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

Randomizeencoder

Main channel

Decoder in the main channel

Legal recipient

Wire-tap channel

Decoder in wire-tap channel

Illegal recipient (eavesdropper)

Sk Xn Yn

Zn

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 WTCCAssumption. 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, Pm=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 (Pm=0 ) then .

c)s,()x,(xx 21 x

)x,x( 21 1x 2xs

k

kn )(fc

kn ck)(n,n

)(~12 xfxs

ss ~

Information leaking over wire-tap channel [2]

(3)

where - is the number of words of the Hamming`s 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)

k

j

k

l PPkk

;ZSII2

1

log11 jj VV

n

i

inw

iwij PPNP

1

1jV

ijN

Pw 10-3 10-2 5·10-2 10-1 2·10-1

Ie 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, Pm>0, Pm<Pw) 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, Pm>0, Pm<Pw)

- 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.

wme CCI eI

)P()P(PPC mmmmm 1log1log1)P()P(PPC wwwww 1log1log1

)(xhsy

xksn

h nxc

r~x

hcr

y kh hxx )( )2( nGF

h

Privacy amplification theorem [4,5]

Where - Renyi information.As n→∞, then ~ , - is entropy function.

3.4. Transformation of the condition Pm>Pw to the condition Pm<Pw .• 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<Pm.• Interaction between legal users:

• More effective methods [6]Next can be used either randomized coding or privacy amplification

technique.

2ln2 )( rktn

eI

222 )1(log1 ww PPnt

t ))(1( wPh))1log()1(log()( wwwww PPPPPh

mm PP `,ABABABAB εxεδxδ,εδx,εδwmww PPPP )1)(1(1`

4. Applications of WTCC in telecommunications1. 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<Pw≤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

R2 < R1 < 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 < P1 < P0 = 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 (Pm)- The wire-tap channel (Pw>Pw')

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 paintingThe 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)

•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 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

6.4. Natural LanguagesNatural 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.

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).

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?

In Section 1 has been considered a solution to the problem:how to transform the model Pm>Pw to the Pm<Pw . 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 , “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.