LNCS 0330 - Proof of Massey's Conjectured Algorithm

PBOOF OF WASSEY'S CONJECTURED ALCORITHH C w s h e n g Ding Department o f Applied Mathematics N o r t h w e s t Telecommunication Engineering I n s t i t u t e X i a n , P e o p l e ' s Republic o f China ABSTRACT: Massey's c o n j e c t u r e d a l g o r i t h m f o r multi-sequence

shift register

s y n t h e s i s i s p r o v e d , a n d i t s s u i t a b i l i t y for t h e minimal r e a l i z a t i o n o f any l i n e a r system is also v e r i f i e d .

I

.

INTRODUCTION

It i s well known that t h e SLFSR(shortest l i n e a r feedback s h i f t r e g i s t e r ) is o f great importance i n p r a c t i c e ( 1)(2 ) . The a l g o r i t h m gives an e f f i c i e n t one( 2). The problem o f s y n t h e -

s y n t h e s i s o f single-sequence Berlekamp-Nassey

s i z i n g m u l t i - s e q u e n c e w i t h LZSR has been g i v e n much concern by many s c h o l a r s in Wassey gave a c o n j e c t u r e d a l g o r i t h m

i n f o r m a t i o n and c o n t r o l s o c i e t y . J.L.

for

t h e SLFSRsyntheais of m u l t i - s e q u e n c e i n 1972. I n 1985 Fen C u e i l i a n g and K.K. Tzeng also gave a n o t h e r o n e ( 3 ) . I n t h i s paper we are g o i n g t o prove Massey's c o n j e c t u r e d a l g o r i t h m , and v e r i f y that it is an u n i v e r s a l one and i s s u i t e d f o r t h e minimal r e a l i z a t i o n o f any l i n e a r system.

I1

.

PROOF OF MASSEY'S CONJECTURED ALGOBITl33

L e t Bi=

and Si-(ali

ail...

aZi

as,

... , M ,

ill,

... sri)t , %(B1

B2

be H sequences of l e n g t h N i n t h e f i e l d F

... B M ) t , Si=S I...S

Then t h e Massey's

i'

c o n j e c t u r e d a l g o r i t h m in F i g . 1 can be s t a t e d as MASSEY'S CONJECTUREr Assume t h a t ( f i , l i ) and d i = f i ( S i + l )

i s t h e ith d i s c r e p a n c y , i - 0 ,

i s t h e SLF'SR which g e n e r a t e s Si,

... , n.

Then

( i ) i f dn=O, t h e n l n + l = land fn+l=fn.

n

'

\,be

( 1 1 ) if d 3 0 , and i s a l i n e a r combination o f di, a basis of

and (kl, k2,

dn =

-

d.

:

... , kr )

2

ui&Ki

OSiSn-1

... ,

i-0,

s u c h t h a t max{n-ki+lki

n-l, l e t

: 1SiSx-r)is minimal

i s m a x i m a l i n a l p h a b e t i c o r d e r . Let

,

I=

ti

: uiko,

%, ,...

16isr)

i=1

C.G. Guenther (Ed.): Advances in Cryptology - EUROCRYPT '88, LNCS 330, pp. 345-349, 1988. 0 Springer-Verlag Berlin Heidelberg 1988

346 ( i i i ) i f dn i s not a l i n e a r combination o f d i , L O ,

n+l and fn+l can be any p o l y n o m i a l i n F[x]

... , n-1,

t h e n ln+l=

o f degree n+1.

F i r s t , w e give some n o t a t i o n s and simple r e s u l t s :

L e t fi= l + f i , l s +

=**

+

fi,li

,Ii,

be a v e c t o r of l e n g t h n+l. Denote Dn+l=(do dl

and Fn+I=(ffO f f l

...

... 0 fiYl ... f i , l i l ... dn) t , An+l-(sl

and ffi-(O

82

...

0

6

a .

0)

n+ 1 )t

f f n ) t . Then it i s e a s y t o know that

(i) Fn+l i s a l o w e r t r i a n g u l a r matrix, and i s i n v e r t a b l e . (1')

Dn+1 = F n + l *n+1- An+l Cn+l Dn+l' -1 and is a l s o a lower triangular matrix.

where Cn+l= Fn+l,

Let us s p l i t t h e m a t r i o e e Fn+l,

Cn+l,

Dn+l

and p a r t i t i o n them by u r i t i n g

[n-L )xn where B-(0

... 0 '4. ...ul) t , 0

c

(0

t h e f o l l o w i n g t h e o r e m 1 holds. Theorem 1. L e t f ( x ) = 1 + ulx +

-

S n + l i f and o n l y if U(n-L)x(n)GnDn

Theorem 2,

... O)t.

By d e f i n i t i o n , it i s a p p a r e n t that

... + uLxL ( L < n + l ) , 0 and BGnD,

f

g,Dn

then (f,L) generates + dn

= 0-

If ( f , L ) can g e n e r a t e S n + l , L d n + l , t h e n t h e r e must e x i s t a v e c t o r

u such t h a t

Theorem 3. A s s u m e that ( f i , L )

i s t h e SLFSR u h i c h g e n e r a t e s S

i

Then ln+l=n+l if and o n l y i f dn i s n o t a l i n e a r combination of di,

, GO, 160,

n*

.a.

... ,

n-1. Theorem 4. Assume t h a t g

c

fn +

ZCl ui

x

n-ki

fki, uifO,

Let 1; be t h e s h o r t e s t L s u c h t h a t ( f i , L ) can g e n e r a t e S

i

. If

i=l,

... ,

B.

(g,L) g e n e r a t e s

347

Sn+',

then we have

Lzmax

4 1;)

, ..., n-ks+%,>

n-kl+l&

, ... , n-ks+\,}

m a { ln, n-kl+\,

I n o r d e r t o prove theorem 4, we now prove t h e following lemma: Lemma: Assume 0

-

m-k f m + ulx ' fk,

, ulfO,

kl,ln-

80

.;1

. Let

1$,

Suppose lksL,j.Put LLGm+kl and

1) i f j+m-k1sY, h(x)-l+hlx+ ..+

h x j , where h f ill, j i' kt ,i'

g(x)=fn+ P ~ X " - ~ ' h( I), js LL

...,

(5,.

... -g(S

Because (g,L) g e n e r a t e s Sm and L 2 1 m , so g(S")= k Thus h(S '

-f(Sbl)-O.

Sk', b u t LL