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4064 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012

Investigations on Bent and Negabent Functions viathe Nega-Hadamard Transform

Pantelimon Stănică, Sugata Gangopadhyay, Ankita Chaturvedi, Aditi Kar Gangopadhyay, and Subhamoy Maitra

Abstract—Parker et al. considered a new type of discrete Fouriertransform, called nega-Hadamard transform.We prove several re-sults regarding its behavior on combinations of Boolean functionsand use this theory to derive several results on negabentness (thatis, flat nega-spectrum) of concatenations, and partially symmetricfunctions. We derive the upper bound for the algebraic de-gree of a negabent function on variables. Further, a characteri-zation of bent–negabent functions is obtained within a subclass oftheMaiorana–McFarland set.We develop a technique to constructbent–negabent Boolean functions by using completemapping poly-nomials. Using this technique, we demonstrate that for each ,there exist bent–negabent functions on variables with al-gebraic degree . It is also demonstrated that thereexist bent–negabent functions on eight variables with algebraic de-grees 2, 3, and 4. Simple proofs of several previously known factsare obtained as immediate consequences of our work.

Index Terms—Bent and negabent functions, Hadamard andnega-Hadamard transforms.

I. INTRODUCTION

L ET be the -dimensional vector space over thetwo-element field . A function from to is called

a Boolean function on variables. The reader is referred toSection I-A for all the basic notations and definitions related toBoolean functions.The Walsh–Hadamard transform (a particular case of a dis-

crete Fourier transform) has been exploited extensively for theanalysis of Boolean functions and used in coding theory andcryptology [3], [4], [6]. For even dimension , the functionsthat attain the largest distance from the set of affine functions(this maximum distance is the nonlinearity) are called bentfunctions. From the perspective of coding theory, these func-tions attain the covering radius of the first-order Reed–Mullercode. Further, a Boolean function on an even number of vari-ables is bent if and only if the magnitude of all the values in its

Manuscript received March 09, 2011; accepted December 20, 2011. Date ofpublication February 03, 2012; date of current version May 15, 2012. This is asubstantially revised and extended version of the conference paper that appearsin [24].We obtained additional results over the conference version; in particular,Section VI is completely new.P. Stănică is with the Department of Applied Mathematics, Naval Postgrad-

uate School, Monterey, CA 93943–5216 USA (e-mail: [email protected]).S. Gangopadhyay, A. Chaturvedi, and A. K. Gangopadhyay are with

the Department of Mathematics, Indian Institute of Technology Roorkee,Roorkee 247667, India (e-mail: [email protected]; [email protected];[email protected]).S. Maitra is with the Applied Statistics Unit, Indian Statistical Institute,

Kolkata 700108, India (e-mail: [email protected]).Communicated by M. G. Parker, Associate Editor for Sequences.Digital Object Identifier 10.1109/TIT.2012.2186785

Walsh–Hadamard spectrum are the same (flatWalsh–Hadamardspectrum). The Walsh–Hadamard transform is an example ofa unitary transformation on the space of all Boolean functions.Riera and Parker [18] considered some generalized bent criteriafor Boolean functions by analyzing Boolean functions thathave flat spectrum with respect to one or more transformschosen from a set of unitary transforms. The transforms chosenby Riera and Parker [18] are -fold tensor products of the

identity mapping , the Walsh–Hadamard transforma-

tion , and the nega-Hadamard transformation

, where . Riera and Parker [18] mention

that this choice is motivated by local unitary transforms thatplay an important role in the structural analysis of pure -qubitstabilizer quantum states. As in the case of the classical discreteFourier transform, a Boolean function whose nega-Hadamardmagnitude spectrum is flat is said to be a negabent function.The research initiated in [18] leads to the natural question

of constructing Boolean functions which are both bent and ne-gabent (these are referred to as bent–negabent functions [16],[23], [24]).First, we concentrate on the nega-Hadamard transform in

more detail. We prove various results in Section II on thebehavior of the nega-Hadamard transform on affine functions,and also on sums and products of functions. Then, we usethis analysis to obtain insights related to the decompositionof negabent functions in Section III. Further, in Section IV,negabent functions, symmetric with respect to two variables,are studied. Our technique renders a simple proof of the mainresult of [21], namely that all symmetric negabent functionsmust be affine. Moreover, a characterization of some bent–ne-gabent functions in the Maiorana–McFarland (MM) class isobtained in Section V, thus complementing some results of[23]. In Theorem 17, we present another method of constructingbent–negabent functions on variables for each algebraicdegree from 2 to , where .In [23, Th. 10], it is proved that if is an MM-type bent func-

tion on variables ( even) which is also negabent then the al-gebraic degree of is at most . Example 6 in [23] describesa technique to construct bent–negabent functions on variablesof algebraic degree ranging from 2 to . Moreover, there is noknown general construction of bent–negabent functions of alge-braic degree greater than , for all .In Section VI, we describe a technique of constructing

bent–negabent functions by using complete mapping poly-nomials on finite fields that constitute a special class ofpermutation polynomials [11], [14]. First, we demonstrate

0018-9448/$31.00 © 2012 IEEE

STĂNICĂ et al.: INVESTIGATIONS ON BENT AND NEGABENT FUNCTIONS VIA THE NEGA-HADAMARD TRANSFORM 4065

the connection between existence of complete mapping poly-nomials over a finite field and the existence of a class ofbent–negabent functions. Then, we demonstrate that for each

, there exist bent–negabent functions on vari-ables with algebraic degree .

A. Definitions and Notations

The set of integers, real numbers, and complex numbers aredenoted by , , and , respectively. The set of all Booleanfunctions on variables is denoted by . Addition over , ,and is denoted by “ ,” while addition over for allis denoted by . If and aretwo elements of , we define the scalar (or inner) productand, the intersection by

The cardinality of a set is denoted by . If ,then denotes the absolute value of , and

denotes the complex conjugate of , where , and. Any can be expressed in algebraic normal

form (ANF) as

The (Hamming) weight of is . Thealgebraic degree of , .Boolean functions having algebraic degrees at most 1 are said tobe affine functions. For any two functions , we definethe (Hamming) distance

.TheWalsh–Hadamard transform of at any pointis defined by

Similarly, the Fourier transform of at any pointis defined by

A function is a bent function if forall . Bent functions (defined by Rothaus [19] more than30 years ago) hold an interest among researchers in this areasince they have maximum Hamming distance from the set of allaffine Boolean functions. Several classes of bent functions wereconstructed by Dillon [8], Dobbertin [9], Rothaus [19], and laterby Carlet [2].The sum is the crosscor-

relation of and at . Taking above, we get the auto-correlation of at . It is known [6] that afunction is bent if and only if for all .For a detailed study of Boolean functions, we refer to [3], [4],

and [6].

The nega-Hadamard transform of at any vectoris the complex valued function:

A function on is said to be negabent if the nega-Hadamardtransform is flat in absolute value, namely for all

. The sum

is the nega-crosscorrelation of and at . We define the nega-autocorrelation of at by

The negaperiodic autocorrelation defined by Parker and Pott[16] is

However, the difference between the aforementioned two defi-nitions is not critical and both definitions can be used.A Boolean function is said to be symmetric if inputs of the

same weight produce the same output, i.e., ,for any permutation .The group of all invertible matrices over is denoted

by . Two Boolean functions are said to beequivalent if there exist , andsuch that for all . Ifand , then and are said to be affine equivalent.

B. Quadratic Boolean Functions

The properties of quadratic Boolean functions, that isBoolean functions having algebraic degree 2, can be found in[15, ch. 15]. If is a quadratic Boolean function on variables,the associated symplectic form of is a mapdefined by

The kernel of is defined as

The set is a subspace of with dimension whereis the rank of . It is known that two quadratic functions

and are equivalent if and only if [15, ch.15, Th. 4]. We recall the following result [1, Prop. A1].

Proposition 1: An element if and only if the function, defined as for all , is

constant. The subspace is said to be the linear kernel of .Since the autocorrelation spectrum of any bent function is

zero at all points except at , the linear kernel of anyquadratic bent function is of dimension 0. Therefore, by [15,


ch. 15, Th. 4] and Proposition 1, all quadratic bent function areequivalent to each other.Suppose , is a permutation and

is any Boolean function. Rothaus proved thata Boolean function defined by

is a bent function. The collection of bent functions of this type iscalled theMM class. If is the identity permutation andfor all , then the functionfor all is a quadratic bent function. Thus, anyquadratic bent function of variables is equivalent to , whichcan be written as (by labeling). From the previous discussions, it is clear that if

is a quadratic bent function, then there exist ,, and such that .

II. PROPERTIES OF THE NEGA-HADAMARD TRANSFORM

It is well known that the inverse of the Walsh–Hadamardtransform of is given by

(1)

for all . The nega-Hadamard transform is also a unitarytransformation. An immediate consequence of the definition ofthe nega-Hadamard transformation of a function in [16]and [18] is the following.

Lemma 2: Suppose . Then

(2)

for all .Next, we prove a theorem that gives the nega-Hadamard

transform of various combinations of Boolean functions. Weshall use throughout the well-known identity (see [15])

(3)

Theorem 3: Let be in . The following statementsare true.(a) For any affine function and ,

.Further, . In particular,

, and, , where are the constant 0, respec-

tively, 1 functions, and is an eighth primitiveroot of 1.

(b) If on , then for ,

(c) If , then.

(d) If , then, where is an

orthogonal matrix over (and so, ).(e) If , and , then

,, where

,.

Moreover, if ,,

.Proof: The first part of is direct from the definition of

the nega-Hadamard transform. The second part of the claimcan be derived from [23, Lemma 1], since

.We show the first identity of (the second follows by sym-

metry). Since

and (see [6, p. 8])

ifif

we obtain (sums are over )

Item is straightforward. The property can be derivedfrom [16, Lemma 2] and [23, Th. 2]. It is to be noted that [23,Th. 2] further proves that the action of the orthogonal group pre-serves the bent–negabentness property of a Boolean function.To show item , we write,


from which we obtain the desired identity. Moreover, if ,and , then , and if

, then , and so

The next result is analogous to the result on the crosscorrela-tion of two Boolean functions [20]. In the nega-Hadamard trans-form context, the basic idea of this result is explained in [7] and[17, eq. (15)].

Lemma 4: If , then the nega-crosscorrelation equals

Proof: We start with the sum at the right hand side.

If we take in the previous lemma, then we obtain

(4)

This is an analogue of the autocorrelation of Boolean functions.It is to be noted that since both Hadamard and nega-Hadamardtransforms are unitary they are energy preserving and hence,Parseval’s theorem holds for both transforms. The classical Par-seval’s identity takes the form

for the Walsh–Hadamard transform. Substituting in (4),we obtain a proof of this fact for the particular case of nega-Hadamard transforms.

Corollary 5 (Nega-Parseval’s Identity): We have

(5)

Lemma 6: A Boolean function is negabent if andonly if for all .

Proof: If is a negabent function, then forall . For all , then by (4), we obtain .The converse also follows from (4).An equivalent result is proved after [17, eq. (15)] and in [16,

Th. 2] for the negaperiodic autocorrelation.

Remark 7: Lemma 6 provides an alternate characterizationof negabent functions.If is an affine function, then for all the nega-

autocorrelation . This implies that any affine functionis negabent. For other proofs, we refer to [23, Lemma 1] and[16, Prop. 1]. A bound on the algebraic degrees of negabentfunctions is an important question. In the following, we providesuch a bound.A Boolean function is said to be near-bent [12] if

its Fourier transform values belong to . It is ratherimmediate that near-bent functions exist only for odd values of. It is known [3] that if is an variable Boolean function( ) and such that the Fourier transform valuesof are divisible by , then has algebraic degree at most

. In case is near-bent, which implies thatthe algebraic degree of is at most .The following theorem (which will be also used later) proved

by Parker and Pott establishes a connection between bent andnegabent functions.

Theorem 8 (see [16, Th. 12]): A functionis negabent if and only if is bent, where

is the elementary sym-metric function of degree 2.

Theorem 9: The algebraic degree of a negabent functionis at most , for any integer .Proof: Suppose is even and is negabent. Then

is bent, and so, its algebraic degree is bounded above by ,therefore the algebraic degree of is at most .Suppose is odd and is a negabent function. Then,

by [16, Remark 1], the Walsh–Hadamard transform values ofbelong to , in other words is a near-bent

function. Therefore, the algebraic degree of is at most .

III. DECOMPOSITION OF NEGABENT FUNCTIONS WITHRESPECT TO CODIMENSION ONE SUBSPACES

Suppose . Then, any function can bethought of as a function from to . For any fixed

, the function is defined asfor all .

Theorem 10: Let be expressed as. Then

Proof: By definition,

(6)


Corollary 11: Suppose is expressed as

where . Then

The functions and are said to have complementary nega-autocorrelation if for all nonzero

The following lemma establishes a connection between thenega-autocorrelations of , and their nega-Hadamard trans-formations.

Lemma 12: Two functions have complementarynega-autocorrelations if and only if

Proof: Let be two functions with complementarynega-autocorrelations. Then

Conversely, suppose for all .Then,

where

ifif .

Thus, the functions and have complementary nega-autocor-relations.

Theorem 13: Suppose is expressed as

where . Then, the following statements are equivalent.1) is negabent.2) and have complementary nega-autocorrelations and

for all with .3) for all and is areal number whenever .

Proof: We show first . Suppose is a negabentfunction. Then for all nonzero .From Corollary 11, we obtain

for all and

which implies for all with.

Conversely, assume that the functions and have comple-mentary nega-autocorrelations and for allwith . Then by Corollary 11,for all nonzero . This implies that is a

negabent function.We now show . The nega-Hadamard transform

of at is

Thus

if

if .(7)

Since is negabent for allwe obtain

(8)

If is negabent, then by Lemma 12 and the equivalence of thefirst two statements proved above, we obtain

Suppose for , . Letand . Then, by (8), we obtain


Therefore, we have , i.e.,

. This proves that is a real

number.Conversely, suppose for all

and is a real number whenever .Without loss of generality, we may first assume, for some . Then, by the aforementioned condition,

. By (7), for all . Nextwe consider the case when . Let

. Then

(9)

Thus, is negabent.

IV. NEGABENT FUNCTIONS SYMMETRIC ABOUT TWOVARIABLES

Suppose is a Boolean function which is symmetricwith respect to two variables, and say. Then there exist func-tions such that

(10)

for all . The Boolean functionis bent if and only if, and are bent and for all

(see [3], [4], [6], and [26]). For negabent functionswe prove the following similar result.

Theorem 14: Suppose is expressed asfor all

. The Boolean function is negabent if and only ifand are negabent and for all .

Proof: The nega-autocorrelation of at is

If is a negabent function then . Therefore, which implies that

for all . Thus, if is a negabent function and sym-metric with respect to the variables and , then it is of the form

, for all. The nega-Hadamard transform of

at is

where varies over . Expanding theabove sum by substituting all possible values of, we obtain

(11)Therefore, for all

. This proves that both and are negabent. Onthe other hand, if and are negabent functions, andfor all , then is also negabent. This shows the

converse.

Corollary 15: A symmetric negabent function is affine.Proof: Let be a symmetric negabent function. Let

us suppose that has algebraic degree greater than or equal to2. Since is symmetric, it is symmetric with respect to any twovariables. Therefore, it is possible to express , for at least onepair of variables, as follows:

where for at least one . But this contradictsthe fact that is negabent. Hence, all symmetric negabent func-tions are affine.

The result of Corollary 15 gives an alternate proof of the factproved in [21]. In fact, the case for even can be immediatelyobtained from Theorem 8.Recall that is the homo-

geneous (i.e., all terms of its ANF are of the same degree), sym-metric and quadratic bent function. Let

, the (only) symmetric linear function. In [22], it is shownthat the only symmetric bent functions are , , ,

.In [21], it is proved (by a long argument) that all the sym-

metric negabent functions are affine. Following [16] and [22],the result in [21] can be achieved in a few lines for even .

Theorem 16: Let be even. A symmetric functionis negabent if and only if it is affine.

Proof: Suppose is a symmetric negabent function.Then is a bent function. Since the direct sum of twosymmetric functions is symmetric, then is a symmetricbent function. The only symmetric bent functions are , ,

, (see [22]). Therefore, can be 0, 1, ,and nothing else. This proves that if is a symmetric negabentfunction on even number of variables, then it is affine.Conversely, it is known that all affine functions are negabent

[23]. Therefore, symmetric functions on even number of vari-ables, if affine, are negabent.


Bent functions do not exist for an odd number of input vari-ables. Thus, there is no equivalent characterization of Theorem8 for odd dimension, and the result in [21] cannot be provedtrivially as before. However, the odd (as well as the even) casehas already been taken care of by Corollary 15.

V. BENT–NEGABENT FUNCTIONS IN MM CLASS

In this section we shall investigate bent functions which arealso negabent in the MM class of bent functions, namely

(12)

where is a permutation satisfying(we call a weight-sum invariant permutation), for all

, and is an arbitrary Boolean function, both on . Weremark that if is orthogonal, that is, withorthogonal ( ), then it satisfies the imposed condition(since , it suffices to show that

; for that, consider). It could be interesting to see if there

are such weight-sum invariant permutations outside of the onesgenerated by the linear orthogonal group.

Theorem 17: A function in (12) on is bent–negabent ifand only if is bent.

Proof: We evaluate

Now, using the fact that is a weight-sum invariant permu-tation, and by (3), we obtain

which implies that

Consequently

that proves our claim.

The following corollary follows easily from our theorem,since bent functions exist for any degree up to half of the (even)dimension. We remark that [23, Th. 10] gives an upper bound

of on the degree of an MM-type bent–negabent functionon variables, but not an existence result.

Corollary 18: If as in (12) is bent–negabent withweight-sum invariant, then the degree of is bounded by .Moreover, there exist bent–negabent functions in the MM classof any degree between 2 and ( is even).

VI. CONSTRUCTION OF A NEW CLASS OF BENT–NEGABENTBOOLEAN FUNCTIONS

It is well known that the maximum degree of a bent functionon variables is and the maximum degree of a negabentfunction is (Theorem 9), which is for even . Thus, it maybe an interesting problem to find out a nontrivial upper bound onthe algebraic degrees of the bent–negabent functions. However,no upper bound strictly less than is known to this date. Parkerand Pott also raised this question in [16, Problem 3, p. 19].So far all the known general constructions of bent–negabent

functions on variables produce functions with algebraic de-grees less than or equal to , where is any positive integerdivisible by 4. In this section, we construct bent–negabent func-tions on variables with algebraic degree equal to ,where is any positive integer greater than or equal to 2.Throughout this section, and is a quadratic

bent function defined as for all. It is known that the quadratic

symmetric Boolean function of the form ,for all is bent. Therefore, by the resultsof Section I-B, the function is equivalent to the quadraticbent function defined previously. Using Theorem 8, provedby Parker and Pott, we obtain the following.

Lemma 19: ABoolean function ( ) is bent–ne-gabent if and only if both and are bent functions.

Proof: Assume that is a bent–negabent function.Since is a negabent function, is a bent function. Thus,and both are bent functions.Conversely let us suppose that and both are bent

functions. Since is a bent function is anegabent function. Therefore, is a bent–negabent function.

The following theorem provides a strategy to constructbent–negabent functions.

Theorem 20: Let for all ,where , and . Supposeis a bent function such that is also a bent function. Then,

defined by

for all is a bent–negabent function.Proof: The function is equivalent to . Therefore,

is bent. The function is affine equivalent to . Sinceis a bent function, is also a bent function. Therefore, byLemma 19, is a bent–negabent function.

Remark 21: Since all quadratic bent functions are equivalent(see Section I-B), if is any quadratic bent function on vari-ables, then there exist , and ,such that for all .


Therefore, if is a bent function and is anyquadratic bent function such that is also a bent function,then by using the strategy described in Theorem 20 we obtain abent–negabent function.Theorem 20 reduces the problem of constructing bent–ne-

gabent functions to characterizing bent functions such thatis also a bent function. We demonstrate below that such

functions can be constructed by using complete mapping poly-nomials.

A. Complete Mapping Polynomials

Let be the field extension of of degree . The finitefield is isomorphic to as a vector space over . Anypermutation of can be identified with a permutation on .Any permutation on can be represented by a polynomialin of degree at most . A polynomial

is said to be a complete mapping polynomial ifand both correspond to permutations on . Fordetails on complete mapping polynomials, we refer to [11] and[14]. The following provides us a strategy to construct bent–ne-gabent functions by using complete mapping polynomials.

Theorem 22: Let . Suppose denotes thepermutation on induced by a complete mapping poly-nomial . Let be definedby for all

, and is a quadratic bent function defined asfor all , such

that for all , where, and . Then, the Boolean

function is an MM-type bent function and

is a bent–negabent function. The algebraic degrees of andare equal.

Proof: Since is a permutation on , the function isa bent function on variables belonging to the MM class.The function

is also a bent function in the MM class since is induced froma complete mapping polynomial. Thus, using Theorem 20, weinfer that is a bent–negabent function.

B. Bent–Negabent Functions on Variables With AlgebraicDegree Greater Than

We consider a particular complete mapping polynomial con-structed by Laigle-Chapuy [11].

Theorem 23 ([11, Th. 4.3]): Let be a prime and. Let be the order of in . Take and a

positive integer coprime to . Assume is such that. Then, the polynomials

are complete mapping polynomials.First, we construct a bent–negabent function on 24 variables

having algebraic degree 7.

Example 24: Following the notations of Theorem 23, let, , and . The order of in is 2, that is .Thus, and . Considerthe polynomial , where

. The last condition guarantees. By using in Theorem 22, we obtain a bent–negabentfunction on 24 variables and algebraic degree 7. The algebraicdegrees of the bent–negabent functions constructed in [23] arebounded above by . Thus, the bent–negabent function con-structed above does not belong to these classes.Certainly, the order of in is , so in general

we have the following result.

Lemma 25: For any , the polynomials

have algebraic degree .Proof: Let . Then

This proves that both and have degree .

Theorem 26: For each , there exist bent–negabent func-tions on variables with algebraic degree .

Proof: For each , it is possible to constructand as in Lemma 25 with and . It isproved in Lemma 25 that and both have alge-braic degree . It is also to be noted that if , we canchoose so that . Therefore,and constructed in this way are complete mapping poly-nomials. If we use Theorem 22 by inducing the permutationwhere , then we obtain bent–negabentfunctions on variables with algebraic de-gree . It is also to be noted that these functions may notbe of MM type but belong to the complete class of MM-typefunctions.

Complete mapping polynomials over of algebraic de-grees 1, 2, and 3, listed in Table I, are obtained by Yuan et al.[25]. In Table I, denotes a primitive element of . Usingthese polynomials and Theorem 22 we can construct bent–ne-gabent functions on eight variables having algebraic degrees 2,3, and 4. Thus, we note that there exist bent–negabent functionson eight variables with maximum possible algebraic degree that


TABLE ICOMPLETE MAPPING POLYNOMIALS OVER

is 4. It is not known whether such is the case for . Thisleads us to state the following as an open problem.

Open Problem: For any , give a generalconstruction of bent-negabent Boolean functions on variableswith algebraic degree strictly greater than .

VII. CONCLUSION

In this paper, we have investigated the nega-Hadamard trans-form of Boolean functions in detail. First, we study the proper-ties of nega-Hadamard transform. Next, we concentrate on de-compositions of negabent functions with respect to codimen-sion one subspaces, and negabent functions that are symmetricabout two variables. A characterization of some bent–negabentfunctions in the MM class allows us to construct bent–negabentBoolean functions on of all degrees up to . Use of com-plete mapping polynomials allows us to further construct newclasses of bent–negabent functions on variables of degreegreater than .

ACKNOWLEDGMENT

The authors gratefully thank the anonymous reviewers andthe associate editor for the detailed and excellent comments, thatsignificantly improved the editorial as well as technical qualityof the paper.

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Pantelimon Stănică received his Master of Science in Mathematics degree in1992 from University of Bucharest, Romania. He completed his Ph.D. in Math-ematics at State University of New York at Buffalo in 1998. Currently, he isa Professor at the Naval Postgraduate School, in Monterey, California. His re-search interests are in Cryptology, Number Theory and Discrete Mathematics.

Sugata Gangopadhyay received his Master of Science in Mathematics degreein the year 1993 from the Indian Institute of Technology Kharagpur. He com-pleted Ph.D. from the Indian Institute of Technology Kharagpur in 1998. Cur-rently he is an Assistant Professor at Indian Institute of Technology Roorkee.His research interests are in Cryptology and Discrete Mathematics.

Ankita Chaturvedi received her Master of Science in Mathematics degree inthe year 2006 from CSJMUniversity Kanpur. Currently she is a Ph.D. student inthe Indian Institute of Technology Roorkee. Her research interest is in DiscreteMathematics and Cryptology.

Aditi Kar Gangopadhyay received her Master of Science in Mathematicsdegree in the year 1995 from the Indian Institute of Technology Kharagpur.She completed Ph.D. from the Indian Institute of Technology Kharagpur in2000. Currently she is an Assistant Professor at Indian Institute of TechnologyRoorkee. Her research interests are in Statistical Inference and DiscreteMathematics.

Subhamoy Maitra received his Bachelor of Electronics and Telecommunica-tion Engineering degree in the year 1992 from Jadavpur University, Calcuttaand Master of Technology in Computer Science in the year 1996 from IndianStatistical Institute, Calcutta. He has completed Ph.D from Indian Statistical In-stitute in 2001. Currently he is a Professor at Indian Statistical Institute. Hisresearch interests are in Cryptology and Security.

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4064 IEEE TRANSACTIONS ON INFORMATION … › pstanica › research › negaIEEE12.pdfHadamard transforms. Corollary 5 (Nega-Parseval’s Identity): We have (5) Lemma 6: A Boolean