Wire-tap Channel Concept in Nature and Society

1
Wire-tap Channel Concept in Nature and Society
(Professor, Dr. Valery Korzhik, University of
Telecommunications, St.Petersburg, Russia
E-mail korzhik_at_spb.lanck.net)
-2008-
2
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

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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-reliabl
e information transmission over the main
channel. N.B. Nothing secret key is distributed
in advance.
Sk
Xn
Yn
Source of information
Randomizeencoder
Main channel
Decoder in the main channel
Legal recipient
Zn
Wire-tap channel
Decoder in wire-tap channel
Illegal recipient (eavesdropper)
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).

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2. WTCC versus cryptography (CR) and
steganography (SG)

• CR and SG require secret key distribution, WTCC
  does not require.
• CR, SG and WTCC satisfy Kerckhoffs 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.

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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
Pm and Pw , respectively. 3.1. Randomized coding
technique (noiseless main channel,
Pm0)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 (Pm0 ) then
.
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Information leaking over wire-tap channel 2


(3) where
- is the number of words of the
Hammings weight i in the jth coset of the
standard array Vn /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)
Pw 10-3 10-2 510-2 10-1 210-1
Ie 0.94 0.605 0.083 0.041 6.210-7
I(xy) 1 0.91 0.77 0.53 0.28
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3.2. Randomized coding technique (noisy main
channel, Pmgt0, PmltPw) as
n?8, secret capacity
(4) where
-
capacity of the main channel
-
capacity of the wire-tap channel 3.2. Privacy
amplification technique (noisy main channel,
Pmgt0, PmltPw) - 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.
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• Privacy amplification theorem 4,5
• Where
  - Renyi information.
• As n?8, then ,
• 
  - is entropy function.
• 3.4. Transformation of the condition PmgtPw to the
  condition PmltPw .
• 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 PmltPm.
• Interaction between legal users
• More effective methods 6
• Next can be used either randomized coding or
  privacy amplification technique.

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4. Applications of WTCC in telecommunications

• 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
• 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?

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

• Presence of redundancy.
• An existing of some randomness connected with
  redundancy (randomized encoder, random hash
  functions).
• 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
  (0ltPw1/2). 3

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Typical dependences for the amount of information
I(Sz) versus the error probability P on BSC for
different code rates R
R2 lt R1 lt R0 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 (P2 lt P1 lt P0 1/2)
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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.
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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).

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• 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).

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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.
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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 (Pm)? -
The wire-tap channel (PwgtPw')
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.
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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.
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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)? - the main (good) channel from
artist to spectators (in other 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
alphabet of abstract painting (what does it
mean such geometric figures as triangle, circle,
oval, waved line)
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• 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?

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Proposal The artist goal is to short gray
area (in other words the number of semi
experts in his (her) painting, because they can
compromise it even more than active opponents.
(It is worth to note that a presence of short
transition area can be found as one of signs
for new directions in a painting). The
property to provide a short transition area is
one of the 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
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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.
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However there may be situation where NL is used
in frame of WTCC. This means that nothing keys
(or any secret information for 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. At
a single glance this setting look as a possession
of the key for legal parties but this key is
rather probabilistic because the 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).
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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 sins should be forgiven
them. (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?
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In Section 1 has been considered a solution to
the problem how to transform the model PmgtPw to
the PmltPw . 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.
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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 markedby
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.
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• 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.

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• 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.

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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 , Point and line to Plane, Dover
Books on Art History . 10. M.Ridley Francis
Crick Discoverer of the Genetic Code ,
Amazon.com. 11. U. Becks-Malory ,Wassily
Kandinsky , 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.