<s>
In	O
sequence	O
design	O
,	O
a	O
Feedback	O
with	O
Carry	O
Shift	O
Register	O
(	O
or	O
FCSR	B-Algorithm
)	O
is	O
the	O
arithmetic	O
or	O
with	O
carry	O
analog	O
of	O
a	O
linear-feedback	B-Architecture
shift	I-Architecture
register	I-Architecture
(	O
LFSR	B-Architecture
)	O
.	O
</s>
<s>
If	O
is	O
an	O
integer	O
,	O
then	O
an	O
N-ary	O
FCSR	B-Algorithm
of	O
length	O
is	O
a	O
finite	O
state	O
device	O
with	O
a	O
state	O
consisting	O
of	O
a	O
vector	O
of	O
elements	O
in	O
and	O
an	O
integer	O
.	O
</s>
<s>
By	O
iterating	O
the	O
state	O
change	O
an	O
FCSR	B-Algorithm
generates	O
an	O
infinite	O
,	O
eventually	O
periodic	O
sequence	O
of	O
numbers	O
in	O
.	O
</s>
<s>
FCSRs	B-Algorithm
have	O
been	O
used	O
in	O
the	O
design	O
of	O
stream	B-Algorithm
ciphers	I-Algorithm
(	O
such	O
as	O
the	O
F-FCSR	B-Algorithm
generator	O
)	O
,	O
in	O
the	O
cryptanalysis	O
of	O
the	O
summation	B-Algorithm
combiner	I-Algorithm
stream	B-Algorithm
cipher	I-Algorithm
(	O
the	O
reason	O
Goresky	O
and	O
Klapper	O
invented	O
them	O
)	O
,	O
and	O
in	O
generating	O
pseudorandom	B-Error_Name
numbers	I-Error_Name
for	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
(	O
under	O
the	O
name	O
Multiply	B-Algorithm
With	I-Algorithm
Carry	I-Algorithm
(	O
MWC	O
)	O
generator	O
-	O
invented	O
by	O
Couture	O
and	O
L'Ecuyer	O
,	O
)	O
generalizing	O
work	O
of	O
Marsaglia	O
and	O
Zaman	O
.	O
</s>
<s>
FCSRs	B-Algorithm
are	O
analyzed	O
using	O
number	O
theory	O
.	O
</s>
<s>
Associated	O
with	O
the	O
FCSR	B-Algorithm
is	O
a	O
connection	O
integer	O
.	O
</s>
<s>
Associated	O
with	O
the	O
output	O
sequence	O
is	O
the	O
N-adic	O
number	O
The	O
fundamental	O
theorem	O
of	O
FCSRs	B-Algorithm
says	O
that	O
there	O
is	O
an	O
integer	O
so	O
that	O
,	O
a	O
rational	O
number	O
.	O
</s>
<s>
There	O
is	O
also	O
an	O
exponential	O
representation	O
of	O
FCSRs	B-Algorithm
:	O
if	O
is	O
the	O
inverse	O
of	O
,	O
and	O
the	O
output	O
sequence	O
is	O
strictly	O
periodic	O
,	O
then	O
,	O
where	O
is	O
an	O
integer	O
.	O
</s>
<s>
There	O
are	O
efficient	O
algorithms	O
for	O
FCSR	B-Algorithm
synthesis	O
.	O
</s>
<s>
This	O
is	O
the	O
problem	O
:	O
given	O
a	O
prefix	O
of	O
a	O
sequence	O
,	O
construct	O
a	O
minimal	O
length	O
FCSR	B-Algorithm
that	O
outputs	O
the	O
sequence	O
.	O
</s>
<s>
If	O
L	O
is	O
the	O
size	O
of	O
the	O
smallest	O
FCSR	B-Algorithm
that	O
outputs	O
the	O
sequence	O
(	O
called	O
the	O
N-adic	O
complexity	O
of	O
the	O
sequence	O
)	O
,	O
then	O
all	O
these	O
algorithms	O
require	O
a	O
prefix	O
of	O
length	O
about	O
to	O
be	O
successful	O
and	O
have	O
quadratic	O
time	O
complexity	O
.	O
</s>
<s>
It	O
follows	O
that	O
,	O
as	O
with	O
LFSRs	B-Architecture
and	O
linear	O
complexity	O
,	O
any	O
stream	B-Algorithm
cipher	I-Algorithm
whose	O
N-adic	O
complexity	O
is	O
low	O
should	O
not	O
be	O
used	O
for	O
cryptography	O
.	O
</s>
<s>
FCSRs	B-Algorithm
and	O
LFSRs	B-Architecture
are	O
special	O
cases	O
of	O
a	O
very	O
general	O
algebraic	O
construction	O
of	O
sequence	O
generators	O
called	O
Algebraic	O
Feedback	O
Shift	O
Registers	O
(	O
AFSRs	O
)	O
in	O
which	O
the	O
integers	O
are	O
replaced	O
by	O
an	O
arbitrary	O
ring	O
and	O
is	O
replaced	O
by	O
an	O
arbitrary	O
non-unit	O
in	O
.	O
</s>
<s>
A	O
general	O
reference	O
on	O
the	O
subject	O
of	O
LFSRs	B-Architecture
,	O
FCSRs	B-Algorithm
,	O
and	O
AFSRs	O
is	O
the	O
book	O
.	O
</s>
