permutation for speech scrambling

Journalof IndianAcad. Mathematics Vol. 25, No. 1 (2003)PP.95-107.

V. Rcw,ichandran, Nuayanan Srinivasan*, M' Jq'qnQla and S. Sivagurunalhan

PERMUTATIONFORSPEECHSCRAMBLING

Abstract : In this paper, we explore the applicationof combinatoricsand permutations in speech communication. specifically speech scrambling. Conditions necessaryfor zero or minimal residual intelligence(RI) are discussedwhich include maximum shift factor, displacementof position. and adjacenc.vof segments. Key words : Permutations.Speechscrambling,Shift factor. AMS Subject ClessificrtionNo. : 68T10.

Introduction In speech scrambling systems, the signal can be encrypted by realranging the speech data using permutations. Such permutation scrambling techniques for speech have been developed in the time domain as well as the frequency domain. While in the time domain a speech signal is divided into small segments which are rearanged, in the frequency domain,

the speech spectrum is divided into a number of sub-bands and these sub-bands are rearranged.In both domains,effective scramblingas measuredby residual intelligibility (RI) [1], is dependenton the displacementof elementsdue to realrangement.Subjectivetests have been performed to test the efficacy of scramblingtechniquesbasedon permutation [3], [6 ], [8 ] The averagedisplacementof all elementsitr a given permutation is known as its shift factor [1]. In permutation scrambling techniques,permutation is the key for encryption and the inverse permutation is the key for decryption. High shift factor is essentialfor a chosen permutation to yield minimum or zero residual intelligence (RI) in the encryptedsignal Il].

V. RAVICHANDRAN,N. SRINIVASAN,M. JAYAMALA & S. SIVAGURU

The perniutationscramblingl2l which use a higher degreepermutationas key, needthe valueof maximumshift factor.Basedon the maximumshift factor,the thresholdandthe key spaceare fixed. Therefore,the computationof maximumshift factor is important. In addition to scramblingin time-domain,permutationscramblinghave also been appliedto power spectrumvaluesobtainedfrom the speechdata.Sakuraliet al. t7l found that RI decreases with increasein Hammingdistancefor scrambledFFT valuesobtained from speech.Initial efforts for generatinga permutationkey sufferedfrom the lack of algorithmsfor generatingpermutationkeys that satisfiedcertainselectioncriteria for key generation.Woo et al. t8l proposedalgorithms for key generationand indexing the permutation keys that met the necessaryselection criteria. However, their algorithms generatedkeys in which all the elementsare displacd out of their naturalplaces.Their algorithms did not maximize the shift factor or producepermutationswith no adjacent segments.In this paper,we considerthree conditionsor selectioncriteria for permutation scramblingand deriveexpressions for the numberof permutationkeysgeneratedfor eachof those conditions.We also considerderangement, i.e., all the elementsin the chosen permutationmust be displacedfrom the original position.In addition,adjacentelementsin the chosenpermutationshouldnot be mappedto adjacentelements.We analyzethesethree conditionsand find the numberof permutationssatisffingtheseconditions.Theseproperties satisfiedby the chosenpermutationsmust also be satisfiedby the inverseto avoid any possiblebreakingof the system.It canbe easilyseenthat the propertiesdiscussed hereare satisfiedby both the given and its inversepermutations. In Prasanna et al. [5], the maximumshift factorfor permutations with degree3 12 are computedby assumingthat the maximumshift factor is given by the permutationswhich sendsi to (n + I - i). A formula for the maximumshift factor basedon the computational result was also proposed.Thoughthis is one of the permutations giving ma"ximumshift factor,this is not verified. The sameauthors[6] haveempiricallystudiedthe propertiesfor ensuringminimum or zero RI and the numberof permutations satisffingtheseproperties were computednumericallyusing computerprograms.In our paperall the permutations giving maximumshift factor are describedand an exactexpressionfor the maximumshift factor is provedin section2. Section2 also presentsa formula for countingthe numberof permutationswith maximum shift factor. Exact formulae(or recurrencerelationsin some cases)for the otherselectioncriteriaarealsodeveloped.Section3 discusses the permutations in which all the elementsaredisplacedfrom their originalposition.ln section4 we countthe numberof permutationsin which no two adjacentelementsaremappedto adjacentelements. permutationsthat sdis0 all the threeconditions. Section5 discusses 2. Permutationswith Maxirur

Shift Factor

PERMUTATIONSFOR SPEECHSCRAMBLING n tb Q z , . . . , Q r o f pl,pz,...,pn A r e a r r a n g e m eQ permutation n. This can degree be written as

i s c a l l e da p e n n u t a t i o on f

( o t P z. . . p , )

p=l

I

[qt

(1)

qz...qr)

For any given permutation p of degree n, let d, represent the position to which the permutationp moves the ith element.The shift factor of the permutationp is given by

a = L' t ',l - a , l In this process, we are interestedonly in the positional changesand not on the original elements. Hence, for the purpose of analysing the effectivenessof the algorithms, it is sufficient to concentrate on the permutation of the integers l, 2, . . . , n. Hence, the permutationswe considerin this paper are the permutationsof the integersand will be of the form

f'

2 nl

(2)

[P' Pz...P") andfor the sakeof simplicitywe canwrite this permutation zspp pz, . . . , pn High shift factoris essentialfor ensuringminimumor zeroRI, sincethe permutations with high shift factordisplacesthe elementsmuchfarther.If the shift factoris takento be the maximumpossiblevalue,we candescribeall the pennutations with the maximumshift factor. The following theoremgives the value for the maximumshift factor amongall permutations of degreen. Theorem 1 : The maximumshift factoramongall permutations of degreen is given by

lt -n l 2 0ro=l l n

(n even)

I

lt-*

(n odd)

Note that the maximumshift factor is approximatelyn I 2. Beforeproving theorem we discussthe set of permutationswith maximumshift factor.

V. RAVICHANDRAN. N. SRINIVASAN. M. JAYAMALA & S. SIVAGURU

98

2.1 Permutations of even degree : Let n be an even integer,say, n:2m, this case,the following permutationshave shift factor n /2:

m > l.ln

(3)

wherepy p2,..., pm is a permutatioo nf m + l, m + 2,...,2m pz^is a pennutationof 1,2, . . . , m.It shouldbe notedthat

and pm+p

pm+2

p,2

i for i 3 m and pi 3 i for i 2 m + l.

Also m

m

m

m

Zp,=E(m+i) andl pm+r=Z i=l

i=l

i=l

i=l

The shift factor cr is given by

n a . =l t - o t l + . . .+

l,

- o ^ l+

-pm++ t . . .+ l* * | lr* nz^l

= pt - I + ... * pm- m + m + | -pm+l* ... + 2m -pzm m

=I

m

m

m

pi-Zi+2,@+,)-I

i=l

= 2;

i=l

i=l

pm+i

i=l

(m + t - ,) -- 2m2

i=I and, therefore, 2m2 2m2 n = d.= m = 2^ i Algorithms for generating all the permutations are discussedin [ ] and have been used fbr speech scrambling [8]. All the permutations of even degree with maximum shift factor can be generatedusing those algorithms. Generatetwo permutatiotrspl, pz, . . . , p^ ffid 8t, Q2, ...,Im.

Then the permutationwith the maximum shift factor is pt * n,

PERMUTATIONSFOR SPEECHSCRAMBLING

99

p z + m ' . . . , P m* f t i ,Q t , Q z , Qm.Inthiscase,theindexforthepermutationcan be takenas an orderedpair of the two indexesof the two permutationsusedfor generating the permutation.Sincethe index is an orderedpair, the individualindexescan be generated by differentalgorithms. 2.2Permutations of odd degree:Letnbe an oddinteger, say,n:zm * l,m ) l.In this case,permutationsin the following classeshavethe shift factor n 2

l 2n

Class I : Consistsof permutationsof the form ( t I

2

m

I

' [Pr Pz ... Pm

m + t

m + 2

m + | Pn+z

where

z m+ r ) (4) nZr*t

)

n + 3 r . . . r 2 m + I 6frdp^+2,

Pm+3,

CaseII

2 . . . .m

m + l

m + 2

2m+

P 2 . . .Pm P m + l P m + 2

P2n+l

'l

(5)

)

whe re P p P 2 , ..., p m + t i s a permutati on of m + l , m + 2, P m + 2 , P m + 3 , . . . , P 2 r + I i s a p e r n u t a t i oonf 1 , 2 , . . , m .

2m + I and

class III : consists of permutationsof the form (5) wherepr, pz, . . ., p^ is a p e r m u t a t ioof nm + 2 , m * 3 , . . . , 2 m + I a n dp m + t ,p m + 2 pz^+1 isa permutation of 1,2, ..., m + l. The permutationsin classI arethosepermutationsin classII and III which fixes z + l' The p€rmutationsof odd degreecan be generated by a proceduresimilar to that discussed in Section2.1. 2'3 Proof of the Formula for the Maximum Shift Fector : The following result regardingpermutationsis usedto prove the formula for the maximumshift factor. The result is of independentinterest;from a given permutationone can get pennutations with higher shift factorsusingthis theorem.

IOO

SIVAGURU V. RAVICHANDRAN.N. SRINIVASAN,M. JAYAMALA & S' Theorem 2 : Let p andp' be permutations given by (t

a

^, - l [.o'"'b

n

c

'\

I

d

e')

c

n )

to)

and (t

a

-, ' - l [r'...d

b

|

(7)

P")

lf max{o' b } < min{" d }, then andu, , aobetheshiftfactorof p, p'respectively. ap < 9.p,.

Proof : We haveto provethatmax {o, b } < min {'' d } implies l a- b l+ l c- d l< l o - d l+ l c b l ' For the proof we considerthe followingcases: (i)

ad

In the first case, la- bl+ lc- dl= b a + d c and l a - d l + 1 "- b l = d - a + c - b ' Hence,the inequalityholds if b - c < c - b or b < c. Proofsforthe remainingthree casesfollow similarlY. '

proof of the theorem I : The proof is given for even and odd degreepermutations otherthan those separately.In the caseof even degreepermutations,for any permutation 2.1, we haveelementssatisffingthe conditionsof the abovetheorem' given in sub-section

PERMUTATIONSFOR SPEECHSCRAMBLING

l0t

Therefore,the new permutationwill havehigher shift factor.This can be seendirectly as follows. If the permutationis not the one in sub-section2.1, then one of the elements d = p i , l < i , p i l m w i l l b e m o v e dt o t h e e l e m e nbt= p j w h e r eI S j , p i 3 m . Correspondingly, one of the elements c = pk, ffi * | S k, pk, < 2m will be movedto t h e e l e m e nf tl = h w h e r em + l < l , p t < 2 m . Now max{a,bl 17n and ) q, min { c, d } m + I andhencemax { b \ < min I c, d} . Therefore,the permutation obtainedby interchangingthe numbersD anddwill havehighershift factor.This provesthat the permutationsgiven in sub-section2.1 arethe permutations with rnaximurnshift factor. In the case of odd permutations,for any permutationother than those given in subsection2.2, we have elementssatisffingthe conditionsof the abovetheorem2 and, therefore,the new permutationwill havehighershift factor.Hence,the permutationsgiven in sub-section2.2 arethe permutationswith maximumshift factor. We closethis sectionwith a countfor the numberof permutations with maximumshift factor. Theorem3 : The numberof permutations of degreen with maximumshift factoris

cn=

I It(;) +)I 1{(

(n even)

(n odd)

The result follows by counting,sincethe permutationswith maximumshift factor are known.For example,when n is odd,we needto countthe distinctpermutationsin classesI, II andIII. Sinceall the permutations in classI belongto boththe classesII andIII, we have c n = m ! ( m+ l ) ! + ( m + l ) ! m l - ( ^ l ) 2

= ( m t ) 2 1 2 m+ l l

=.[ (q I

By Stirling'sasymptoticformula nt x r{e-n(2nn)l/2,

the percentageof permutations with ma: