Existence of Primitive Normal Pairs with One Prescribed Trace over Finite Fields
aa r X i v : . [ m a t h . N T ] J a n Existence of Primitive Normal Pairs with OnePrescribed Trace over Finite Fields
Hariom Sharma, R. K. Sharma
Department of Mathematics, Indian Institute of Technology Delhi, NewDelhi, 110016, India
Abstract
Given m, n, q ∈ N such that q is a prime power and m ≥ a ∈ F q ,we establish a sufficient condition for the existence of primitive pair( α, f ( α )) in F q m such that α is normal over F q and Tr F qm / F q ( α − ) = a ,where f ( x ) ∈ F q m ( x ) is a rational function of degree sum n . Further,when n = 2 and q = 5 k for some k ∈ N , such a pair definitely existsfor all ( q, m ) apart from at most 20 choices. Keywords:
Finite Fields, Characters, Primitive element, Normal ele-ment2010 Math. Sub. Classification: 12E20, 11T23 Given the positive integers m and q such that q is a prime power, F q denotesthe finite field of order q and F q m be the extension of F q of degree m . A gen-erator of the cyclic multiplicative group F ∗ q m is known as a primitive element of F q m . For a rational function f ( x ) ∈ F q m ( x ) and α ∈ F q m , we call a pair( α, f ( α )) a primitive pair in F q m if both α and f ( α ) are primitive elementsof F q m . Further, α is normal over F q if the set { α, α q , α q , · · · , α q m − } formsa basis of F q m over F q . Also, the trace of α over F q , denoted by Tr F qm / F q ( α )is given by α + α q + α q + · · · + α q m − . emails: [email protected] (Hariom), [email protected] (Rajendra) α, f ( α )) in F q for the rational function f ( x ) = x + a, a ∈ F q . Many more researchersworked in this direction and proved the existence of primitive pair for moregeneral rational function [8, 2, 14, 3]. Additionally, in the fields of even order,Cohen[5] established the existence of primitive pair ( α, f ( α )) in F q n such that α is normal over F q , where f ( x ) = x +1 x . Similar result has been obtainedin [2] for the rational function f ( x ) = ax + bx + cdx + e . Another interesting problemis to prove the existence of primitive pair with prescribed traces which havebeen discussed in [13, 10, 15].In this article, we consider all the conditions simultaneously and provethe existence of primitive pair ( α, f ( α )) in F q m such that α is normal over F q and for prescribed a ∈ F q , Tr F qm / F q ( α − ) = a , where f ( x ) is more generalrational function. To proceed further, we shall use some basic terminologyand conventions used in [8]. To say that a non zero polynomial f ( x ) ∈ F q m [ x ]has degree n ≥ f ( x ) = a n x n + · · · + a , where a n = 0 andwrite it as deg( f ) = n . Next, for a rational function f ( x ) = f ( x ) /f ( x ) ∈ F q m ( x ), we always assume that f and f are coprime and degree sum of f = deg( f ) + deg( f ). Also, we can divide each of f and f by the leadingcoefficient of f and suppose that f is monic. Further, we say that a rationalfunction f ∈ F q m ( x ) is exceptional if f = cx i g d for some c ∈ F q m , i ∈ Z (setof integers) and d > q m − f ( x ) = x i for some i ∈ Z such thatgcd( q m − , i ) = 1 . Finally, we introduce some sets which have an important role in thisarticle. For n , n ∈ N , S q,m ( n , n ) will be used to denote the set of nonexceptional rational functions f = f /f ∈ F q m ( x ) with deg( f ) ≤ n anddeg( f ) ≤ n , and T n ,n as the set of pairs ( q, m ) ∈ N × N such that forany given f ∈ S q,m ( n , n ) and prescribed a ∈ F q , F q m contains a normalelement α with ( α, f ( α )) a primitive pair and Tr F qm / F q ( α − ) = a . Define S q,m ( n ) = S n + n = n S q,m ( n , n ) and T n = T n + n = n T n ,n . By [4], for m ≤ α such that Tr F qm / F q ( α − ) = 0.Therefore, we shall assume m ≥ n ∈ N , we take f ( x ) ∈ S q,m ( n ) a general rationalfunction of degree sum n and a ∈ F q , and prove the existence of normalelement α such that ( α, f ( α )) is a primitive pair in F q m and Tr F qm / F q ( α − ) = a . To be more precise, in section 3, we obtain a sufficient condition forthe existence of such elements in F q m . In section 4, we further improve thecondition by proving a generalization of sieving technique due to Anju andCohen[6]. In section 5, we demonstrate the application of the results ofsection 3 and section 4 by working with the finite fields of characteristic 5and n = 2. More precisely, we get a subset of T . In this section, we provide some preliminary notations, definitions and resultswhich are required further in this article. Throughout this article, m ≥ q is an arbitrary prime power and F q is a finite field of order q .For each k ( > ∈ N , ω ( k ) denotes the number of prime divisors of k and W ( k ) denotes the number of square free divisors of k . Also for g ( x ) ∈ F q [ x ],Ω q ( g ) and W ( g ) denote the number of monic irreducible(over F q ) divisors of g and number of square free divisors of g respectively, i.e., W ( k ) = 2 ω ( k ) and W ( g ) = 2 Ω q ( g ) .For a finite abelian group G , a homomorphism χ from G into the multi-plicative group S = { z ∈ C : | z | = 1 } is known as a character of G . The setof all characters of G forms a group under multiplication, which is isomorphicto G and is denoted by b G . Further, the character χ , defined as χ ( g ) = 1for all g ∈ G is called the trivial character of G . The order of a character χ is the smallest positive integer r such that χ r = χ . For a finite field F q m ,the characters of the additive group F q m and the multiplicative group F ∗ q m are called additive characters and multiplicative characters respectively. Amultiplicative character χ ∈ b F ∗ q m is extended from F ∗ q m to F q m by the rule χ (0) = ( χ = χ χ = χ . For more fundamentals on characters, primitiveelements and finite fields, we refer the reader to [12].For a divisor u of q m −
1, an element w ∈ F ∗ q m is u - free , if w = v d , where v ∈ F q m and d | u implies d = 1. It is easy to observe that an element in F ∗ q m is ( q m − free if and only if it is primitive. A special case of [16, Lemma10], provides an interesting result. 3 emma 2.1. Let u be a divisor of q m − , ξ ∈ F ∗ q m . Then X d | u µ ( d ) φ ( d ) X χ d χ d ( ξ ) = ( uφ ( u ) if ξ is u -free , otherwise.where µ ( · ) is the M ¨ obius function and φ ( · ) is the Euler function, χ d runsthrough all the φ ( d ) multiplicative characters over F ∗ q m with order d . Therefore, for each divisor u of q m − ρ u : α θ ( u ) X d | u µ ( d ) φ ( d ) X χ d χ d ( α ) , (2.1)gives a characteristic function for the subset of u - free elements of F ∗ q m ,where θ ( u ) = φ ( u ) u .Also, for each a ∈ F q , τ a : α q X ψ ∈ b F q ψ ( Tr F qm / F q ( α ) − a )is a characterstic function for the subset of F q m consisting elements withTr F qm / F q ( α ) = a . From [12, Theorem 5.7] every additive character ψ of F q can be obtained by ψ ( a ) = ψ ( ua ), where ψ is the canonical additivecharacter of F q and u is an element of F q corresponding to ψ . Thus τ a ( α ) = 1 q X u ∈ F q ψ ( Tr F qm / F q ( uα ) − ua )= 1 q X u ∈ F q ˆ ψ ( uα ) ψ ( − ua ) , (2.2)where ˆ ψ is the additive character of F q m defined by ˆ ψ ( α ) = ψ ( Tr F qm / F q ( α )).In particular, ˆ ψ is the canonical additive character of F q m .The additive group of F q m is an F q [ x ]-module under the rule f o α = k P i =1 a i α q i ; for α ∈ F q m and f ( x ) = k P i =1 a i x i ∈ F q [ x ]. For α ∈ F q m , the F q -orderof α is the unique monic polynomial g of least degree such that g o α = 0.Observe that g is a factor of x m −
1. Similarly, by defining the action of F q [ x ] over b F q m by the rule ψ o f ( α ) = ψ ( f o α ), where ψ ∈ b F q m , α ∈ F q m and4 ∈ F q [ x ], b F q m becomes an F q [ x ]-module, and the unique monic polynomial g of least degree such that ψ o g = χ is called the F q -order of ψ . Further thereare Φ q ( g ) characters of F q -order g , where Φ q ( g ) is the analogue of Euler’sphi-function on F q [ x ](see [12]).Similar to above, for g | x m − α ∈ F q m is g - f ree , if α = h o β ,where β ∈ F q m and h | g implies h = 1 . It is straightforward that an elementin F q m is ( x m − f ree if and only if it is normal. Also, for g | x m − g - f ree elements is given by κ g : α Θ( g ) X h | g µ ′ ( d )Φ q ( h ) X ψ h ψ h ( α ) , (2.3)where Θ( g ) = Φ q ( g ) q deg ( g ) , the internal sum runs over all characters ψ h of F q -order h and µ ′ is the analogue of the M¨obius function defined as µ ′ ( g ) = ( ( − s if g is a product of s distinct monic irreducible polynomials , Lemma 2.2. [ , T heorem . Let f ( x ) ∈ F q d ( x ) be a rational function.Write f ( x ) = Q kj =1 f j ( x ) n j , where f j ( x ) ∈ F q d [ x ] are irreducible polynomialsand n j are non zero integers. Let χ be a multiplicative character of F q d .Suppose that the rational function Q d − i =0 f ( x q i ) is not of the form h ( x ) ord ( χ ) in F q d ( x ) , where ord ( χ ) is the order of χ , then we have (cid:12)(cid:12) X α ∈ F q ,f ( α ) =0 ,f ( α ) = ∞ χ ( f ( α )) (cid:12)(cid:12) ≤ ( d k X j =1 deg( f j ) − q . Lemma 2.3. [ , T heorem . Let f ( x ) , g ( x ) ∈ F q m ( x ) be rational func-tions. Write f ( x ) = Q kj =1 f j ( x ) n j , where f j ( x ) ∈ F q m [ x ] are irreduciblepolynomials and n j are non zero integers. Let D = P kj =1 deg( f j ) , let D = max (deg( g ) , , let D be the degree of denominator of g ( x ) , and let D be the sum of degrees of those irreducible polynomials dividing denominator f g but distinct from f j ( x )( j = 1 , , · · · , k ) . Let χ be a multiplicative char-acter of F q m , and let ψ be a non trivial additive character of F q m . Suppose g ( x ) is not of the form r ( x ) q m − r ( x ) in F q m ( x ) . Then we have the estimate (cid:12)(cid:12) X α ∈ F qm ,f ( α ) =0 , ∞ g ( α ) = ∞ χ ( f ( α )) ψ ( g ( α )) (cid:12)(cid:12) ≤ ( D + D + D + D − q m . Let l , l ∈ N be such that l , l | q m −
1. Also, a ∈ F q , f ( x ) ∈ S q,m ( n ) and g | x m −
1, then N f,a,n ( l , l , g ) denote the number of elements α ∈ F q m suchthat α is both l - free and g - f ree , f ( α ) is l - free and Tr F qm / F q ( α − ) = a .We now prove one of the sufficient condition as follows. Theorem 3.1.
Let m, n and q ∈ N such that q is a prime power and m ≥ .Suppose that q m − > ( n + 2) W ( q − W ( x m − . (3.1) Then ( q, m ) ∈ T n .Proof. To prove the result, it is enough to show that N f,a,n ( q m − , q m − , x m − > f ( x ) ∈ S q,m ( n ) and prescribed a ∈ F q . Let f ( x ) ∈ S q,m ( n )be any rational function and a ∈ F q . Let U be the set of zeros and poles of f ( x ) in F q m and U = U ∪ { } . Assume l , l be divisors of q m − g bea divisor of x m −
1. Then by definition N f,a,n ( l , l , g ) = X α ∈ F qm \ U ρ l ( α ) ρ l ( f ( α )) τ a ( α − ) κ g ( α )now using (2.1), (2.2) and (2.3), N f,a,n ( l , l , g ) = θ ( l ) θ ( l )Θ( g ) q X d | l ,d | l h | g µ ( d ) φ ( d ) µ ( d ) φ ( d ) µ ′ ( h )Φ q ( h ) X χ d ,χ d ,ψ h χ f,a ( d , d , h ) , (3.2)where χ f,a ( d , d , h ) = P u ∈ F q ψ ( − au ) P α ∈ F qm \ U χ d ( α ) χ d ( f ( α )) ψ h ( α ) ˆ ψ ( uα − ).Since ψ h is an additive character of F q m and ˆ ψ is canonical additive character6f F q m , therefore there exists v ∈ F q m such that ψ h ( α ) = ˆ ψ ( vα ). Hence χ f,a ( d , d , h ) = P u ∈ F q ψ ( − au ) P α ∈ F qm \ U χ d ( α ) χ d ( f ( α )) ˆ ψ ( vα + uα − ).At this point, we claim that if ( d , d , h ) = (1 , , F q [ x ], then | χ f,a ( d , d , h ) | ≤ ( n + 2) q m +1 . To see the claim, firstsuppose d = 1, then χ f,a ( d , d , h ) = P u ∈ F q ψ ( − au ) P α ∈ F qm \ U χ d ( α ) ˆ ψ ( vα + uα − ). Here, if vx + ux − = r ( x ) q m − r ( x ) for any r ( x ) ∈ F q m ( x ) then byLemma 2.3 | χ f,a ( d , d , h ) | ≤ q m +1 + ( | U | − q ≤ ( n + 2) q m +1 . Also, if vx + ux − = r ( x ) q m − r ( x ) for some r ( x ) ∈ F q m ( x ) then following [Comm.Anju], it is possible when u = v = 0, which implies, | χ f,a ( d , d , h ) | ≤ | U | q < ( n + 2) q m +1 .Now suppose d >
1. Let d be the least common multiple of d and d . Then [12] suggests that there exists a character χ d of order d such that χ d = χ d/d d . Also, there is an integer 0 ≤ k < q m − χ d = χ kd .Consequently, χ f,a ( d , d , h ) = P u ∈ F q ψ ( − au ) P α ∈ F qm \ U χ d ( α k f ( α ) d/d ) ˆ ψ ( vα + uα − ). At this moment, first suppose vx + ux − = r ( x ) q m − r ( x ) for any r ( x ) ∈ F q m ( x ). Then Lemma 2.3 implies that | χ f,a ( d , d , h ) | ≤ ( n + 2) q m +1 . Also, if vx + ux − = r ( x ) q m − r ( x ) for some r ( x ) ∈ F q m ( x ), then following [15] we get u = v = 0. Therefore, χ f,a ( d , d , h ) = P u ∈ F q ψ ( − au ) P α ∈ F qm \ U χ d ( α k f ( α ) d/d ).Here, if x k f ( x ) d/d = r ( x ) d for any r ( x ) ∈ F q m ( x ), then using Lemma 2.2 weget | χ f,a ( d , d , h ) | ≤ nq m +1 < ( n + 2) q m +1 . However, x k f ( x ) d/d = r ( x ) d forsome r ( x ) ∈ F q m ( x ) gives that f is exceptional(see [8]).Hence, from the above discussion along with (3.2), we get N f,a,n ( l , l , g ) ≥ θ ( l ) θ ( l )Θ( g ) q ( q m − | U | − (( n + 2) q m +1 )( W ( l ) W ( l ) W ( g ) − ≥ θ ( l ) θ ( l )Θ( g ) q ( q m − ( n + 1) − (( n + 2) q m +1 )( W ( l ) W ( l ) W ( g ) − ≥ θ ( l ) θ ( l )Θ( g ) q ( q m − ( n + 2) q m +1 W ( l ) W ( l ) W ( g )) (3.3)Thus, if q m − > ( n + 2) W ( l ) W ( l ) W ( g ), then N f,a,n ( l , l , g ) > f ( x ) ∈ S q ( n ) and prescribed a ∈ F q . The result now follows by taking l = l = q m − g = x m −
1. 7
Sieving Results
Here, we state some results, their proofs have been omitted as they followon the lines of the results in [10] and have been used frequently in [13, 8, 10,14, 2].
Lemma 4.1.
Let k and P be co-prime positive integers and g, G ∈ F q [ x ] beco-prime polynomials. Also, let { p , p , · · · , p r } be the collection of all primedivisors of P , and { g , g , · · · , g s } contains all the irreducible factors of G .Then N f,a,n ( kP, kP, gG ) ≥ r X i =1 N f,a,n ( kp i , k, g ) + r X i =1 N f,a,n ( k, kp i , g )+ s X i =1 N f,a,n ( k, k, gg i ) − (2 r + s − N f,a,n ( k, k, g ) . Lemma 4.2.
Let l, m, q ∈ N , g ∈ F q [ x ] be such that q is a prime power, m ≥ and l | q m − , g | x m − . Let c be a prime number which divides q m − but not l , and e be irreducible polynomial dividing x m − but not g . Then | N f,a,n ( cl, l, g ) − θ ( c ) N f,a,n ( l, l, g ) | ≤ ( n + 2) θ ( c ) θ ( l ) Θ( g ) W ( l ) W ( g ) q m , | N f,a,n ( l, cl, g ) − θ ( c ) N f,a,n ( l, l, g ) | ≤ ( n + 2) θ ( c ) θ ( l ) Θ( g ) W ( l ) W ( g ) q m and | N f,a,n ( l, l, eg ) − Θ( e ) N f,a,n ( l, l, g ) | ≤ ( n + 2) θ ( l ) Θ( e )Θ( g ) W ( l ) W ( g ) q m . Theorem 4.1.
Let l, m, q ∈ N , g ∈ F q [ x ] be such that q is a prime power, m ≥ and l | q m − , g | x m − . Also, let { p , p , · · · p r } be the collection ofprimes which divides q m − but not l , and { g , g , · · · g s } be the irreduciblepolynomials dividing x m − but not g . Suppose δ = 1 − r P i =1 1 p i − s P i =1 1 q deg( gi ) , δ > and ∆ = r + s − δ + 2 . If q m − > ( n + 2)∆ W ( l ) W ( g ) then ( q, m ) ∈ T n . Now, we present a more effective sieving technique than Theorem 4.1,which is an extension of the result in [6]. For this, we adopt some notationsand conventions from [6] as described. Let Rad( q m −
1) = kP L , where k is the product of smallest prime divisors of q m − L is the product of8arge prime divisors of q m − L = l · l · · · l t , and rest of theprime divisors of q m − P and denoted by p , p , · · · , p r . Similarly,Rad( x m −
1) = gGH , where g is the product of irreducible factors of x m − H whichare denoted by h , h , · · · , h u and rest lie in G and denoted by g , g , · · · , g s . Theorem 4.2.
Let m, q ∈ N such that q is a prime power and m ≥ . Usingabove notations, let Rad ( q m −
1) = kP L , Rad ( x m −
1) = gGH , δ = 1 − r P i =1 1 p i − s P i =1 1 q deg( gi ) , ǫ = t P i =1 1 l i , ǫ = u P i =1 1 q deg( hi ) and δθ ( k ) Θ( g ) − (2 ǫ + ǫ ) > .Then q m − > ( n +2)[ θ ( k ) Θ( g ) W ( k ) W ( g )(2 r + s − δ )+( t − ǫ )+(2 / ( n +2))( u − ǫ )+ ( n/ ( n + 2))(1 /q m/ )( t + u − ǫ − ǫ )] / [ δθ ( k ) Θ( g ) − (2 ǫ + ǫ )] (4.1) implies ( q, m ) ∈ T n .Proof. Clearly, N f,a,n ( q m − , q m − , x m −
1) = N f,a,n ( kP L, kP L, gGH ) ≥ N f,a,n ( kP, kP, gG )+ N f,a,n ( L, L, H ) − N f,a,n (1 , , . (4.2)Further, by Lemma 4.1 N f,a,n ( kP, kP, gG ) ≥ δN f,a,n ( k, k, g )+ r X i =1 { N f,a,n ( kp i , k, g ) − θ ( p i ) N f,a,n ( k, k, g ) } + r X i =1 { N f,a,n ( k, kp i , g ) − θ ( p i ) N f,a,n ( k, k, g ) } + s X i =1 ( N f,a,n ( k, k, gg i ) − Θ( g i ) N f,a,n ( k, k, g )). Using (3.3) and Lemma 4.2, we get N f,a,n ( kP, kP, gG ) ≥ δθ ( k ) Θ( g ) (cid:0) q m − − ( n + 2) W ( k ) W ( g ) q m (cid:1) − ( n + 2) θ ( k ) Θ( g ) W ( k ) W ( g ) (cid:0) r X i =1 θ ( p i ) + s X i =1 Θ( g i ) (cid:1) q m = θ ( k ) Θ( g ) (cid:0) δq m − − ( n + 2)(2 r + s − δ ) W ( k ) W ( g ) q m (cid:1) . (4.3)9gain, by Lemma 4.1 N f,a,n ( L, L, H ) − N f,a,n (1 , , ≥ t X i =1 N f,a,n ( l i , ,
1) + t X i =1 N f,a,n (1 , l i , u X i =1 N f,a,n (1 , , h i ) − (2 t + u ) N f,a,n (1 , , t X i =1 { N f,a,n ( l i , , − θ ( l i ) N f,a,n (1 , , } + t X i =1 { N f,a,n (1 , l i , − θ ( l i ) N f,a,n (1 , , } + u X i =1 { N f,a,n (1 , , h i ) − Θ( h i ) N f,a,n (1 , , } − (2 ǫ + ǫ ) N f,a,n (1 , ,
1) (4.4)By (3.2), for a prime divisor l of q m − | N f,a,n ( l, , − θ ( l ) N f,a,n (1 , , | = θ ( l ) φ ( l ) q | P χ l χ f,a ( l, , | , where | χ f,a ( l, , | = | X u ∈ F q ψ ( − au ) X α ∈ F qm \ U χ l ( α ) ˆ ψ ( uα − | ≤ q m +1 + nq. Hence, | N f,a,n ( l, , − θ ( l ) N f,a,n (1 , , | ≤ θ ( l )( q m + n ) . Similarly, | χ f,a (1 , l, | = | X u ∈ F q ψ ( − au ) X α ∈ F qm \ U χ l ( f ( α )) ˆ ψ ( uα − | ≤ ( n + 1) q m +1 , which further implies | N f,a,n (1 , l, − θ ( l ) N f,a,n (1 , , | ≤ ( n + 1) q m .Also, for an irreducible divisor h of x m − | χ f,a (1 , , h ) | = | X u ∈ F q ψ ( − au ) X α ∈ F qm \ U ψ h ( α ) ˆ ψ ( uα − | = | X u ∈ F q ψ ( − au ) X α ∈ F qm \ U ˆ ψ ( vα + uα − | ≤ q m +1 + nq. Therefore, | N f,a,n (1 , , h ) − Θ( h ) N f,a,n (1 , , | ≤ Θ( h )( q m + n ). Using thesebounds in (4.4), we have N f,a,n ( L, L, H ) − N f,a,n (1 , , ≥ − t P i =1 θ ( l i )( q m +10 ) − t P i =1 θ ( l i )( n + 1) q m − u P i =1 Θ( h i )(2 q m + n ) − (2 t + u ) N f,a,n (1 , , N f,a,n (1 , , ≤ q m − together with t P i =1 θ ( l i ) = ( t − ǫ ) and u P i =1 = ( u − ǫ )implies N f,a,n ( L, L, H ) − N f,a,n (1 , , ≥ −{ ( n + 2)( t − ǫ ) + 2( u − ǫ ) } q m − n ( t + u − ǫ − ǫ ) − (2 ǫ + ǫ ) q m − . (4.5)Now using (4.3) and (4.5) in (4.2) we get, N f,a,n ( q m − , q m − , x m − ≥ { δθ ( k ) Θ( g ) − (2 ǫ + ǫ ) } q m − − θ ( k ) Θ( g )( n +2)(2 r + s − δ ) W ( k ) W ( g ) q m −{ ( n +2)( t − ǫ )+2( u − ǫ ) } q m − n ( t + u − ǫ − ǫ )= q m (cid:2)(cid:0) δθ ( k ) Θ( g ) − (2 ǫ + ǫ ) (cid:1) q m − − ( n +2) { θ ( k ) Θ( g )(2 r + s − δ ) W ( k ) W ( g ) − { ( t − ǫ ) + (2 / ( n + 2))( u − ǫ ) } − ( n/ ( n + 2))(1 /q m/ )( t + u − ǫ − ǫ ) } (cid:3) Thus q m − > ( n +2)[ θ ( k ) Θ( g ) W ( k ) W ( g )(2 r + s − δ )+( t − ǫ )+(2 / ( n +2))( u − ǫ )+ ( n/ ( n + 2))(1 /q m/ )( t + u − ǫ − ǫ )] / [ δθ ( k ) Θ( g ) − (2 ǫ + ǫ )]implies N f,a,n ( q m − , q m − , x m − > q, m ) ∈ T n .It is easy to observe that Theorem 4.1 is a special case of Theorem 4.2and can be obtained by setting t = u = ǫ = ǫ = 0. However the results discussed above are applicable for arbitrary natural num-ber n and the finite field F q m of any prime characteristic. Though to demon-strate the application of above results and make the calculations uncompli-cated we assume that q = 5 k for some k ∈ N and n = 2, and work on the set T . Precisely, in this section, we prove the following result. Theorem 5.1.
Let q = 5 k for some k ∈ N and m ≥ is an integer. Then ( q, m ) ∈ T unless one of the following holds: . q = 5 , , , , , , , and m = 3 ;2. q = 5 , , , and m = 4 ;3. q = 5 , and m = 5 , q = 5 and m = 7 , , , . We shall divide it in two parts, in first part we shall work on m ≥ m = 3 ,
4. For further calculation work and toapply the previous results we shall need the following lemma which can alsobe developed from [5, Lemma 6.2].
Lemma 5.1.
Let M be a positive integer, then W ( M ) < × M / . In this part, we assume m ≥ m = m ′ j , where j ≥ ∤ m ′ . Then Ω q ( x m −
1) = Ω q ( x m ′ −
1) which further implies W ( x m −
1) = W ( x m ′ − • m ′ | q − • m ′ ∤ q − Case 1. m | q − q ( x m ′ −
1) = m ′ . Let l = q m − g = 1 in Theorem 4.1 then ∆ = q +( a − q +2( a − q +1 , where a = q − m ′ , whichfurther implies ∆ < q . Hence ( q, m ) ∈ T if q m − > W ( q m − . However,by Lemma 5.1, it is sufficient if q m − > · (4515) , which holds for q ≥ m ≥
28. In particular, for q ≥
125 and for all m ′ ≥
28. Next, weexamine all the cases where m ′ ≤
27. For this we set l = q m − g = 1in Theorem 4.1 unless mentioned. Then δ = 1 − m ′ q and ∆ = 2 + ( m ′ − qq − m ′ m ′ = 1 . Here m = 5 j for some integer j ≥ q m − > · · W ( q m − . Again Lemma 5.1 implies( q, m ) ∈ T if q m − > · (4515) i.e., q j − > · (4515) , which holds for allchoices of ( q, m ) except (5 , , (5 , ) , (5 , , (5 , ) , (5 , , (5 , , · · · , (5 , q m − > · · W ( q m − di-rectly by factoring q m − , , (5 , , (5 , , ,
5) and (5 , m ′ = 2. In this case, m = 2 · m j for some j ≥ qq − < q · j − > · (4515) , which istrue except the 9 pairs (5 , , (5 , , (5 , , (5 , , · · · , (5 , q m − > · · W ( q m − for these pairs yield the only possibleexceptions as (5 ,
10) and (5 , m ′ ≤
27 we getthat there is no exception for many values of m ′ . Values of m ′ with possibleexceptional pairs is as below. m ′ = 4 . (5 , m ′ = 6 . (5 , , (5 ,
6) and (5 , . m ′ = 8 . (5 , , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , ,
8) Theorem 4.1 holds for some choice of l and g (see Table 1).Hence, only left possible exceptions in this case are (5 , , (5 , , (5 , , q, m ) l r g s δ > ∆ < W ( g ) W ( l ) < ,
5) 2 5 1 1 0 . . ,
5) 6 6 1 1 0 . . ,
5) 6 9 1 1 0.390631 48.079201 61554 (5 ,
10) 6 6 1 2 0.503329 27.828038 35625 (5 ,
20) 6 6 x + β x + β ,
6) 6 6 1 6 0.476599 37.669274 48227 (5 ,
6) 6 9 1 6 0.330094 71.677019 91758 (5 ,
8) 6 4 1 8 0.401942 39.318735 5033where β is a primitive element of F . Case 2. m ′ ∤ q − q mod m ′ be denoted by b . Then b ≥ x m ′ − F q is less than or equal to b . Let M denotes the number of distinct irreducible factors of x m − F q of de-gree less than b . Also let ν ( q, m ) denotes the ratio ν ( q, m ) = Mm . Then, mν ( q, m ) = m ′ ν ( q, m ′ ). 13or the further progress, we need the following two results which are thedirectly implied by Proposition 5 . Lemma 5.2.
Let k, m, q ∈ N be such that q = 5 k and m ′ ∤ q − . In thenotations of Theorem 4.1, let l = q m − and g is the product of irreduciblefactors of x m − of degree less than b , then ∆ < m ′ . Lemma 5.3.
Let m ′ > and m = gcd( q − , m ′ ) . Then following boundshold.1. For m ′ = 2 m , ν ( q, m ′ ) = ;
2. for m ′ = 4 m , ν ( q, m ′ ) = ;
3. for m ′ = 6 m , ν ( q, m ′ ) = ;
4. otherwise, ν ( q, m ′ ) ≤ . At this point we note that m ′ = 1 , q − q = 5 k andhave been discussed in above case, whereas m ′ = 5 is not possible. Therefore,in this case we need to discuss m ′ = 3 and m ′ ≥ m ′ = 3. Then m = 3 · j for some integer j ≥
1. Also, m ′ ∤ q − q = 5 k then k is odd and x m ′ − W ( x m −
1) = W ( x m ′ −
1) = 2 = 4and (3 .
1) implies ( q, m ) ∈ T if q m − > · W ( q m − . By Lemma 5.1,it is sufficient if q m − > · (4515) , which hold for q = 5 and m ≥ q = 125 and m ≥ q ≥ and m ≥
14. Thus, only possible exceptions are(5 ,
15) and (125 , q m − > · W ( q m − directly by factoring q m − , m ′ = 3 is (5 , m ′ ≥
6. At this point, in Theorem 4.1 let l = q m − g be the product of irreducible factors of x m − b . Therefore, Lemma 5.2 along with Theorem 4.1 implies ( q, m ) ∈ T if q m − > · m ′ · W ( q m − · m ′ ν ( q,m ′ ) . By Lemma 5.1, it is sufficient if q m − > · m · (4515) · mν ( q,m ′ ) . (5.1)Further, we shall discuss it in four cases as follows. m ′ = 2 m , m , m . Here, Lemma 5.3 implies ν ( q, m ′ ) = . Using this in (5.1) we get ( q, m ) ∈ T q m − > · m · (4515) · m , which holds for q m ≥ . Next, for q m ≤ ,we verified q m − > · m · W ( q m − · m by factoring q m − , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , , (5 , m ′ = 2 m . In this case, ν ( q, m ) = . Therefore, (5.1) implies ( q, m ) ∈ T if q m − > · m · (4515) · m , which holds for q = 5 and m ≥
466 while for q ≥ m ≥
56. Here, for q = 5, we have m ′ = 8 only. Thuspossible exception for q = 5 are (5 , , (5 ,
40) and (5 , q ≥
25 and q m < along with above three possible excep-tions we checked q m − > · m · W ( q m − · m and got it verified except(5 , , (5 ,
40) and (5 , m ′ = 4 m . Here, ν ( q, m ) = . Again, (5.1) gives ( q, m ) ∈ T if q m − > · m · (4515) · m ,which is true for q m ≥ . On the other side, verification of q m − > · m · W ( q m − · m for q m < provides only possible exception as(5 , m ′ = 6 m . Similar to the above case, we have ν ( q, m ) = and q m − > · m · (4515) · m holds for q m ≥ . Also, for q m < , q m − > · m · W ( q m − · m holds for all ( q, m ) except (5 , q, m ) l r g s δ > ∆ < W ( g ) W ( l ) < ,
11) 2 1 1 3 0.799359 7.004009 2252 (5 ,
13) 2 1 1 4 0.795199 8.287731 2663 (5 ,
14) 2 4 x + 1 3 0.059683 169.55170 54264 (5 ,
17) 2 2 1 2 0.795110 8.288442 2665 (5 ,
18) 6 5 1 6 0.061578 245.59029 314366 (5 ,
19) 2 3 1 3 0.789208 12.136745 3897 (5 ,
21) 2 4 1 5 0.689908 19.393614 6218 (5 ,
22) 2 5 x + 1 5 0.014867 943.67119 301989 (5 ,
27) 2 7 1 4 0.561470 32.277659 103310 (5 ,
30) 6 9 x + 1 3 0.110695 182.67531 2338311 (5 ,
36) 6 9 x − ,
7) 2 4 1 3 0.219683 47.520125 152113 (5 ,
9) 6 5 1 5 0.421578 35.208505 450714 (5 ,
11) 2 5 1 3 0.176146 70.124930 224415 (5 ,
6) 6 5 1 4 0.525578 26.734639 342316 (5 ,
6) 6 9 10 4 0.390055 55.838482 714817 (5 ,
15) 2 5 1 2 0.473298 25.241167 80818 (5 ,
40) 6 9 x + β x + β ,
8) 6 6 1 6 0.454072 39.438940 504920 (5 ,
16) 6 4 x + 1 7 0.038742 363.35624 4651021 (5 ,
24) 6 6 x − , , , , , , , , , , , , , , , , , , , , , possible exceptions in the case m ′ ∤ q − , , , , , . In this part we shall consider m = 3 , . Following result will be required forfurther calculation, which follows on the lines of [6, Lemma 51].
Lemma 5.4.
Let k ∈ N such that ω ( k ) ≥ . Then W ( k ) < k . W ( x m − ≤
16. Now, first assume ω ( q m − ≥ q, m ) ∈ T if q m − > · q m i.e., q m − >
64 or q m > m m − , sufficient if q m > , which is true for ω ( q m − ≥ ≤ ω ( q m − ≤ g = x m − l to be the product of least 88 primesdividing q m − W ( l ) = 2 . Then r ≤ δ will be at least its valuewhen { p , p , · · · , p } = { , , · · · , } . This gives δ > . < . × , hence 4∆ W ( g ) W ( l ) < . × = R (say). ByTheorem 4.1 ( q, m ) ∈ T if q m − > R or q m > R mm − . But m ≥ mm − ≤
6. Therefore, if q m > R or q m > . × then ( q, m ) ∈ T .Hence, ω ( q m − ≥
152 gives ( q, m ) ∈ T . Repeating this process of Theorem4.1 for the values in Table 3 implies ( q, m ) ∈ T if q m − > m = 3 it is sufficient if q > (889903387) and for m = 4 we need q > , , (5 , , · · · , (5 ,
3) and(5 , , (5 , , · · · , (5 , , , (5 , , (5 , , (5 , , · · · , (5 ,
3) and (5 , , (5 , , · · · , (5 , possible exceptions here are (5 , , (5 , , · · · , (5 ,
3) and (5 , , , (5 , , · · · , (5 , a ≤ ω ( q m − ≤ b W ( l ) δ > ∆ < W ( g ) W ( l ) < a = 17 , b = 151 2 . . . × a = 9 , b = 51 2 . . . × a = 7 , b = 37 2 . . a = 7 , b = 36 2 . . a = 7 , b = 34 2 . . a = 7 , b = 33 2 . . q, m ) l r g s δ > ∆ < W ( g ) W ( l ) < ,
3) 2 7 1 2 0.801533 20.714128 6632 (5 ,
3) 2 4 1 2 0.925433 11.725177 3763 (5 ,
3) 6 9 1 3 0.330478 62.518314 80034 (5 ,
3) 2 4 1 2 0.910167 11.888295 3815 (5 ,
3) 6 10 1 3 0.508443 45.269297 57956 (5 ,
3) 2 10 1 2 0.603902 36.773815 11777 (5 ,
3) 6 9 1 3 0.368379 56.291827 72068 (5 ,
3) 2 6 1 2 0.930565 15.970005 5129 (5 ,
3) 6 12 1 3 0.499055 54.098369 692510 (5 ,
3) 2 5 1 2 0.924693 13.895837 44511 (5 ,
3) 6 15 1 3 0.183646 176.24807 2256012 (5 ,
3) 2 9 1 2 0.822416 25.102645 80413 (5 ,
3) 6 10 1 3 0.522529 44.102865 564614 (5 ,
3) 2 7 1 2 0.920550 18.294603 58615 (5 ,
3) 6 14 1 3 0.296682 103.11815 1320016 (5 ,
3) 2 14 1 2 0.666688 45.498589 145617 (5 ,
4) 6 6 1 4 0.485944 32.867712 420818 (5 ,
4) 2 6 1 4 0.105913 143.62473 459619 (5 ,
4) 2 7 1 4 0.054494 313.95724 1004720 (5 ,
4) 6 9 1 4 0.330476 65.544620 839021 (5 ,
4) 6 9 1 4 0.568640 38.930216 498422 (5 ,
4) 2 8 1 4 0.039829 479.03888 1533023 (5 ,
4) 6 9 1 4 0.368379 59.006421 7553Further, for all the left possible exceptions we checked Theorem 4.2 andgot it verified in case of (5 , , (5 ,
4) and (5 ,
9) for the values in Table 5.Table 5Sr.No. ( q, m ) k P L f G H R ′ < ,
9) 2 589 829 x − x + x + 1 x + x +1 2692 (5 ,
3) 2 229469719 519499 x − x + x + 1 2623 (5 ,
4) 6 216878233 9161 x +1 x + x + β x + β R ′ represent the right hand side value of (4.1). Hence, all the resultsfrom part 1 and part 2 collectively implies Theorem 5.1.18 eferences [1] G. B. Agnew, R. C. Mullin, I. M. Onyszchuk, and S. A. Vanstone.An implementation for a fast public-key cryptosystem. J. Cryptology ,3(2):63–79, 1991.[2] Anju and R. K. Sharma. Existence of some special primitive normalelements over finite fields.
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