Journalof IndianAcad.Mathematics Vol.25, No.2 (2003)PP.251-259.

V. Ravichandran and Narayanan Srinivasan

.

MEASURESFORDISPLACEMENTOF PERMUTATIONSUSEDFOR SPEECH SCRAMBLING

Abstract : In speech scrambling systems,the signal is encryptedby rearyangingthe speechdata using permutations.Effective scramblingas Two on rearrangement. measuredby residualintelligibility (Rl) is dependent (i) the are They parameterscan be used for measuringthe displacement. places an order of displacement(OD) which is the minimum numberof elementis displacedor the distancefrom its naturalpositions,(ii) the shift factor or the mean permutationdistance(MPD) which is the average displacementof all elementsin a given permutation.The permutation scramblersrvhichusea higherdegreepermutationaskey, needthe valueof shift factor,the. maximumshift factor or MPD. Basedon the ma:
*.t i",#,t;rtJ"T*

No. :1047' AMS SubjectClassification l. Introduction In speech scrambling systems, the signal can be encrypted by realranging the speech have been data using permutations. Such permutation scrambling techniques for speech a developed in the time domain as well as the frequency domain. While in the time domain domain frequency in the speech signal is divided into small segmentswhich are realranged, are the speech spectrum is divided into a number of sub bands and these sub-bands (RI) intelligibiltty residual rearranged.In both domains, effective scrambling as measuredby to rearrangement.Subjectivetests have [l], is dependenton displacementof elementsdue been performed to test the efficacy of scrambling techniquesbased on pennutation [4]' [7].

252

V. RAVICHANDRAN & N. SRINIVASAN

The averagedisplacement of all elementsin a given permutationis known as its shift factor tll or mean permutationdistancet7l. In permutationscramblingtechniques, permutationis the key for encryptionandthe inversepermutationis the key for decryption. High shift factor is essentialfor a chosenpermutationto yield minimumor zero residual intelligence(RI) in the encryptedsignal[]. The permutationacrambling[2] which usea higher degreepennutationas key, needthe value of maximumshift factor. Basedon the maximumshift factor,the thresholdandthe key spacearefixed. Therefore,the computation of maximumshift factor is important. In additionto scrambling in time-domain,permutationscramblingshavealsobeen appliedto powerspectrumvaluesobtainedfrom the speechdata.Sakuraiet al.16l foundthat RI decreases with increasein Hammingdistancefor scrambledFFT valuesobtainedfrom speech.Woo et al. [7] proposedalgorithmsfor key generationin which they generated permutationsin which each speechsub-bandvalueswere displacedfrom their original places.The miniryumdistancebetweenan element'soriginalpositionandits new displaced positionis known as order of displacemenent (OD). A derangement is a key with an OD of atleastone.They proposedan algorithmto generate(n - l) ! derangements. They compared this set of derangementswith the set of all derangements basedon two parameters: one is the averageOD value for all the derangements and the other is the averagevalue of the meanpermutationdistance(MPD). They found for n rangingfrom 5 to 12,the difference betweenboth the parameters were smallor nill. They alsoconjectured that this similarity holdsfor all n. While Woo er al. l7l calculatedvalues for the averageOD valueand the average valueof the MPD for n rangingfrom 5 to 12,they did not computethesevaluesfor all n. In this paperwe computethe averageMPD valuefor bothsetsof permutations discussed in [7]. We alsocomputethe maximumOD valuefor the setof all permutations of degreen. 1.1Preliminaries : A rearrangement Qb Qz,..., Qnof pt, pz, ..., pn is calleda permutationof degreen. This pennutationcanbe wriffenas

o=

( o, pz ln, qz

e,l q,)

(l)

For any given permutati on p of degreen, let d, representthe position to which the permutation p movesthe fth element.The shift factorof the permutation p is givenby n

uo MPD{n) =

}

I i= I

l , -o , l

.

f

-

,

MEASUREFOR DISPLACEMENT OF PERMUTATION USED

253

(OD) is given by The orderof derangement

o D ( p=), T i ll,, : a , l . t

t

Clearly OD(p) < MPD(p). In this process,we are interestedonly in the positional changesand not on the original elements.Hence,it is sufficientto concentrateon the permutation of the integers1,2, ..., n. Hence,the permutations we considerin this paperare the permutationsof the integersandwill be of the form ( r ' 2 I

t

[Pt

Pz

n ) l.' P, )

t

Q)

andfor the sakeof simplicrtywe canwrite this permutation dspp p2, . . . , pn. The following theoremgives the value for the maximumshift factor amongall permutationsof degreen. Theorem I : [3] The maximumshift factor amongall permutations of degreen is given by

(n even)

fl :2 MPDmax=0,n.*=.{

(nodd)

l*-l2n 12

The numberof permutationsof degreen with maximumshift factor is (r/

. 12

llt*lrll L ( . 2ii cn=1

(naen)

rtl

lt ,Ll\ ( L+ l) ,I l ' ( n o d d ) Note that the maximumshift factor is approximatelynl2. Let n be an eveninteger,say,n = 2m,m > l. In this case,the followingpermutations haveshift factornl2: ( t

l [Pt

2 Pz

l

m

m + l

Pm

P m+ |

2 r \

(

P zm )

3

)

t

-

t

V. RAVICHANDRAN& N. SRINIVASAN

254

p^+1, Pm+2,

w h e r ep 1 , p 2 , . . . , P n i sa p e r m u t a t i oonf m + l , m + 2 , . . ' , 2 m a n d . . ., Pz, is a Pennutationof l, 2, ...' m.

1et n be an odd integer,say, n = 2m + l, ffi 2 l. In this case,permutationsin the following classeshave the shift factor

n 2

l 2n

of the form ClassI : consistsof permutations ( t

I [Pt

z Pz

m P,,

m + I m+ I

m + 2 Pm+Z

where p1, P2, . . ., Pm is a permutationof m + 2, P m + ZP , m + 3 '. . ' , P z r + l i s a p e r m u t a t i o 1n ,o2f , " ' , m '

2m + t')

I

Pz*+t )

m + 3 r . . . r 2 m+ l

of the form ClassII : consistsof permutations '., [,

(4)

m

m + l

m + 2

2m*

Pm

Pm+ l

Pn+2

P2m+l

w h e re p 1 , p 2 , ..., Pm+ t i s a pennutati on of m + l , m + 2 , P m + 2 , P m + 3 , . ' . , P z r + I i s a p e r m u t a t i o1n, 2 , " ' , i l '

'l

and

(5)

) ,2m + | and

of the form (5) wherePb P2, .- ., P^ l s a Class III : consistsof permutations p e r m u t a t i o fn m + 2 , m I 3 , . . . , 2 m + | a n dp - + 1 , P m + 2 ,. . . , P z n t + l i s a permutation of 1,2, ...,m * l. The permutationsin class I arethose permutationsin classII and III which fix m*1. i.e., permutationswith displaced Other than maximumshift factor,derangement, RI. Woo e/ al. [7] haveproposedalgorithmsto generate positionsare shownto decrease in whichall the permutation keyswith displacedpositions.We now discussthepermutations is givenby positions.Thatis, if the permutation elementsaredisplacedfrom their respective with g, * pi . Equationl, thenwe discussthe permutations

255

USED OF PERMUTATTON FOR DISPLACEMENT MEASURE

with all the elementsdisplacedfrom Theorem 2 : l3l The numberof permutations their originalpositionis (alsosee[5]) -2)!dn=nCz(n

n C z ( n- 3 ) ! + " ' + ( - l ) ' n C n ( 0 ! ) '

equalto n tl e wheree : 2.718is the base For very valueof n, this numberis approximately of the natural logarithm. Also the percentagewith this property is approximately 100/e = 36.78. 2. Mean PermutationDistance of 1, 2, of MPD valuesoverthe setProf all permutations Theorem3 : The average . . . , n i s ( n 2- l ) / ( 3 n ) . thenpicanbe1,2, ...,ni eachof them of a permutation, Proof :li p;is theith element nt/n = (n - I ) !. andthis numberis, therefore, occurin equalnurnberof permutations is givenby Therefore,the sumof MPD over all the permutations

I M p D (-pt )+ ,z1

0.P,,

[

n

n

] i ' r - t l + . . .+ ) l r - t l f .

li=r

i=l

,|

Since

l r - i l + . . . + l- rr l =" 5 " . # _ 2 t2 2(l + n)i + n(, +-f) 2 and the sum of the first n naturalnumbersand the sum of the squaresof first n natural n u m b e rasr e n ( n + l ) / 2 a n dn ( n + l ) ( 2 n + l ) / 6 r e s p e c t i v ewl ye,h a v e

l . . .+ l n - t l ) I t l l - , 1+ 1 2 - t + i=I

V. RAVICHANDRAN & N. SRINIVASAN

256 Therefore, I

I

r rPDD ^ /( -p\=)_ M nt I n ^Y ! ,7p, \r

This provesthatthe averageis ( t

(n -l)!

n

n(*-l) = 3

_12-l

3n

- | )/(3 n) .

of Theorem 4 : The averageof MPD valuesover the set $n of all derangements l ) / 3 . 1,2,...,nis(n Proof : By theorem2, the numberof permutationin 5" ir dn lf pi is the rth element i n a p e r m u t a t i o n b e l o n g i nSgrt,ot h e n p i c a n b el , 2 , . . . , i - l , i + l , . . . , n ; e a c h of themoccurin equalnumberof permutations andthis numberis, therefore,dn/(n - l). Therefore,the sumof MPD over 5, is givenby r

(

I M P D ( p n) (,=nd- ln) r , l i

pez,,

ll-rl+....i

[,=t,,*,

d

n

ln-il

i=r,i+n

n

=i1r?ry,?,''t-t+ l " ' + l n- t l ) ' As in previous theorem, we have d I MPD(p)=e*Tw=d,+ Pe5,

This provesthatthe averageis (n + l) / 3. We now compute the averageMPD value of all permutationsgeneratedby an algorithmproposedby Woo et al. that generates(n - l) ! derangements. We begin with a discussionof their algorithm.Thei; algorithmis basedon anotheralgorithmby ltuuth [5J that generates all permutations. The main idea in thesealgorithmsis to representa given integerin factorialnumbersystem. A l g o r i t h ml : F o r e v e r yi n t e g e r giln t h e r a n g e g s g r < ( n - l ) f , a u n i q u e derangement of n elements(Ut, ..., Un)is generated. l . I n i t i a l i zteh es e q u e n c( U e1, .. ., Un) to ( 1, . . ., n). 2. For i : 2 to n, do the following:

USED MEASURE FOR DISPLACEMENT OF PERMUTATION ( a ) s eftf i i= ( r , -

257

-t )] 1 m o c t i- r 1 ) * l ; & = [ A - r / ( r

(b) ExchangeU^, and U,. For Woo's algorithm,each integer91 with 0 s & < (n - I ) ! is represented follows: gl = Qll

+ d42l+ d53!+ ... * dn(n -2)t, 0 < di< i -2.

for m,: d, + | : By takingdz: 0, we havethe followingpossibilities f f i z = l ; m l = l , 2 ; m 4= l , 2 , 3 i . . . t f l n = l , 2 , . . . , f r - l . Therefore,thereare(n - l) ! tuplesof the form (m2,nt3r...tmn) andfor eachof these is generated.It shouldbe noted that the algorithmgeneratesa tuples, a derangement d e r a n g e m e n t,( U U 1z , . . . , U i ) o f 1 , 2 , . . . , i i n ( i - l ) s t s t a g e ( i . e . w hne: ni ) . l n the first stage,(2, l) is generatedand in the secondstage either(3, l, 2) or (2, 3, l) is generated generated. inthe (i - l)st In general, if (pt, p2, ..., pi-r ) is a derangement is generated: stage,then in the rth stageone of the following derangements ( . ( \ . . . , P i _ t ,P t\) , t , . . . , P i _ 1P , Z ) ,. . . tt,PZ, \pr, It is alsoclearthat eachof the numbersl, 2, ..., i - l, i + l, ...,r?occursin the rth positionin (n - 2) ! permutations. of 1,2, ..., Theorem5 : The average of MPD valuesoverthe setof all derangements by Algorithm I is (n + l) I 3. n generated The proof of the aboveresultis similarto the proof of theorem4. By theorems 4 and 5, it follows that the averageof the MPD values over all the derangementsand over those generatedby Algorithm I are equal. This is a conjectureof Woo er al. [71.

3. Order of Displacement In this sectionwe computethe maximumOD valueoverall permutations of degreen. If n is even,our resultshowsthat the maximumMPD valueand maximumOD valueare equal.

& N. SRINIVASAN V. RAVICHANDRAN

258

Theorem 6 : The maximum value of OD over the setof all permutationsof degree /r ( > l) is givenby

(( nn e v e n )

lry lI ;

OD** = I

(nodd)

lt +

Proof : (i) For the caseof evenn, the permutation (

2

m

m + l

m + 2

2 m

I

|

2 ^ \

p=l [ r * r

(6)

|

m )

. . .

with has OD value -- m : n / 2. Note that this permutationis one of the permutations we permutation, maximumMPD value.If OD > n / 2 for somepermutationp, then,for that have

- o' l, l , ; = M P D m u r . M P D ( p=)+ ' , 7i r 1l , which is not possible. (ii) If n : 2m + I is odd,thenthe permutations (

|

Pt=l [ t * l (

|

Pz=l \ m + 2

2

m

m + l

m + 2

2 * \

m + 2

2 m

2 m +I

I

m )

2

m

m + l

m + 2

m + 3

2 m

|

2

haveoD values= m = L-]-. that permutationwe have

n p, oD , Now, if for somepermutatio

M P D=(: p; 1)' ", - o , =l * , i , [ + . r ) = r )' = * t ' 2n

- ' ' * ' ' '. 2

l. (7)

z * \ n

+,

|

(8)

) then for

MEASURE FOR DISPLACEMENTOF PERMUTATIONUSED

259

Since

u i o ^= + u n a I f r + we havep suchthatMP\p) 2 llfPD*axwhichis not possible' 4. Conclusions We have computedthe MPD values for the permutationsin (i) the set of all generatedby and (iii) the set of derangements permutations(ii) the set of all derangements Woo's algorithml. Our resultprovesa conjectureby Woo et al. [7]. Also we havecomputed maximumOD valueover the set of all permutationsof degreen. REFERENCES l.

(1982).ppt.327'370. Becker,H. J. andPiper,F. C. : Speechsecuritysystemin Ciphersystems,

2. Becker,H. J. and Piper,F. C. : Frequency,time and dimensionalspeechscramblingalgorithmsin (1985)'pp. 102-258. SecureSpeechCommunications, for S. : Permutations V. and Sivagurunathan, 3. Jayamala,M., Srinivasan,Narayanan,Ravichandran, speechscrambling,PrePrint. S. W. and Quinn,A. M. : A comparisonof four 4. Jayant,N. S., McDermott,B. J., Christensen, Trans.Commun.,Vol. COM-29,(1981),pp. l8' privacy, IEEE speech methodsfor analog 23, Jan. MA:Addison-Wesley, Reading, programming,Yol.2,2nded. 5. Knuth,D. E. : Theart of computer (le8l). 6. Sakurai,A., Kog4 K. and Muratani,T. : A speechscramblerusing the fast fourier transform Vol. SAC-2(1984),pp. 434-442,May. IEEEJ. Select.Areas,Commun., technique, speech 7. Woo, R. W. and Leung, C. S. : A new key generationmethodfor frequency-domain pp. July. ( 49-7 52, 7 I 997), Vol. coM45, Commun., Trans. I EEE scramblers .

Departmentof Matlrematicsand ComputerApplications Departmentof Electronicsand Engineering Communications collegeof Engineering Sri Venkateswara 105,India. Pennalur,Sriperumbudur-602

(ReceivedDec. 16, 2002)