A General Approach to Construction and Determination of the Linear Complexity of Sequences Based on Cosets

Abstract

We give a general approach to N-periodic sequences over a finite field

\mathbb Fq\mathbb F_q constructed via a subgroup D of the group of invertible elements modulo N. Well known examples are Legendre sequences or the two-prime generator. For some generalizations of sequences considered
in the literature and for some new examples of sequence constructions we determine the linear complexity.

Full text

Page 1

A general approach to construction and
determination of the linear complexity of
sequences based on cosets
Ay¸ ca C ¸e¸ smelio˘ glu and Wilfried Meidl
Faculty of Engineering and Natural Sciences,
Sabancı University, Tuzla, 34956,˙Istanbul, Turkey
Abstract. We give a general approach to N-periodic sequences over a
finite field Fq constructed via a subgroup D of the group of invertible
elements modulo N. Well known examples are Legendre sequences or the
two-prime generator. For some generalizations of sequences considered in
the literature and for some new examples of sequence constructions we
determine the linear complexity.
1Introduction
A sequence S = s0,s1,... with terms in a finite field Fdwith d elements is said
to be N-periodic if si= si+Nfor all i ≥ 0. The linear complexity L(S) of an N-
periodic sequence S over Fdis the smallest nonnegative integer L for which there
exist coefficients c1,c2,...,cLin Fdsuch that S satisfies the linear recurrence
relation si+ c1si−1+ ··· + cLsi−L= 0 for all i ≥ L. If d and N are relatively
prime and θ is a primitive Nth root of unity in some extension field of Fd, and
S(x) = s0+ s1x + ··· + sN−1xN−1then
L(S) = N − |{a : S(θa) = 0,0 ≤ a ≤ N − 1}|.
The linear complexity is considered as a primary quality measure for periodic se-
quences and plays an important role in applications of sequences in cryptography
and communication (see for instance [13] and the references therein).
In this paper we point to a general approach to N-periodic sequences over
a finite field Fd defined via a subgroup D of the group Z∗
elements modulo N. Well-known basic examples are the Legendre sequences and
its generalizations and the two-prime generator. We describe a uniform approach
to obtain results on the linear complexity for such sequence constructions that
comprise also the known proofs [3–7] for the above mentioned examples. We
apply this approach to some further examples of sequences and determine their
linear complexity. The first example can be seen as a natural generalization of
earlier constructions, the further examples are different, some - otherwise than
the sequences mentioned above - are based on subgroups D of Z∗
factor group Z∗
(1)
Nof the invertible
Nfor which the
N/D is not cyclic.

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2Ay¸ ca C ¸e¸ smelio˘ glu, Wilfried Meidl
2 A general construction of sequences based on cosets
Let N be an odd integer, ∆ be a divisor of ϕ(N), where ϕ denotes Euler’s totient
function, and let D = D0be a subgroup of index ∆ of Z∗
elements modulo N. Denote the elements of the factor group G = Z∗
{D0,D1,...,D∆−1}. Naturally this defines a partition of Z∗
we will write n ∈ Djif nD0= Djfor an integer n ∈ Z∗
S = s0,s1,... over a finite field Fdsatisfying
sn= ξjwhenever n mod N ∈ Dj
is then called a coset sequence. We remark that the sequence terms sn for
gcd(n,N) ?= 1 have to be defined separately.
In order to obtain (almost) balanced sequences over Fdone may prefer to con-
sider subgroups D0 of index d and to assign every field element ξj ∈ Fd to
precisely one coset Dj.
If the period N = p is prime and ∆ is a divisor of p − 1, then the (only)
subgroup D0of index ∆ of Z∗
D0= {g∆s: s = 0,1,...,(p − 1)/∆ − 1}
for a primitive element g modulo p. The cosets Dj= gjD0, 0 ≤ j ≤ ∆ − 1, are
then called the cyclotomic classes of order ∆. Trivially the factor group Z∗
is then cyclic.
Some well-known examples of coset sequences are the following:
Legendre sequences and its generalizations: To describe this class of se-
quences in its most general form we have to fix an ordering of the elements of
the finite field Fd, d = rtfor a prime r. Given a basis {β0,β1,...,βt−1} of Frt
over Frwe fix an ordering of the elements of Frt by
ξj= j0β0+ j1β1+ ··· + jt−1βt−1
if (j0,j1,...,jt−1)r is the r-ary representation of the integer j. If t = 1 this
reduces to the conventional ordering 0,1,...,r − 1 of the prime field Fr (with
β0= 1).
Let N = p be a prime, ∆ = d = rta prime power divisor of p − 1 and D0
be the group of the dth powers modulo p. The generalized Legendre sequence is
then the N-periodic sequence over Fddefined by
sn= ξjif n mod p ∈ Dj,
For d = 2 the sequence (4) is known as the classical Legendre sequence, its
linear complexity is determined in [5]. In [6] and [4] the linear complexity of (4)
is presented for d prime and for d = rt, r prime and gcd(t,r) = 1.
Hall’s sextic residue sequence: Let N = p be prime congruent 1 modulo 6,
D0,...,D5be the cyclotomic classes of order 6 defined as in (2). The N periodic
binary coset sequence given by
?1
N, the group of invertible
N/D0by
N, regarding to which
N. An N-periodic sequence
Nis the set of ∆th powers
(2)
N/D0
(3)
and
sn= 0 if n ≡ 0 mod p.
(4)
sn=
:
:
n mod N ∈ D0∪ D1∪ D3,
otherwise0

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Linear complexity of sequences based on cosets3
is called Hall’s sextic residue sequence (see [10] for its linear complexity).
Two-prime generator: For two odd primes p and q let D0be the subgroup of
index 2 of Z∗
modulo both primes p and q. Denoting the two elements of the corresponding
factor group by D0and D1, the two-prime generator is the binary pq-periodic
sequence given by sn+pq= snand for 0 ≤ n < pq
sn= j if n ∈ Dj, sn= 0 if n ∈ Q ∪ {0} and sn= 1 if n ∈ P,
where here and in the following P = pZ∗
{q,2q,...,(p−1)q}. The linear complexity of the two-prime generator has been
determined in [7] for gcd(p − 1,q − 1) = 2. In [9] the generalization to arbitrary
prime fields has been analysed.
pqconsisting of the elements which are either squares or nonsquares
q= {p,2p,...,(q −1)p} and Q = qZ∗
p=
In [1,15] the subgroup D of Z∗
a square modulo q has been used to define a pq-periodic binary sequence. As
pointed out in [12] where a generalization to arbitrary prime fields was consid-
ered, these sequences essentially are only concatenations of p Legendre sequences
of period q. Similar constructions leading to binary sequences of period qmand
2qmwith much similarity to concatenated Legendre sequences of period q have
been considered recently in [14,16].
pqwhich consists of all elements which are
3Basic results
In what follows N will always be an odd integer, d a prime power divisor of
ϕ(N), D0a subgroup of Z∗
of D0. If Z∗
suppose that DiDj= Di+j mod d.
Let S be a coset sequence of period N over Fdwith sn= ξjif n ∈ Dj. (At this
position ξj does not necessarily refer to the ordering in (3).) The polynomial
S(x) corresponding to S can then be written as S(x) = U(x) + T(x) with
Nof index d, and D0,D1,...,Dd−1denote the cosets
N/D0is cyclic, which always applies when d is prime, then we can
U(x) =
?
n∈ZN\Z∗
N
snxnand T(x) =
d−1
?
j=0
ξjfj(x) where fj(x) =
?
i∈Dj
xi.
(5)
We collect some simple basic properties which partly had been shown in the
literature for different concrete examples of coset sequences (see e.g. [4–7]). In
what follows we suppose that d = rt, r prime, gcd(N,r) = 1, and we let θ be a
primitive Nth root of unity over Fd.
Lemma 1. (i) If a,¯ a ∈ Difor some 0 ≤ i ≤ d − 1 then T(θ¯ a) = T(θa).
(ii) For all 0 ≤ a ≤ N − 1 we have fj(θa) ∈ Frd, 0 ≤ j ≤ d − 1. If d ∈ D0then
fj(θa) ∈ Fd, 0 ≤ j ≤ d−1, and T(θa) ∈ Fdfor all 0 ≤ a ≤ N −1. If also r ∈ D0
then fj(θa) ∈ Fr, 0 ≤ j ≤ d − 1, for all 0 ≤ a ≤ N − 1.

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4Ay¸ ca C ¸e¸ smelio˘ glu, Wilfried Meidl
(iii) If a ∈ Dkthen T(θa) =?d−1
(iv)?d−1
Proof. (i),(ii) are straightforward, we also may refer to [4].
(iii) T(θa) =
j=0ξj
?d−1
xϕ(N)−1in the Nth cyclotomic polynomial QN. With QN=?
QN=(xa1− 1)···(xa1− 1)
j=0ξj?kfj(θ) where j ? k = l if Dj= DkDlin
Z∗
N/D0.
j=0fj(θ) = µ(N), where µ denotes the M¨ obius function.
?d−1
?
i∈Djθai=
?d−1
Nθkis the negative of the coefficient of
j=0ξj
?
i∈aDjθi=
?d−1
j=0ξjfj⊕k(θ) =
j=0ξj?kfj(θ).
(iv) Observe that?d−1
(see [11, Theorem 3.27]) we obtain
j=0fj(θ) =?
k∈Z∗
c|N(xN/c−1)µ(c)
(xb1− 1)···(xbs− 1)= (xA−xA−a1+···±1) : (xB−xB−b1+···±1),
where ai,bj run through the divisors c of N for which N/c is squarefree, we
choose a1 and b1 to be the minimum of the ai and bj, respectively, and put
A = a1+···+arand B = b1+···+bs. As obvious, A−B = ϕ(N). Performing
the division we then get
QN= xϕ(N)± xϕ(N)−min(a1,b1)± ··· + 1,
where the coefficient of xϕ(N)−min(a1,b1)is ”1” if a1> b1and ”−1” if a1< b1. As
µ(N) = 0 implies min(a1,b1) > 1, the coefficient of xϕ(N)−1in QNis zero in this
case. If µ(N) = 1 then min(a1,b1) = a1= 1, if µ(N) = −1 then min(a1,b1) =
b1= 1, which completes the proof.
As generally known the possible values for the linear complexity of an N-periodic
sequence over Fd depend on the degrees of the polynomials in the canonical
factorization of xN− 1 over Fd. The following proposition indicates that for
many classes of coset sequences the order of the coset Dj which contains d in
the factor group Z∗
Proposition 1. Let D0 be a subgroup of Z∗
B = ?Dj? be the subgroup of G generated by Dj. For a corresponding coset
sequence over Fd let T(x) be defined as in (5). If T(θa) = 0 for a ∈ Dk then
T(θb) = 0 for all b ∈ BDk.
Proof. Let s be the order of d modulo N, then the minimal polynomial of θa
over Fd is given by m(x) =?s−1
(depending on the order of Dj in G elements in this set repeat), with Lemma
1(i) we have T(θb) = 0 for all b ∈ BDk.
Remark 1. If U(θa) = c ∈ Fdis constant for all a ∈ Z∗
consequently Proposition 1 also holds for S(x).
?
N/D0decides on the possible values for the linear complexity
N, G = Z∗
N/D0, d ∈ Dj and let
l=0(x − θadl). Consequently if T(θa) = 0 then
T(θadl) = 0 for 0 ≤ l ≤ s − 1. Since B = ?Dj? = {D0,dD0= Dj,...,ds−1D0}
?
Nthen Lemma 1(i) and
If Z∗
N/D0is cyclic (as in the sequence constructions in the literature , see [1,
4–7,12,15]) then we can naturally employ the ordering defined as in (3) to define

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Linear complexity of sequences based on cosets5
a coset sequence. Following the objective of the paper to give a general approach
to N-periodic sequences constructed via subgroups D0of Z∗
classes of factor groups that are not cyclic. We concentrate hereby on factor
groups whose order is a prime power.
For an odd integer N and a prime r let D0be a subgroup of Z∗
is isomorphic to Zrt1 ×Zrt2 ×···×Zrtw (with the componentwise addition) for
some positive integers ti, 1 ≤ i ≤ w. The cardinality of Z∗
with t = t1+t2+...+tw, and we can easily define an N-periodic coset sequence
over Fdwhich is close to be balanced.
Example. Let N = pq for two odd primes p and q, let D(p)
set of squares modulo p and q, and consider
Nwe consider further
Nsuch that Z∗
N/D0
N/D0is then d = rt
0
and D(q)
0
denote the
D0= {j | 1 ≤ j ≤ pq − 1,j mod p ∈ D(p)
As obvious D0is a subgroup of Z∗
For the definition of a sequence we again employ the ordering (3) of the elements
of Frt. In order to assign the elements of Frt to the rtcosets of D0we also need an
ordering of the elements of Z∗
?w
Ψ(D) = (J1,J2,...,Jw) with J1+ J2rρ1+ J3rρ2+ ··· + Jwrρw−1= j.
Based on the orderings (3), (6), N-periodic coset sequences over Frt with
0,j mod q ∈ D(q)
pq/D0isomorphic to Z2× Z2.
0},
pqwith Z∗
N/D0. We put ρ0= 0,ρ1= t1,ρ2= t1+t2,...,ρw=
i=1ti= t, and let Ψ be the isomorphism from Z∗
For 0 ≤ j ≤ rt− 1 we then denote the coset D of D0by Djfor which
N/D0to Zrt1×Zrt2×···×Zrtw.
(6)
sn= ξjif n ∈ Dj
can be defined. We remark that DkDl= Dk⊕lwhen we define
?
according to the operation in Zrt1 × Zrt2 × ··· × Zrtw.
The following Lemma generalizes [4, Lemma 10] shown for the generalized
Legendre sequence (4).
k ⊕ l = h if k =
w
i=1
Kirρi,l =
w
?
i=1
Lirρiand h =
w
?
i=1
(Ki+ Limod rti)rρi, (7)
Lemma 2. Let N be squarefree, D0 a subgroup of Z∗
with gcd(r,t) = 1, and let
N, d = rta prime power
1. Z∗
2. Z∗
N/D0be a cyclic group of order d, or
N/D0be isomorphic to Zrt1 × Zrt2 × ··· × Zrtw with t1+ ··· + tw= t.
Consider a coset sequence over Fdsatisfying sn= ξj if n ∈ Dj, where ξj refers
to the ordering (3) of the elements of Fd, the cosets Djare naturally ordered in
case 1 and ordered as in (6) in case 2. Then T(θa?) ?= T(θa) if a??≡ a mod D0.
Proof. For this proof we denote by k ⊕ l the addition modulo d in case 1 and
the addition (7) in case 2. Let a ∈ Dkand a?∈ Dk?, let k ? k?= δ and suppose