof. This is Theorem 2 of [13]. □ Lemma 2.5. Suppose that = q p n is an odd prime power, α is a primitive element of the finite field q and > d 1 be a divisor of − q 1. Let  * q denote the set of all multiplicative characters of * q and = * Proof. This is Lemma 1 of [14]. □ Lemma 2.6. Suppose that T is a field and ( ) ∈ [ ] g x T x is a non-zero polynomial, then it can be written in the form and the multiplicity of each zero of ( ) * g x in T is less than k.
Proof. This is Lemma 6 of [15].   It is well known that

Write
is a representation of the unique interpolating polynomial ( ) 1 is a non-zero polynomial, by Lemma 2.6, ( ) g x can be written in the form q and the multiplicity of each zero of ( ) * g x in ¯q is less than d. Let It follows that By   which proves Theorem 1.3.