Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F

Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F

and over F

for linear recurring Contents lists available at ScienceDirect

Finite Fields and Their Applications

www.elsevier.com/locate/ffa

Linear complexity over F q and over F q ^m for linear recurring sequences

Wilfried Meidl ^a , Ferruh Özbudak ^{b , ∗}

a

Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, 34956, ˙Istanbul, Turkey

Department of Mathematics, Middle East Technical University, ˙Inönü Bulvarı, 06531, Ankara, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:

Received 7 August 2008 Revised 25 September 2008 Communicated by Gary L. Mullen

Keywords:

Joint linear complexity

Generalized joint linear complexity Multisequences

Linear recurring sequences

Since the F

-linear spaces F

and F

are isomorphic, an m-fold multisequence S over the finite field F

with a given characteristic polynomial f ∈ F

[ ^x ] , can be identified with a single sequence S over F

with characteristic polynomial f . The linear complexity of S , which will be called the generalized joint linear complexity of S , can be significantly smaller than the conventional joint linear complexity of S . We determine the expected value and the variance of the generalized joint linear complexity of a random m- fold multisequence S with given minimal polynomial. The result on the expected value generalizes a previous result on periodic m- fold multisequences. Moreover we determine the expected drop of linear complexity of a random m-fold multisequence with given characteristic polynomial f , when one switches from conventional joint linear complexity to generalized joint linear complexity.

© 2008 Elsevier Inc. All rights reserved.

1. Introduction

A sequence S = ^s 0 , s 1 , . . . with terms in a finite field F q with q elements (or over the finite field F q ) is called a linear recurring sequence over F q with characteristic polynomial

f ( x ) =

 l i = ⁰

c i x ⁱ ∈ F q [ ^x ]

* Corresponding author.

E-mail addresses:

(W. Meidl),

(F. Özbudak).

1071-5797/$ – see front matter © 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.ffa.2008.09.004

of degree l, if

 l i = ⁰

c _i s _{n + i} = ^{0 for n} = ⁰ , 1 , . . . .

Without loss of generality we can always assume that f ( x ) is monic, i.e. c _l = 1. In accordance with the notation in [2] we denote the set of sequences over F q with characteristic polynomial f by M ⁽ q ^{1 )} ( f ) . The minimal polynomial of a linear recurring sequence S ∈ M ⁽ q ^{1 )} ( f ) is defined to be the (uniquely determined) monic polynomial d ( x ) ∈ F q [ ^x ] of smallest degree such that S ∈ M q ^{( 1 )} ( d ) . We remark that then d is a divisor of f . The degree of d is called the linear complexity L ( S ) of the sequence S.

Motivated by the study of vectorized stream cipher systems (see [1,3]) we consider the set of m parallel sequences over F q , each of them being in M ⁽ q ^{1 )} ( f ) . As usual we call this set the set of m-fold multisequences over F q with joint characteristic polynomial f and denote it by M ⁽ q ^{m )} ( f ) . The joint minimal polynomial of an m-fold multisequence S = ( σ 1 , σ 2 , . . . , σ m ) ∈ M ⁽ q ^{m )} ( f ) is then defined to be the (uniquely determined) monic polynomial d of least degree which is a characteristic polynomial for all sequences σ r , 1  ^r  m. The joint linear complexity L ⁽ _q ^{m )} ( S ) of S is then the degree of d.

Let S = ( σ 1 , σ 2 , . . . , σ m ) ∈ M ⁽ q ^{m )} ( f ) and suppose that σ r = ^s r , 0 s r , 1 s r , 2 . . . , 1  ^r  m. Then there exist unique polynomials g _r ∈ F q [ ^x ] ^{with deg} ( g _r ) < deg ( f ) and g _r / f = ^s r , 0 + ^s r , 1 x + ^s r , 2 x ² · · · ^{, 1}  r  m. By [7, Lemma 1] this describes a one-to-one correspondence between the set M ⁽ q ^{m )} ( f ) and the set of m-tuples of the form ( g 1 / f , g 2 / f , . . . , g m / f ) , g r ∈ F q [ ^x ] ^{and deg} ( g r ) < deg ( f ) for 1  ^r  ^m.

If S ∈ M ⁽ q ^{m )} ( f ) corresponds to ( g 1 / f , g 2 / f , . . . , g m / f ) , then the joint minimal polynomial d of S is the unique polynomial in F q [ ^x ] for which there exist h 1 , . . . , h m ∈ F q [ ^x ] ^{with g} r / f = ^h r / d for 1  r  ^{m, and gcd} ( h 1 , . . . , h m , d ) = 1. Therefore the joint linear complexity of S is then given by

L ⁽ _q ^{m )} ( S ) = ^deg ( f ) − ^deg 

gcd ( g ₁ , g ₂ , . . . , g _m , f )  .

Since the F q -linear spaces F ^m q and F q

are isomorphic, the multisequence S can be identified with a single sequence S having its terms in the extension field F q

, namely S = S( ^S , ξ ) = ^s 0 , s 1 , . . . with

s n = ξ 1 s 1 , n + · · · + ξ m s m , n ∈ F q

, n  ⁰ , (1.1) where ξ = (ξ 1 , . . . , ξ m ) is an arbitrary but fixed ordered basis of F q

over F q . This describes a one-to- one correspondence between the sets M ⁽ q ^{m )} ( f ) and M ⁽ _q ¹

⁾ ( f ) .

Let S ∈ M ⁽ q ^{m )} ( f ) correspond to ( g 1 / f , g 2 / f , . . . , g m / f ) , then it is easily seen that the single se- quence S ∈ M ⁽ _q ¹

⁾ ( f ) defined as in ( 1 . 1 ) corresponds to the 1-tuple ( G / f ) with

G ( x ) = ^g 1 ξ 1 + ^g 2 ξ 2 + · · · + ^g m ξ m .

The minimal polynomial of S ^{is then d} = ^f / gcd ( G , f ) ∈ F q

[ ^x ] and the linear complexity of the sequence S , which we will call the generalized joint linear complexity of S and denote by L _q

_,ξ ( S ) , is given by

L q

,ξ ( S ) = ^deg ( f ) − ^deg 

gcd ( G , f )  ,

where the greatest common divisor is now calculated in F q

[ ^x ] . The dependence of the generalized

joint linear complexity L _q

,ξ ( S ) on the ordered basis ξ follows from the definition (cf. [5, Example 3]).

Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F

and over F

for linear recurring Clearly we always have L q

,ξ ( S )  ^{L (} q ^{m )} ( S ) , in some cases L q

,ξ ( S ) can be considerably smaller than L ⁽ _q ^{m )} ( S ) . However in [5, Theorem 2] it has been pointed out that

L _q

_,ξ ( S ) 

 k i = ¹

a _i deg ( r i ) gcd ( deg ( r i ), m )

if S ∈ M ⁽ q ^{m )} ( f ) , f = ^{r e} ₁

^r ₂ ^e

· · · ^r _k ^e

is the canonical factorization of f into irreducibles over F q , and the joint minimal polynomial of S is d = ^{r a} ₁

^{r a} ₂

· · · ^r _k ^a

^{, 0}  ^a i  ^e i for 1  ⁱ  k. As one consequence we will always have L _q

_,ξ ( S ) = ^{L (} q ^{m )} ( S ) if gcd ( deg ( r i ), m ) = ^{1 for i} = ¹ , 2 , . . . , k (cf. [5, Theorem 1]). In [4, Theorem 3] the expected value for the generalized joint linear complexity of a random m-fold multisequence S with minimal polynomial x ^N − 1 for a given integer N has been determined. In this article with a different method we obtain much more general results and present expected value and variance for the generalized joint linear complexity of a random m-fold multisequence S with an arbitrary given minimal polynomial. Moreover we present results on the expected value of D ( S ) :=

L

(

)− ^L

(

)

L

(

) , the difference of joint linear complexity and generalized joint linear complexity in relation to the value for the joint linear complexity of an m-fold multisequence S , which estimates the expected drop of linear complexity if one switches from conventional joint linear complexity to generalized joint linear complexity.

The rest of the paper is organized as follows. In Section 2 we fix some notation and we give some basic results that we use later. We obtain our main results in Section 3.

2. Preliminaries

We first recall an important function on the set of monic polynomials in F q [ ^x ] and some of its properties (see [2, Section 2]). For a monic polynomial f ∈ F q [ ^x ] and a positive integer m we let Φ q ^{( m )} ( f ) denote the number of m-fold multisequences over F q with joint minimal polynomial f . Then we have [2, Lemmas 2.1 and 2.2]



d | ^f

Φ _q ^{( m )} ( d ) = ^{q m deg ( f )} , (2.1)

Φ q ^{( m )} ( f ₁ f ₂ ) = Φ q ^{( m )} ( f ₁ )Φ q ^{( m )} ( f ₂ ) if gcd ( f ₁ , f ₂ ) = ¹ . (2.2) Let N q ^{( m )} ( f ) denote the subset of M q ^{( m )} ( f ) consisting of multisequences S ∈ M ⁽ q ^{m )} ( f ) such that L ⁽ _q ^{m )} ( S ) = ^deg ( f ) . It is clear that

N q ^{( m )} ( f )  = Φ _q ^{( m )} ( f ).

For an ordered basis ξ = (ξ 1 , . . . , ξ m ) of F q

over F q let N  _q ⁽

¹ _,ξ ⁾ ( f ) be the subset of M ⁽ _q ¹

⁾ ( f ) given by

N  _q ⁽

¹ ,ξ ⁾ ( f ) = 

S = S( ^S , ξ ) : S ∈ N q ^{( m )} ( f )  .

It is obvious that |  N _q ⁽

^{1 )} ,ξ ( f )| = |N q ^{( m )} ( f )| = Φ q ^{( m )} ( f ) .

Proposition 2.1. Let f ∈ F q [ ^x ] be a monic polynomial with deg ( f )  1 and suppose that

f = ^{r e} ₁

^r ₂ ^e

· · · ^r _k ^e

is the canonical factorization of f into irreducibles over F q . Let ξ = (ξ 1 , . . . , ξ m ) be an ordered basis of F q

over F q , let S be a sequence in M ⁽ _q ¹

⁾ ( f ) and let d ∈ F q

[ ^x ] be its minimal polynomial. Then S ∈  N _q ⁽

¹ _,ξ ⁾ ( f ) if and only if d is of the form

d = ^d 1 d ₂ · · · ^d k , where d 1 , d 2 , . . . , d k ∈ F q

[ ^x ] ^{and d} 1 | ^{r e} ₁

, d 2 | ^r ₂ ^e

, . . . , d k | ^{r e} _k

^{, and}

d 1  ^{r e} ₁

^{− 1} , d 2  ^{r e} ₂

^{− 1} , . . . , d _k  ^{r e} _k

^{− 1} .

Proof. Suppose that S corresponds to G / f and let g 1 , g 2 , . . . , g m be the unique polynomials in F q [ ^x ] for which G = ξ 1 g 1 + ξ 2 g 2 + · · · + ξ m g m . If d is the minimal polynomial of S then trivially d is of the form d = ^d 1 d ₂ · · · ^d k , where d ₁ , d ₂ , . . . , d _k ∈ F q

[ ^x ] ^and

d 1 | ^{r e} ₁

, d 2 | ^{r e} ₂

, . . . , d _k | ^{r e} _k

.

Suppose that without loss of generality d 1 | r ^e ₁

^{− 1} . Then r 1 divides f / d, and consequently G / f = G 1 / d implies that r 1 divides G = ^G 1 f / d. With [5, Proposition 1.2] we obtain that r 1 divides g 1 , g 2 , . . . , g m , thus ( g 1 , g 2 , . . . , g m , f ) = 1 and f is not the minimal polynomial of S ∈ M ⁽ q ^{m )} ( f ) for which we have S = S( S , ξ ) .

Suppose conversely that d _i  ^{r e} _i

^{− 1 for i} = ¹ , 2 , . . . , k, but ( g ₁ , g ₂ , . . . , g _m , f ) = ^{1. Then r} i divides g _j , 1  ^j  m, for an integer i, 1  ⁱ  k. Consequently by [5, Proposition 1.2] r i divides G, and d =

f / gcd ( G , f ) (where the greatest common divisor is calculated over F q

) contradicts d i  r ^e _i

^{− 1} . 2 Remark 2.2. Note that Proposition 2.1 implies that, amongst others, N  _q ⁽

¹ ,ξ ⁾ ( f ) is independent from the choice of the ordered basis ξ , and we can simply write N  _q ⁽

^{1 )} ( f ) instead of N  _q ⁽

¹ ,ξ ⁾ ( f ) . Similarly the expectation  E q

( f ) and the variance  V ar q

( f ) are independent from the choice of the ordered basis ξ , and hence in the following we will not include ξ in the notations  E q

( f ) and  V ar q

( f ) for the expected value and the variance.

The following definitions are useful.

Definition 2.3. Let f ∈ F q [ ^x ] be a monic polynomial with canonical factorization f = ^{r e} ₁

^{r e} ₂

· · · ^{r e} _k

into irreducibles over F q . We define  S q

, 1 ( f ) and  S q

, 2 ( f ) as

 S _q

_{, 1} ( f ) = 

d

| ^r

d

 r



d

| ^r

d

 r

· · · 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d ₁ d ₂ · · · ^d k ) deg ( d ₁ d ₂ · · · ^d k ),

and

 S q

, 2 ( f ) = 

d

| ^r

d

 ^r



d

| ^r

d

 ^r

· · · 

d

| r

d

 r

Φ _q ^{( 1}

⁾ ( d 1 d 2 · · · ^d k ) 

deg ( d 1 d 2 · · · ^d k )  2

,

where the summations are over monic polynomials d _i ∈ F q

[ ^x ] such that d _i | ^{r e} _i

^{and d} i  ^{r e} _i

^{− 1 .}

Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F

and over F

for linear recurring The identities in the following lemma will be used in Section 3.

Lemma 2.4. Let r 1 , r 2 , . . . , r _k be distinct irreducible polynomials in F q [ ^x ] ^{and e} 1 , e 2 , . . . , e _k be positive inte- gers. We have

 S q

, 1 ( r ^e ₁

r ^e ₂

· · · ^r _k ^e

) _k

i = ¹ ( q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ ) =

 k i = ¹

 S _q

_{, 1} ( r ^e _i

)

q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ , (2.3) and

 S q

, 2 ( r ^e ₁

r ^e ₂

· · · ^r _k ^e

) k

i = ¹ ( q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ )

=

 k i = ¹

 S _q

_{, 2} ( r _i ^e

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

+ ² 

1  ⁱ < j  ^k

 S q

, 1 ( r _i ^e

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

 S q

, 1 ( r ^e _j

)

q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ . (2.4)

Proof. The identities are trivial if k = 1. Assume that k = ^{2. With} ( 2 . 2 ) we have

 S _q

_{, 1}  r ^e ₁

r ^e ₂



= 

d

| ^r

d

 r



d

| ^r

d

 r

Φ _q ^{( 1}

⁾ ( d ₁ )Φ _q ^{( 1}

⁾ ( d ₂ ) 

deg ( d ₁ ) + ^deg ( d ₂ )  .

Then we get

 S _q

_{, 1}  r ₁ ^e

r ^e ₂



= 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 1 ) deg ( d 1 ) 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 2 )

+ 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 2 ) deg ( d 2 ) 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 1 )

= ^S q

, 1

 r ^e ₁



q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾  + ^S q

, 1

 r ^e ₂



q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ 

, (2.5)

where the identity



d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d i ) = 

q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ 

(2.6)

for i = ¹ , 2 follows from ( 2 . 1 ) . Dividing both sides of (2.5) by 2

i = ¹ ( q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ ) we obtain (2.3) for k = 2. We complete the proof of (2.3) by induction on k using similar arguments.

Again for k = ^{2, we have}

 S q

, 2

 r ^e ₁

r ^e ₂



= 

d

| r

d

 ^r



d

| r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 1 )Φ _q ^{( 1}

⁾ ( d 2 )

× 

deg ( d 1 )  2

+ 

deg ( d 2 )  2

+ ^{2 deg} ( d 1 ) deg ( d 2 )  .

Then we get

 S q

, 2

 r ₁ ^e

r ^e ₂



= 

d

| r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 1 ) 

deg ( d 1 )  2 

d

| r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 2 )

+ 

d

| r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 2 ) 

deg ( d 2 )  2 

d

| r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 1 )

+ ² 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 1 ) deg ( d 1 ) 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d 2 ) deg ( d 2 ),

and hence

 S q

, 2

 r ^e ₁

r ^e ₂



= ^S q

, 2

 r ^e ₁



q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾  + ^S q

, 2

 r ₂ ^e



q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ 

+ ²  S q

, 1

 r ^e ₁

 S q

, 1

 r ^e ₂



. (2.7)

Dividing both sides of (2.7) by 2

i = 1 ( q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ ) we obtain (2.4) for k = 2. We complete the proof of (2.4) by induction on k using similar arguments. 2

3. Main results

In [2, Theorem 3.4] exact formulas for the expected value E ^{( m )} ( f ) and the variance Var ^{( m )} ( f ) of the joint linear complexity of a random m-fold multisequence S ∈ M ⁽ q ^{m )} ( f ) has been presented: Let f = ^{r e} ₁

^r ₂ ^e

· · · ^r _k ^e

be the canonical factorization of f into monic irreducible polynomials over F q , then

E ^{( m )} ( f ) = ^deg ( f ) −

 k i = ¹

1 − α _i ^{− e}

α i − ^{1 deg} ( r _i ), (3.1)

Var ^{( m )} ( f ) =

 k i = ¹

deg ( r i ) 1 − α _i ^{− 1}

 2 

( 2e _i + ¹ ) 

α ⁻ _i ^e

^{− 2} − α ⁻ _i ^e

^{− 1 } − α _i ^{− 2e}

^{− 2} + α _i ^{− 1 } ,

where α i = ^{q m deg ( r}

^{) for 1}  ⁱ  k. In this section we present the expected value  E _q

( f ) and the variance  V ar q

( f ) for the generalized joint linear complexity of a random m-fold multisequence S ∈ M ⁽ q ^{m )} ( f ) with the maximal possible joint linear complexity deg ( f ) . The result on the expected value generalizes the result on N-periodic multisequences given in [4, Theorem 3].

Theorem 3.1. Let r 1 , r 2 , . . . , r k be distinct irreducible polynomials in F q [ ^x ] ^{and e} 1 , e 2 , . . . , e k be positive inte-

gers. We have

Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F

and over F

for linear recurring

 E q

 r ^e ₁

r ₂ ^e

· · · r _k ^e



=

 k i = ¹

 E q

 r ^e _i

 ,

and

 V ar q

 r ₁ ^e

r ^e ₂

· · · ^{r e} _k



=

 k i = 1

 V ar q

 r _i ^e

 .

Proof. With Definition 2.3, ( 2 . 6 ) and ( 2 . 3 ) we obtain

 k i = ¹

 E _q

 r ^e _i



=

 k i = ¹

 S q

, 1 ( r ^e _i

)

q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ =  S q

, 1 ( r ^e ₁

r ^e ₂

· · · ^r _k ^e

) k

i = ¹ ( q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ ) . (3.2) Hence it remains to show that

 k i = ¹

 q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ 

= 

d

| ^r

d

 r



d

| ^r

d

 r

· · · 

d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d ₁ d ₂ · · · ^d k ). (3.3)

For k = ^{2 with} ( 2 . 6 ) and ( 2 . 2 ) we obtain  2

i = ¹

 q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ 

=  2 i = ¹



d

| ^r

d

 ^r

Φ _q ^{( 1}

⁾ ( d _i ) = 

d

| ^r

d

 r



d

| ^r

d

 r

Φ _q ^{( 1}

⁾ ( d ₁ d ₂ ).

We complete the proof on the expectation by induction on k. Next we consider the variance. With Definition 2.3, ( 2 . 4 ) , ( 3 . 2 ) and ( 3 . 3 ) we obtain

 V ar q

 r ^e ₁

r ₂ ^e

· · · ^r _k ^e



=  S q

, 2 ( f )

Φ q ^{( m )} ( f ) −  S q

, 1 ( f ) Φ q ^{( m )} ( f )

 2

=

 k i = ¹

 S _q

_{, 2} ( r _i ^e

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

+ ² 

1  ⁱ < j  ^k

 S q

, 1 ( r _i ^e

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

 S q

, 1 ( r ^e _j

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

− _k



i = ¹

 S q

, 1 ( r ^e _i

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

 2

=

 k i = ¹

 S _q

_{, 2} ( r _i ^e

)

q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾ −

 S _q

_{, 1} ( r ^e _i

) q ^me

^{deg ( r}

⁾ − ^{q m ( e}

^{− 1 ) deg ( r}

⁾

 2 

=

 k i = ¹

 V ar q

 r ^e _i



. 2

Before we present formulas for  E q

( f ) and  V ar q

( f ) if f = ^{r e , r} ∈ F q [ ^x ] irreducible, we recall some definitions and identities from [2]: For a monic polynomial f ∈ F q [ ^x ] ^{, let}

S q , 1 ( f ) = 

d | f

Φ _q ^{( 1 )} ( d ) deg ( d ), (3.4)

and

S q , 2 ( f ) = 

d | ^f

Φ _q ^{( 1 )} ( d )  deg ( d )  2

,

where the summation is over monic polynomials d ∈ F q [ ^x ] dividing f . If f = ^{r e} ₁

^r ₂ ^e

· · · ^r _k ^e

is the canon- ical factorization of f into monic irreducible polynomials over F q and α i = ^{q deg ( r}

⁾ , then (see [2, Proposition 3.2])

S _{q , 1} ( f ) = ^{q deg ( f )}

 k i = ¹

S _{q , 1} ( r _i ^e

)

α _i ^e

^{, (3.5)}

S _{q , 2} ( f ) = ^{q deg ( f )} _k



i = ¹ S _{q , 2} ( r ^e _i

)

α _i ^e

^{+ 2}



1  i < j  k S _{q , 1} ( r _i ^e

)

α _i ^e

S q , 1 ( r ^e _j

) α ^e _j



. (3.6)

Particularly, if f = ^{r e with r} ∈ F q [ ^x ] irreducible, then (see [2, Eqs. (3.9) and (3.12)])

S q , 1 ( r ^e )

α ^{e = deg ( r )}

e − ¹ − α ^{− e} α − ¹

, (3.7)

S _{q , 2} ( r ^e ) α ^{e =}

deg ( r ) α − ¹

 2 

e ² α ² ₋ ^ 2e ² + ^2e − ¹ 

α _{+ (} e + ¹ ) ² − α ^{1 − e} ₋ α ^{− e } , (3.8)

where α ₌ q ^{deg ( r )} .

Proposition 3.2. Let r be an irreducible polynomial in F q [ ^x ] ^{, and e} , m be positive integers, and suppose that u = ^gcd ( deg ( r ), m ) . Then with β = ^q

^{deg ( r ) we have}

 E q

 r ^e 

= ^{e deg} ( r ) − ^deg ( r )

1 − β ^{− e}

β − ¹ − ¹ − β ^{− e} β ^u − ¹ 

, (3.9)

and

 V ar q

 r ^e 

= ¹ u

deg ( r ) (β − ¹ )( 1 − β ^{− u} )

 2 

β − ( ^2e + ¹ )β ^{1 − e} + ( ^2e + ¹ )β ^{− e} − β ^{− 2e} + β ^{− u} 

β ^{− e} − ¹ 

2 β + β ^{− e} + β ^{2 − e} − ^u (β − ¹ ) ² 

β ^{− e} − ¹ 

+ ^2e β ^{− e}  β ² − ¹ 

+ β ^{− 2u} 

β − ( ^2e − ¹ )β ^{2 − e} + ( ^2e − ¹ )β ^{1 − e} − β ^{2 − 2e} 

. (3.10)

Proof. Note that

 E q

 r ^e 

=  S q

, 1 ( r ^e )

q ^{me deg ( r )} − ^{q m ( e − 1 ) deg ( r ) (3.11)}

Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F

and over F

for linear recurring and

 S q

, 1

 r ^e 

= ^S q

, 1

 r ^e 

− ^S q

, 1

 r ^{e − 1} 

. (3.12)

By [6, Theorem 3.46] the canonical factorization of r into irreducibles over F q

is of the form

r = ρ 1 ρ 2 · · · ρ u , (3.13) where deg ( ρ i ) = ^deg ( r )/ u. With ( 3 . 5 ) we have

S q

, 1

 r ^e 

= ^{q me deg ( r )}

 u i = 1

S q

, 1 ( ρ _i ^e )

β ^e . (3.14)

For 1  ⁱ  ^{u, from E ( 1 )} ( ρ _i ^e ) = ^S q

, 1 ( ρ _i ^e )/β ^e and ( 3 . 1 ) we get S _q

_{, 1} ( ρ ^e _i )

β ^e = ^deg ( r ) u

e − ¹ − β ^{− e} β − ¹

. (3.15)

Using (3.14) and (3.15) we obtain that

S q

, 1

 r ^e 

= ^{q me deg ( r ) deg} ( r )

e − ¹ − β ^{− e} β − ¹

, (3.16)

and similarly

S q

, 1

 r ^{e − 1} 

= ^{q m ( e − 1 ) deg ( r ) deg} ( r )

e − ¹ − ¹ − β ^{− e + 1} β − ¹

. (3.17)

Note that

e − ¹ − ¹ − β ^{− e + 1}

β − ¹ =

e − ¹ − β ^{− e} β − ¹

−  1 − β ^{− e} 

. (3.18)

Combining (3.16)–(3.18) we get

S _q

_{, 1} ( r ^e ) − ^S q

, 1 ( r ^{e − 1} )

q ^{me deg ( r )} − q ^{m ( e − 1 ) deg ( r )} = ^{e deg} ( r ) − ^deg ( r ) ¹ − β ^{− e}

β − ¹ + ^deg ( r ) ¹ − β ^{− e}

β ^u − ¹ . (3.19) We complete the proof of (3.9) using (3.11), (3.12) and (3.19).

Next we consider the variance  V ar q

( r ^e ) . Note that

 V ar q

 r ^e 

=  S q

, 2 ( r ^e )

β ^ue ( 1 − β ^{− u} ) −  S q

, 1 ( r ^e ) β ^ue ( 1 − β ^{− u} )

 2

(3.20)

and as in (3.12) we have

 S _q

_{, 2}  r ^e 

= ^S q

, 2

 r ^e 

− ^S q

, 2

 r ^{e − 1} 

. (3.21)