Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

We consider homogeneous linear recurring sequences over a finite field Fq, based on an irreducible characteristic polynomial of degree n and order m. Let t = (qn − 1)/m. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when t has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.


Introduction
Let F q be the finite field with q elements where q = p m for prime p.Let k be a positive integer, and let a 0 , a 1 , . . ., a k−1 be given elements of F q .A sequence s 0 , s 1 , . . . of elements of F q satisfying the relation is called a kth-order homogeneous linear recurring sequence in F q .The terms s 0 , s 1 , . . .s k−1 , which determine the rest of the sequence uniquely, are referred to as the initial values.Let s 0 , s 1 , . . .be a kth order homogeneous linear recurring sequence in F q satisfying the linear recurrence relation in (1), where a j ∈ F q for 0 ≤ j ≤ k − 1.The polynomial is called a characteristic polynomial of the linear recurring sequence.For the homogeneous linear recurring sequence s 0 , s 1 , . . . in F q , m(x) ∈ F q [x] is said to be the minimal polynomial of the sequence if it has the following property: a monic polynomial f (x) ∈ F q [x] of positive degree is a characteristic polynomial of s 0 , s 1 , . . .if and only if m(x) divides f (x).
Definition Let f ∈ F q [x] be a non zero polynomial.If f (0) 0, then the least positive integer e for which f (x) divides x e − 1 is called the order of f which is denoted by ord( f ).
Theorem 1. (Lidl & Niederreiter, 1994) Let s 0 , s 1 , ... be a homogeneous linear recurring sequence in F q with minimal polynomial m(x) ∈ F q [x].Then the least period of the sequence is equal to ord(m(x)).
Discussions on linear recurring sequences took place for many years with a substantial development in the area of examining zeros and determining effective bounds for the set of zeros over infinite fields (Everest, Poorten, Shparlinski & Ward, 2003).Linear recurring sequences over finite fields have appeared sporadically over the years in a variety of contexts in Cryptography, mainly in the area of linear shift registers where determining the exact number of zeros is of higher importance (Lidl & Niederreiter, 1994).Let S be a homogeneous linear recurring sequence over F q and let f (x) ∈ F q [x] be the irreducible minimal polynomial of S with degree n and order m and let t = q n −1 m .Kottegoda and Fitzgerald (2017) provided an accurate bound for the number of zeros of S within its least period, also providing formulas for the exact number of zeros when t has the form q 2a − q a + 1 where a ∈ N. Here, we will give the exact number of zeros when t = q a + 1 using applications in quadratic forms over finite fields.
In section 2, we will describe some known results on one-term trace forms over finite fields of even and odd characteristics by Klapper (1993Klapper ( , 1997) ) and include proofs for the simpler formulation of Klapper's results which were stated by Mullen and Panario (2013)7.2without proofs.
In section 3, we give our main theorem by providing formulas for the exact number of occurrences of zeros of S within its least period for the case where t = q n −1 m takes the form q a + 1 where a ∈ N, using the quadratic form results from section 2. Hence we will also prove that the cardinality of the set of zeros in this case is 2.

Results on Quadratic Forms
Let F = F q and K = F q n where q is a prime power.Let Q : K → F be a quadratic form and N(Q = u) be the number of solutions for Q(x) = u in K. General information and the main definitions on quadratic forms can be obtained by Mullen and Panario (2013)7.2.
We now recall what is known about N(Q = u) over finite fields of even and odd characteristic.
Proposition 1. (Klapper, 1993) Every quadratic form Q of rank m in n variables over F q for even q is equivalent to one of the following 3 standard types under a change of co-ordinates : (Klapper, 1997) For any quadratic form Q of rank m in n variables over F q for odd q is equivalent under a change of coordinates to precisely one of the following quadratic forms: where b ∈ {1, a} and 1 a ∈ F * q /(F * q ) 2 .and the determinants of Q (i.e det(Q)) are as follows: for Type I: 2 −1 for Type I and Type III q n−1 + η((−1) x is a square −1, x is a not a square 0, x = 0 The following formulation for N(Q = 0) is much simpler to use for computations.
where r = dim rad(Q) and Λ(Q) is an invariant defined in terms of the discriminant (if q is odd) or the Arf invariant (if q is even).Let ν 2 (x) denote the 2-adic valuation of x.Consider the trace form Q(x) = T r K/F (γx q a +1 ).Set d = (n, a).
Hence if n is even then a is even resulting n d to be odd.If n is odd then n d is odd.Therefore by Theorem 4.1 (Klapper, 1993) Here n is even and ν 2 (a) < ν 2 (n).Hence n d is even.Since ν 2 (2d) = 1 + ν 2 (a) = ν 2 (n), n 2d is odd.Therefore by Theorem 4.1 (Klapper, 1993), Q(x) is of Type III with rank n if γ is not a q a + 1th power in K and Q(x) is of Type I with rank n − 2d if γ is not a q a + 1th power in K. Hence r take the values 0 and 2d respectively and Λ(Q) take the values -1 and 1 respectively.
Here n is even and ν 2 (n) ≥ 2. If a is odd, then d is odd and hence both n d and n 2d are even.If a is even, then d is even and ν 2 (d) = ν 2 (a) < ν 2 (n).Hence both n d and n 2d are even.Therefore by Theorem 4.1 (Klapper, 1993), Q(x) is of Type I with rank n if γ is not a q a + 1th power in K and Q(x) is of Type III with rank n − 2d if γ is a q a + 1th power in K. Hence r take the values 0 and 2d respectively and Λ(Q) take the values 1 and -1 respectively.Proposition 4. (Mullen & Panario, 2013) For odd q let Q(x) = T r K/F (γx q a +1 ) and γ ∈ K. Let ω be a primitive element of K and write γ = ω g for some 0 ≤ g < q n − 1.

Zeros of Homogeneous Linear Recurring Sequences
When the characteristic polynomial of a homogeneous linear recurring sequence is irreducible, each term of the sequence can be expressed in terms of a suitable trace function as given in the following theorem.
Theorem 2. (Lidl & Niederreiter, 1994) Let s 0 , s 1 , . . .be a kth-order homogeneous linear recurring sequence in F = F q whose characteristic polynomial f (x) is irreducible over F. Let α be a root of f (x) in the extension field K = F q k .Then there exists a uniquely determined θ ∈ K such that s n = Tr K/F (θα n ) for n = 0, 1, . . .