Concept in Nature and

blessinghomoeopathΤεχνίτη Νοημοσύνη και Ρομποτική

30 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

99 εμφανίσεις

Wire
-
tap Channel
Concept in Nature and
Society

(Professor, Dr. Valery Korzhik, University
of Telecommunications, St.Petersburg,
Russia; E
-
mail: korzhik@spb.lanck.net)

-
2008
-

Outlook of the lecture:




Definition of wire
-
tap channel concept (WTCC)




WTCC versus cryptography and steganography




ABC of WTCC




Applications of WTCC in communications




The main features of WTCC which can be important for

their
tracing in nature and society




Some examples of a WTCC presence in nature and society:



camouflage in animal kingdom



genetic code



modern painting



natural languages




Conclusion




References

1.

Definition of WTCC [1]

WTCC

is

mathematical

model

in

which

two

communication

channels

are

presented
:

the

main

channel

(less

noisy)

and

the

wire
-
tap

channel

(more

noisy)
.

Along

this

model

special

randomized

encoding

procedure

is

used

with

a

goal

to

prevent

information

leakage

over

wire
-
tap

channel

(even

after

optimal

decoding)

and

simultaneously
-
reliable

information

transmission

over

the

main

channel
.

N
.
B
.

Nothing

secret

key

is

distributed

in

advance
.

Source of
information

Randomize
encoder

Main
channel

Decoder in
the main
channel

Legal
recipient

Wire
-
tap
channel

Decoder in
wire
-
tap
channel

Illegal recipient
(eavesdropper)

S
k

X
n

Y
n

Z
n

Fig.1. Scheme of the wire
-
tap channel concept

Remark
.

Another

possible

distinction

of

the

main

and

the

wire
-
tap

channels

are
:



main

channel

has

a

feedback,



wire
-
tap

channel

does

not

have

any

feedback

to

legal

user

(eavesdropper

is

passive)
.

2.

WTCC versus cryptography (CR) and
steganography (SG)



CR

and

SG

require

secret

key

distribution,

WTCC

does

not

require
.



CR,

SG

and

WTCC

satisfy

Kerckhoff’s

assumption
.



CR

have

to

be

used

with

noiseless

channels

(block

ciphers)

or

both

with

noiseless

and

noisy

channels

(stream

ciphers)
.


CR

is

basically

computational

secure,

whereas

SG

and

WTCC

are

mostly

information
-
theoretic

secure

(like

perfect

one
-
time

pad

ciphers)
.


CR

is

used

mostly

without

reducing

of

date

rate,

whereas

WTCC

reduces

date

rate
.


Both

CR

and

WTCC

do

not

hide

of

the

fact

that

some

secret

information

there

exists,

whereas

SG

does

it
.



3.

ABC of WTCC

Assumption
.

Both

the

main

channel

and

the

wire
-
tap

channels

are

binary

symmetric

channels

without

memory

(BSC)

with

symbol

error

probably

P
m

and

P
w

,

respectively
.

3
.
1
.

Randomized

coding

technique

(noiseless

main

channel,

P
m
=
0
)[
2
]












(
1
)

Where

-

is

the

output

of

encoder
.



-

concatenation

of

vectors

and



-

is

the

binary

message

string

of

the

length

k


-

truly

random

binary

string

of

the

length




is

the

function

mapping

information

string

of

the

length

to

the

check

string

of

the

length

for

some

chosen

in

advance

binary

linear

-

code

V
.


-

is

bit
-
wise

modulo

two

addition




(
2
)

If

the

main

channel

is

noiseless

(
P
m
=
0

)

then

.





c)
s
,
(
)
x
,
(x
x
2
1




x
)
x
,
x
(
2
1
1
x
2
x
s
k

k
n

)
(

f
c


k
n

c
k)
(n,n


)
(
~
1
2
x
f
x
s


s
s

~
Information

leaking

over

wire
-
tap

channel

[
2
]



(
3
)


where





-

is

the

number

of

words

of

the

Hamming`s

weight

i

in

the

j
th

coset

of

the

standard

array

V
n

/V
.

Remark
.

It

is

a

problem

to

calculate

coset

spectra

of

arbitrary

linear

codes
.

They

are

known

only

for

some

classes

of

such

codes

[
3
]
.

Example
.

(
V

is

(
31
,
26
)



Hamming

code)
















k
j
k
l
P
P
k
k
;Z
S
I
I
2
1
log
1
1
j
j
V
V









n
i
i
n
w
i
w
ij
P
P
N
P
1
1
j
V
ij
N
P
w

10
-
3

10
-
2

5∙10
-
2

10
-
1

2∙10
-
1

I
e

0.94

0.605

0.083

0.041

6.2∙10
-
7

I(
x;y
)

≈1

0.91

0.77

0.53

0.28

3
.
2
.

Randomized

coding

technique

(noisy

main

channel,

P
m
>
0
,

P
m
<P
w
)


as

n→∞
,



secret

capacity

(
4
)

where



-

capacity

of

the

main

channel


-

capacity

of

the

wire
-
tap

channel


3
.
2
.

Privacy

amplification

technique

(noisy

main

channel,

P
m
>
0
,

P
m
<P
w
)




-

is

the

binary

message

string

of

the

length



-

truly

random

binary

sequence

of

the

length



-

truly

random

binary

string

of

the

length



-

check

string

of

the

length

to

for

given

error

correcting

code

V


-

check

string

of

the

length

to

for

given

error

correcting

code

V


In

a

particular

case

the

string

is

absent

(key

distribution)
.


where





is

multiplication

in

the

field

,

[

]
k



keeping

of

k

least

significant

bits
.

w
m
e
C
C
I


e
I
)
P
(
)
P
(
P
P
C
m
m
m
m
m





1
log
1
log
1
)
P
(
)
P
(
P
P
C
w
w
w
w
w





1
log
1
log
1
)
(
x
h
s
y


x
k
s
n
h
n
x
c
r
~
x
h
c
r
y


k
h
h
x
x


)
(

)
2
(
n
GF
h
Privacy

amplification

theorem

[
4
,
5
]



Where

-

Renyi

information
.

As

n
→∞
,

then

~

,


-

is

entropy

function
.



3
.
4
.

Transformation

of

the

condition

P
m
>P
w

to

the

condition

P
m
<P
w

.




The

use

of

repetition

code

(each

of

bits

is

repeated

s

times
.

Legal

user

accepts

it

only

of

the

number

of

zeros

or

ones

occurs

more

than

some

threshold)
.
It

can

result

under

appropriated

choice

of

s

in

condition

P
`
m
<P
m
.




Interaction

between

legal

users
:







More

effective

methods

[
6
]


Next

can

be

used

either

randomized

coding

or

privacy

amplification

technique
.



2
ln
2
)
(
r
k
t
n
e
I









2
2
2
)
1
(
log
1
w
w
P
P
n
t




t
))
(
1
(
w
P
h

))
1
log(
)
1
(
log
(
)
(
w
w
w
w
w
P
P
P
P
P
h





m
m
P
P









`
,
AB
AB
AB
AB
ε
x
ε
δ
x
δ
,
ε
δ
x
,
ε
δ
w
m
w
w
P
P
P
P





)
1
)(
1
(
1
`
4. Applications of WTCC in telecommunications

1.
Protection

against

leakage

of

secret

information

over

side

channels

(electromagnetic

radiation,

side

and

back

petals

of

antenna

diagrams

for

indoor

radio

links,

monitoring

of

optical

fiber

lines,

against

a

presence

of

eavesdropping)

[
7
]

2.
Key

sharing

between

legal

users

in

presence

of

passive

or

active

eavesdroppers

(the

use

of

trust

party,

quantum

cryptography,

mobile

communications)
.


It

seems

to

be

naturally

to

apply

WTCC

to

enhance

security

in

technical

world
.

But

in

nature

and

society

?


5.

The main features of WTCC which can be
important for their “tracing” in nature and society

1.
Presence

of

redundancy
.

2.
An

existing

of

some

randomness

connected

with

redundancy

(randomized

encoder,

random

hash

functions)
.

3.
An

existing

of

two

channels
:


-

main

(legal)

channel

that

is

small

noisy,


-

wire
-
tap

(illegal)

channel

that

is

larger

noisy
.


In

reality

the

situation

can

be

not

so

strong

(not

exactly

“white”

and

“black”

channels)
.


Consider

the

channels

with

varying

noise

(
0
<P
w

1
/
2
)
.

[
3
]

Typical dependences for the amount of information
I
(S;z)
versus the error probability
P

on BSC for different code
rates
R

R
2

< R
1

< R
0

= 1

4. The more is redundancy (the less code rate) the more
«sharp» is a transition between reliable and unreliable
information transmission.

5. The more is redundancy the closer to zero is the probability
for that the information transmission is negligible small value

(P
2

< P
1

< P
0

= 1/2)

Consider

an

example

of

WTCC

that

demonstrates

all

properties

given

above

but

that

does

not

belong

to

technical

area
.


#
1


Ordinary

written

word
.

It

is

excellently

readable
.

#
2

Ordinary

written

word

with

“graphical

noise”

(crossing

by

stripe

lines)
.

It

is

still

readable
.

#
3

Word

written

in

“wave
-
wise

randomized”

manner
.

It

is

good

readable
.

#
4

Word

written

in

“waved
-
wise

randomized”

manner

with

“graphical

noise”

(crossing

by

waved

randomize

lines)
.

It

is

not

readable
.

6.

Some examples of a WTCC presence in
nature and society

6.1
.Camouflage in animal Kingdom


Remark:


We do not consider «trivial» camouflage
(hare

on snow or grasshopper in the grass), but so called
«challenging painting» (like giraffe or zebra).


Features of WTCC:



redundancy (draw on the skin)




randomness

(random spots or strips on the skin of
animals)




existing of two channels: very noisy (observation from
a long distance through natural plants) and small noisy
(observation from a short distance inside of one herd)




the need in a short «transitions distance» between
good and bad recognition in order to provide better
security against predators (say, lions).

Proposal:

The goal of any camouflage procedure is to
reduce the surface (skin of animals) to the contour.


Proofs by Samsung TV
-
advertising:
Sea
-

slope looks as
contour of its body on sea bottom. (See next slide).



Remark:

This phenomenon cannot be explained as only a
matching of «noise» with «signal» (speaking in
telecommunication language).


The difference rely on the following:


Expansion of error (incorrect recognition of some details
may result in increasing of incorrect recognition of complete
objects)



The more is redundancy (size of skin) the worse is
recognition procedure. (This is not the case for the
phenomenon a matching between signal and noise, where
the efficiency of signal detecting is determined by signal
-
to
-
noise
-
ratio).

6.2. Genetic code




Short definition:

Genetic code
(GC) transmits inherited
information from individuals to their descendants. It is triplet,
four
-
letter code, where each of triplet tuple is compared with
amino
-
acids.

GC has a redundancy that is called by its
regeneration
. But
there may be a presence of redundancy on higher levels
(codons of groups of codons).


Mutations
-

random changing of GC structure.

Mutations play positive role providing progressive evolution of
biological species. Very strong mutations can result in death of
individual or even in degeneration of all population after some
time period.


Let us compare the features of GC and WTCC

Genetic code

-

Redundancy in GC

-

Randomness in GC
(randomized coding)


-

Small mutations

-

Strong mutations

WTCC

-

Redundancy in WTCC

-

Randomness in construction
of WTCC

-

The main channel (
P
m
)


-

The wire
-
tap channel (
P
w
>P
w'
)

The goals of both constructions

Provide a transmission of
genetic information to next
descendants in the case of
small mutation and to
eliminate (to result in death)
of those individuals that
have been subject to strong
mutations.

Provide reliable information
transmission over the main
channel and except any
information transmission over
the wire
-
tap channel.

Hypothesis:


The most danger is a «grey area» (see sl.
11), when the number of errors due to mutations is not so large
in order to result to death of individuals, but is too large for
progressive evolution and they can result in degradation of all
population in the future. Solution to this GC
-

problem can be in
expansion of errors that result in death of individuals who are
unable to transfer «forged» information to next descendants.
This can be done namely by the technique similar to those used
in WTCC.

6.3
. Modern painting

The features of WTCC which can be found in paintings:

-

redundancy (not direct information transmission! : «Night
watch» by Rembrandt )


-

randomness (with point of view of spectators)


-

the‏main‏(“good”)‏channel‏from‏artist‏to‏spectators‏(in‏other‏
words for those of them who is already prepared to a
comprehension of this direction in a painting)


-

the wire


tap‏(“bad”)


channel from artist to spectators (in
other words for those of them who is an amateur in this painting
and is not yet prepared to a positive comprehension)



Example in abstract painting:

Experts know at least
“alphabet‏of‏abstract‏painting”‏(what‏does‏it‏mean‏such‏
geometric figures as triangle, circle, oval, waved line)


Good recognition of painting details result in a good
comprehension of complete picture and produces a feeling of
satisfaction whereas a misunderstanding of painting details by
amateurs results in expansion of misunderstanding regarding to
all work that in turn discourages spectators and forms a very
negative impression.



«Experts» and «amateurs» in painting are not fixed set of
spectators. They can be exchanged to their members in time.
(There are many historical examples when very famous experts
in painting did not accept some abstract works and only many
years later they were transformed from opponents to proponents
of this art)




The question arises: If the artist intentionally uses WTCC in his
(or her) activity, and if «yes», for what does it?

Proposal:

The artist goal is to short «gray area» (in other
words‏the‏number‏of‏“semi‏experts”‏in‏his‏(her)‏painting,‏
because they can compromise it even more than active
opponents. (It is worth to note that a presence of short
“transition‏area”‏can‏be‏found‏as‏one‏of‏signs‏for‏new‏directions‏
in a painting).



The‏property‏to‏provide‏a‏short‏“transition‏area”‏is‏one‏of‏the‏
main features for WTCC (see sl.
11
). But of course artists do
not perform randomized coding (as it was described above) but
they do it intuitively using his (her) technique? experience and
artistic feeling.


Examples:





-

Painting of Vasilij Kandinski (next slide)






-

French structuralists

6.4. Natural Languages

Natural languages (NL) can be considered not only as a
technical mean to exchange information between the members
of some community (tribe, nation) but as a measure to
prevent
information transmission

to «outsider».


(Remember a fact from the second World War when two
English men were speaking in rare language
Urdu
to prevent an
interception by German).


But this NL application has rather the features of cryptography
than WTCC.



NL can be used also as primitive steganography if some words
or phrases are looking as completely innocent ones but in fact
they have some hidden information.


The NL applications considered above are based on the keys
distributed between «legal users» in advance.

However there may be situation where NL is used in frame of
WTCC.

This‏means‏that‏nothing‏“keys”‏(or‏any‏secret‏information‏for‏
outsiders) are distributed between legal participants of
conversation in advance.

A leakage of information to illegal participants is prevented by
special
language
-
based encoding
under the condition that
parties have worse language channels than legal ones.

Legal parties are carriers of some NL, whereas illegal are not.
At‏a‏single‏glance‏this‏setting‏look‏as‏a‏possession‏of‏the‏“key”‏
for‏legal‏parties‏but‏this‏“key”‏is‏rather‏probabilistic‏because‏the‏
difference in the knowledge of NL depends on individuals and
on subject of discussion.


Example of randomized encoding with NL:


The use
phrases containing humor. (It is know that humor is very hard
recognizable by non
-
carriers of NL).

Cover sense in Bible's stories:


And he said unto them, unto you it is given to know the
mystery of the kingdom of God: but unto them that are without,
all these things are done in parables: That seeing they may
see, and not perceive; and hearing they may hear, and not
understand; lest at any time they should be converted, and their
sins‏should‏be‏forgiven‏them.”

(The Gospel according to St. Mark).


The hidden sense is understandable for so called consecrated.
They are those individuals who know more than others or even
who are closer on spirit to author of parables.

For non
-
consecrated parables are looking as very redundant
and decorated by random facts stories without any hidden
sense. If parables are not example of WTCC?

In Section
1
has been considered a solution to the problem:

how to transform the model P
m
>P
w
to the P
m
<P
w
. The same
problem can be solved by means of language dialogue between
legal parties, say Alice and Bob, in the presence of illegal party,
say Eve:





-

Alice asks Bob some questions chosen randomly from very
large list (this list (but not answers!) can be known both Bob
and Eve in advance)


-

Bob replies to Alice «Yes» or «No» depending on that if he are
sure or not in correct answer.

-

After many iterations of two previous steps Alice and Bob can
share a list of correct answers in the most positions one to
another.

Next these list are quantized in order to form digital strings
considered as a provision for joint secret key between Alice and
Bob.















After interception of all rounds of the presented above protocol
Eve be able to form her list of questions but it will coincide with
correct list of Alice answers in less number of positions.

This statement is based on the fact that Bob may answer «yes»
only to the questions to which he knows answers for sure,
whereas Eve has to fix answers to the questions «marked»by
«yes» with Bob.

Even so Eve is more smart than Bob it does not present to her
advantage against or for feed back this protocol.

It would be not the case if Alice asks Bob directly without any
feedback in the presence of very smart Eve.

(We do not discuss how to correct disagreement in the final
Alice and Bob strings and to remove information about shared
joint key obtained by Eve. This can be done as usually by check
symbol transmission and application of privacy a amplification).

We remark that conversation between Alice and Bob should be
authenticated.

Conclusion


WTCC has been proposed by A. Wyner (1975) and later
developed by many researchers (see list of references).

WTCC executes special randomized encoding in order to
prevent information leaking over the wire
-
tap channel while
maintaining information transmission over the main channel.
The difference between the wire
-
tap and the main channel is in
different level of noises in these channels or in an existing of
feedback channel as a part of the main channel.


WTCC has many applications in the communication
engineering as a method to prevent a leaking of information
over side channels (like electromagnetic radiation from cables
or devices into environments) or to solve key distribution
problem for legal users under the presence of adversary.


But the goal of this presentation was not a further developing
of WTCC with technical application.


We

try

«go

back

to

nature»

in

order

to

find

some

examples

of

the

WTCC

presence

in

nature

and

society
.



Unfortunately we were unable to present strict proofs of a
presence WTCC in mentioned above phenomenons. We try
only to shed some light on this problem and motivate listeners
to find some new examples of the WTCC presence in nature
and society.

We hope that in positive case the results at WTCC
investigations in technical area can be implemented to
phenomenon nature and society and back


some phenomenon
in nature and society allow to get advance in communication
engineering.

References*

1
.

A.D. Wyner, «The wire


tap channel», Bell Syst.Techn.J.
54
,
№‏
8
, pp.
1355
-
1387
,(
1975
)


2
.

V.I.Korzhik, V.A.Yakovlev, «Nonasymptotic bounds on the efficiency of noisy coding in
one channel», Problems of Information Transmission,
17
,№
4
, pp.
11
-
18
, (
1981
)


3
.

V.I.Korzhik, V.A.Yakovlev, «Capacity of a communication channel with inner random
coding»,Problems of Information Transmission,
28
,№
4
, pp.
317
-
328
, (
1992
)


4
.

C.H.Bennet, et al. «Generalized privacy amplification», IEEE Trans. on IT, vol.
41
,‏№‏
6
, pp.
1915
-
1923
, (
1995
)


5
.

V.I.Korzhik, et al. «Privacy amplification theorem for noisy main channel», LNCS, vol
2200
, pp.
18
-
26
, (
2001
)


6
.

U. Maurer, «Secret key agreement by public discussion from common information»,
IEEE Trans. on IT, vol.
39
,‏№‏
3
, pp.
733
-
742
, (
1993
)


7
.

V.I.Korzhik et al. «The wire


tap channel concept against eavesdropping of indoor
radio telephone», International Symposium on Personal, Indoor and Mobile Radio
Communications, proc. pp.
477
-

480
, (
1997
)


8
.

E.L. Feinberg, «Cybernetics, logic, art», Moscow,
1981
, (in Russian)


9
.

W.‏Kandinsky‏,‏“Point‏and‏line‏to‏Plane”,‏Dover‏Books‏on‏Art‏History‏.

10
.

M.Ridley‏“Francis‏Crick‏:Discoverer‏of‏the‏Genetic‏Code‏“‏,‏Amazon.com.

11
.

U. Becks
-
Malory‏,”Wassily‏Kandinsky‏,‏
1866
-
1944
:The Journey to Abstract (Big Art
Series).


*) We are quoting only such of many references devoted to WTCC that are directly
needed for our presentation.