Several New Classes of Bent Functions From Dillon ... - IEEE Xplore

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 3, MARCH 2013

Several New Classes of Bent Functions From Dillon Exponents Nian Li, Tor Helleseth, Fellow, IEEE, Xiaohu Tang, Member, IEEE, and Alexander Kholosha

Abstract—Several new classes of binary and -ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of which have close relations with the well-known Kloosterman sums. Index Terms—binary bent function, Dickson polynomial, Kloosterman sum, -ary bent function.

I. INTRODUCTION

L

ET be the ring of integers modulo . An -variable boolean function from to is bent if it has maximal Hamming distance to the set of affine boolean functions, which was initially introduced by Rothaus [28]. Boolean bent functions have attracted much attention due to their important applications in coding theory, cryptography, and sequence design. As a logical extension of Rothaus’ notion of a bent function, Kumar et al. generalized it to -ary bent function from to [21], where is an integer. From now on, assume that are positive integers with , is a prime, and is the finite field with elements. Until now, several classes of bent functions have been found in various papers, for example, the binary bent functions in [1], [4], [5], [7], [9], and [30] and the -ary bent functions in [13]–[17]. Notice that most of these known bent functions are quadratic functions, monomial functions, or binomial functions. In this paper, we investigate a class of bent functions of the form (1) where , , positive integer satisfying

, and

, is the smallest , and

Manuscript received April 03, 2012; revised August 28, 2012; accepted November 07, 2012. Date of publication November 22, 2012; date of current version February 12, 2013. N. Li and X. Tang were supported in part by the National Science Foundation of China under Grants 61171095 and 61201243 and in part by the Funds for the Excellent Ph.D. Dissertation of Southwest Jiaotong University. T. Helleseth and A. Kholosha were supported by the Norwegian Research Council. N. Li is with the Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu 610031, China, and also with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway (e-mail: [email protected]). X. Tang is with the Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]). T. Helleseth and A. Kholosha are with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway (e-mail: [email protected]; [email protected]). Communicated by N. Kashyap, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2012.2229782

is the trace function from to its subfield , where . For the binary case, i.e., , Dillon characterized the bentness of monomial function for [7], and Leander [22] and Charpin and Gong [3] generalized it to any with . In [3], the authors also considered the functions of form (1) with multiple trace terms based on some exponential sums and Dickson polynomials. For an odd prime , Helleseth and Kholosha [14] studied the monomial Dillon function with and proved that such bent functions exist for . This is the unique known -ary bent function of form (1) with . For the case , several classes of binomial functions can be found in [23] and [29] for and in [19] and [31] for , respectively. In this paper, for any prime , we give a necessary and sufficient condition concerning the bentness of defined by (1). Then, together with some partial exponential sums studied in [24] and [19], we derive several classes of bent functions of form (1) with Dillon exponents which are not previously studied both for even and odd , where is an integer. This kind of Dillon exponents leads us to new detailed characterizations for the bentness of from a simple approach. The bentness of these functions has close connections to the well-known Kloosterman sum. From our method, new binomial, trinomial, and quadrinomial bent functions of this type and bent functions with multiple trace terms are characterized. The remainder of this paper is organized as follows. Section II gives some preliminaries. In Section III, we derive a necessary and sufficient condition for the bentness of . In Section IV, several classes of new binary bent functions are obtained according to some partial exponential sums. By the same techniques used to construct binary bent functions, new -ary bent functions of form (1) are obtained in Section V, and the concluding remarks are given in Section VI. II. PRELIMINARIES A. Bent Functions In this paper, we mainly consider the following two kinds of bent functions. 1) Boolean bent function: Let be a boolean function in variables. The Walsh transform of is defined by

Then, a boolean function .

0018-9448/$31.00 © 2012 IEEE

is bent if

for all

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

2) -ary bent function: Let be a -ary function in variables, and is a primitive th root of unity. The Walsh transform of is

Then, is -ary bent if for all . It is well known that a binary bent function only exists for even and its maximal degree is [28]. Moreover, for an odd prime , a -ary bent function is called regular , for some function if for all mapping into . A -ary bent function is called weakly regular if there exists a complex having unit magnitude such for all . that The degree of a -ary bent function had been characterized as follows. Lemma 1 [18]: If is a bent function from to , of satisfies , then the degree is (weakly) regular bent, then . and if For any prime , one can verify that almost all the bent functions investigated in this paper have maximal degree. B. Binary Dickson Polynomials The Dickson polynomial over [26]

of degree

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Throughout this paper, let , , and be a positive divisor of . Denote the cyclic subgroup of by

The cyclic subgroup

where the cyclic group For

can be decomposed into

and

is a generator of

. and

, define

We discuss for and 1) : For this case, if every nonzero element , where for some

as follows. , it is well known that . Notice that has a unique representation as and . Thus, for any and , we have

is defined by

Dickson polynomials can also be defined by the following recurrence relation:

This implies that we only need to consider for . For the case of , Mesnager gave a more detail discussion of and obtained the following result. Lemma 2 [24]: For

and

, we have

with initial values where Some properties of Dickson polynomials are given as follows: P1 ; P2 ; P3 [8] If and , then . P4 [26] Let for any , and , then if if if

and and

.

C. Partial Exponential Sums For fined by

, the Kloosterman sum

over

is de-

For , some results on are obtained in [29]. By using the properties of Dickson polynomials, we can obtain some results for the general later. 2) : The only known result on is for . For , the Kloosterman sum also has close relations with the partial exponential sums and in this paper we mainly use the following result. For more details, the reader is referred to [14] and [19]. Let be a primitive element in and for . Note that if . Thus, we can define for and for as follows:

and define where

is a primitive th root of unity.

(2)

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Lemma 3 [19]: For

Case 2:

, we have if otherwise

and if otherwise if

where

, and

otherwise. III. BENTNESS OF

OF

FORM (1)

In this section, we consider the bentness of functions of form (1) for any prime as below. Note that for any prime , every nonzero element has a unique representation as , where and , where is a primitive element of . Then by the definition, for any , one can derive (3), shown at the bottom of the page. We discuss (3) as follows: Case 1: : By (3), one gets

since

runs through

when

: For any fixed , one can claim that has a unique solution when varies from 0 to . This can be verified by the facts that has solutions in and implies for any . Assume that , , is the unique solution of , then by (3) one can deduce (4), shown at the bottom of the page. Therefore, we can obtain the following result according to the above discussions and the method used in [31]. For convenience, define

varies from 0 to .

Theorem 1: For any prime and , the function defined by (1) is bent if and only if . Moreover, if holds for an odd prime , then is regular bent, and its Walsh coefficient is given by if if where , , is the unique solution of when varies from 0 to . Proof: We prove it for and , respectively. 1) : If , it can be easily verified that is bent. Conversely, if is bent, then which implies since is an integer for .

(3)

(4)

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

2) : If , the bentness and Walsh coefficients of can also be verified by the discussions in Cases 1 and 2. On the other hand, if is bent, for some and then we can assume that since (see Property 8 in [21]). Let for . Then one has and . These together with

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2) For any

, one has

. Then, one can derive

since is a permutation on . By using the following relation between

imply

and

(5) i.e.,

since by Eisenstein’s criteria, is irreducible over the rational numbers, and hence, it is the minimal polynomial of over the field of rational numbers. Then by , one has . Thus, we can discuss it as follows: 1) Suppose , i.e., . Note that this only occurs if since there are at least indexes such that according to (5). For , again by (5) one has . Hence, this is impossible . 2) Suppose , then when . By (5), one can claim that since otherwise there exists some such that . Therefore, one gets and then for any , which shows . This completes the proof. In what follows, we determine several classes of bent functions of form (1) by Theorem 1 and the choices of and for and , respectively.

(6) proved by Delsarte and Goethals [6], we can derive the result as follows. Proposition 2: Let and . If

, where , then

Proof: By the definition and the fact deduce that

, one can

Then by (6), and the property of Dickson polynomials P2, we have

IV. SEVERAL CLASSES OF BINARY BENT FUNCTIONS In this section, assume that and we consider the binary functions of form (1) in the finite field . First, we give some results on for general with . Define an exponential sum related to Dickson polynomial as follows:

Proposition 1: Let , where . Then, we have 1) ; 2) . Proof: Since , we can deduce that 1) By the definition, one has

Since , one has that is a permutation on These together with P3 imply

and then according to P4.

and

since

if

. Hence

This finishes the proof. For any given

.

, when ranges over

runs through . This implies

, Remark 1: This result generalizes the case in [29] for and some special cases can also be similarly discussed as in [29]. For the general case, it seems difficult to determine for and .

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A. First Class of Binary Bent Functions Let

and be a positive integer with ; in this section, new binary bent functions of the

form (7)

can be obtained, where , , and is the smallest integer satisfying and . By Theorem 1, in order to investigate the bentness of , we need to consider whether the exponential sum

equals to 1 or not since . Notice that if , where and is a generator of the cyclic group . This implies if and (8), shown at the bottom of the page. For any , one has (9) since and then ranges over when runs through , shown at the bottom of the page. For the sake of simplicity, for , define (10) where

and

Theorem 2: Let , and

where

In what follows, we give more concrete characterizations on defined as (7) by suitably choosing the bentness of , , and . (I) Binomial Bent Functions: : Note that for this case, the bentness of this function had been discussed by Charpin and Gong for [3], and generalized by Mesnager [23] for , , and and by Wang et al. for the case of , , and in [29], respectively. In this paper, we consider the general case for and . By Theorem 2, one can know that the bentness of is determined by the values of and for . However, for the general , the values of are not determined yet. Especially, for the case of , is a permutation on for any ; thus, we can get the following result by (9), (10), Theorem 2, and Proposition 2. From now on, we always assume that is a primitive element of , , and . , Theorem 3: Let , where and ; then, defined by (7) with only if

Example 1: Let and , , function defined by (7) with

, and with is bent if and

; then , and the is

. be defined by (7) with . Then, is bent if and only if

,

1) If one takes and gram, one can get

, then by a Magma pro, , and . This implies

are defined by (10).

(8)

(9)

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

2) If one takes program, one can get

and

, then by a Magma , and . This implies

,

Computer experiments show that

and

2) If one takes program, one can get and implies

and

, and is

, then by a Magma , . This

,

, then by a Magma , . This

are bent for both cases.

Remark 2: The bent functions of form (7) with indeed exist for the case of . For example, , , then . (II) Trinomial Bent Functions: and : Note that implies is odd. For this case, by Theorem 2, the binary function

is bent if and only if

, one has that

since and is a root of . Similarly, Thus, the result follows from Theorem 2 and Lemma 2. For any , it is well known that

where , and are given in (10), and is a positive integer with . Note that for , the values of are determined by and due to Lemma 2. Especially, from Theorem 2, we can get the following corollary. Corollary 1: Let

be defined by (7) with , , , is bent if and only if if otherwise.

,

, , and

.

(11) Using this together with Theorem 2, we can derive the following results. Corollary 2: Assume that is defined by (7) with , , , . Then for odd , we have 1) for any ; 2) with and cannot be bent if ; 3) with cannot be bent if . Proof: These results can be verified as follows: 1) If one takes and , then by Theorem 2, the function with is bent if and only if . However, cannot be bent if [3], which implies that cannot hold for any . 2) For this case, by Theorem 2, is bent if and only if if , and otherwise. But by (11), one has that and . Thus, with and cannot be bent if . 3) In this case, by Theorem 2, is bent if and only if , and by (11), one gets . Then, the desired result is obtained if . This completes the proof. Remark 3: Some other special cases of Theorem 2 can also be discussed as above, and more results can be obtained if one considers the relation between and . For the explicit evaluation of , the reader is referred to [2]. Some examples of bent functions obtained as above are given as follows. Example 3: Let . Thus,

be a primitive element and . Let and . We consider the functions of form (7) with and as follows: 1) If one takes , , , and , then , , , and the function defined by (7) is . Then, by (10) and a Magma program, one can get , , and , i.e., , , and , , . Then by Lemma 2, one has that . of

. Then, is equal to

and ,

; then

,

Computer experiments show that

Proof: By

are bent for both cases.

and Example 2: Let , , the function defined by (7) with

1) If one takes program, one can get and implies

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and

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2) If one takes

, , , and , then , , , and the function defined by (7) is . Then, by (10) and a Magma program, one can get , , and , i.e., , , and , , . Then by Lemma 2, one has that . Computer experiments show that both the functions here are bent, which are consistent with the results given in Theorem 2. Example 4: Let and be a primitive element . Thus, and . We consider the functions of form (7) with , and in as follows: 1) If one takes , , and , then , , , and the function defined by (7) is . By a Magma program, one can get , , , and , i.e., . 2) If one takes , , and , then , , , and the function defined by (7) is . By a Magma program, one can get , , and , i.e., . Computer experiments show that both the functions here are bent, which are compatible with the results given in Corollary 1. (III) Quadrinomial Bent Functions: and : For and , one has and then by Theorem 2, the binary function of

is bent if and only if

, , and for , and are defined by (10), and are given as in Lemma 2. Note that new trinomial (respectively, binomial) bent functions can also be obtained if one takes one (respectively, two) of equals to 0 according to Theorem 2. The case for had been discussed in [23] and other cases are novel. For the case of , the values of for are discussed for some special cases in [29], and for larger , we can only derive the expression of in this paper. Thus, up to now, for the case of , only some special choices of the

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 3, MARCH 2013

coefficients can be considered such that by (7) is bent according to Theorem 2.

defined

B. Second Class of Binary Bent Functions In this section, several classes of bent functions with multiple trace terms of the form (12) can be obtained according to Theorem 1, where , , are integers, is a subset of , and is the smallest positive integer such that and . According to Theorem 1, the function defined by (12) is bent if and only if (13) In what follows, we discuss the cases for respectively. (I) Bent Functions With Multiple Trace Terms: we can obtain the following results. Theorem 4: Let , are integers with with , 1) If

2) If

in

Proof: Note that since

if

and

and

, : First,

, then

, where and . Then is bent if and only if

, then

is bent if and only if

and

has a unique solution . Thus

. This implies

where

If through

, then . Therefore

ranges over

when

runs

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

If runs through

, then . Thus

ranges over

times when

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This together with (6) implies

This finishes the proof. and be two functions of the form Remark 4: Let in Theorem 4 with the pairs and . Similar conditions as in Theorem 4 can be derived such that is bent if and are suitably chosen. , , and be a primitive Example 5: Let element of . Thus, and , where . Several examples of bent functions in Theorem 4 are given as follows. 1) Let ; if one takes and , then one has , , and , i.e., . Thus, is bent if . 2) Let ; if one takes and , then one can obtain , , and with the help of computer, i.e., . Thus, is bent if . These binary functions are verified by a Magma program, which are compatible with the results given in Theorem 4. In what follows, we present another two families of binary bent functions with multiple trace terms. Theorem 5: Let and

with is bent if and only if

. Then,

Proof: Note that for

and

, one has

since and , if . where in the last second identity, we use Then, the result follows from Theorem 5 and the fact if . Theorem 6: Let and

In particular, if bent if and only if

. Then,

and

be the function defined as in Theorem Corollary 3: Let 5 with and . Then, is bent if and only if . Proof: By Theorem 5, if and , using for any , one has

, then

is

(14) Proof: For

, one can similarly derive

Then by Theorem 1, the first assertion is obtained. If , then one can derive

This together with (6) implies Then, the result follows from Theorem 1. For , we can get the following corollary.

with is bent if and only if

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where the last identity holds since

implying . This finishes the proof.

Remark 5: It is very interesting that equality (14) is related to a conjecture proposed by Helleseth [12] about the cross correlation between -sequence and its decimation. Helleseth conjectured that is always a cross correlation value between -sequence of period and its -decimation if . If (14) holds for some with , then the function is bent and in this case Helleseth’s conjecture also is true. For some small values of , we can have the following. be the function defined as in Theorem Corollary 4: Let 6 with . Then, we have the following. 1) If , then cannot be bent. 2) If , then is bent for all . 3) If and is odd, then is bent for all , where . Proof: By Theorem 6 and the fact , we can have the following. , then 1) If . This implies , and then case. 2) If

, then

(II) Bent Functions With Multiple Terms: and : For this case, since , and new binary bent functions can also be obtained if (13) holds. Assume that and . By the fact that if and , one is bent for this case if and only if can conclude that

On the other hand, one has

cannot be bent for this and

Thus, if , then when ranges over . Then, we can obtain the following result. be defined by (12) with , , where and is bent if and only if

Theorem 7: Let , ,

if 3) For

. Thus, the result is obtained. and is odd, one has

runs through

, , and . Then,

and

where Lemma 2 for if . This completes the proof. Remark 6: Note that the functions in Corollary 4-2) and 3) actually had been studied in [11] by using the property of Dickson polynomials. In this paper, we obtain these results by a different approach. Remark 7: From the above discussions, one can know that 1) The Walsh coefficient of in Corollary 4-1) takes at most 3 values for given nonzero . 2) Let be a subset of , then the function of the form with can be similarly discussed as in this section and some new characterizations on the bentness of can be obtained if , and are suitably chosen.

and

are given as in

.

Remark 8: From above discussion, we can have that 1) for any other suitably chosen , similar result can be obtained as above; 2) For general , similar result can also be obtained. Especially for the case of , new bent functions can be obfor . tained by the results in [29] about , if one takes , and For with , where and . for any , and thus Then, is involved in the bentness of . This case only can also be determined. to To end this section, we compare our results of those exposed in [25]. Mesnager and Flori [25] studied a class of functions of the form , where , is a set of representatives , , , of the cyclotomic classes modulo

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

, and . They characterized the bentness of by Dickson polynomials and some partial exponential sums, and then considered further relations among the , or . partial exponential sums for some special choices of However in this section, we mainly consider the functions of (1) which are with Dillon exponents of the form not previously studied. This leads us to different characterizafrom a tions and more detailed results on the bentness of different approach.

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A class of -ary trinomial functions over

is defined as (16)

where

, , and is a positive integer with . . Then, one can Note that is odd since conclude that both and permutate . By Theorem 1, defined by (16) is bent if and only if

V. SEVERAL CLASSES OF -ARY BENT FUNCTIONS In this section, it is always assumed that is an odd prime. We can obtain several classes of -ary bent functions by using the same techniques in constructing binary bent functions. Note is bent, then it is regular bent. Thus, that in Theorem 1, if all bent functions obtained in this section are regular. A. First Class of -ary Bent Functions Let over

and as follows:

; define a class of -ary functions

Using this together with Lemma 3, we can determine the values of and for . For the sake of , . Recall that simplicity, let

(15)

where , with such that and By Theorem 1,

, , is a positive integer , and is the smallest positive integer . defined by (15) is bent if and only if

which implies if and only if sin

Similar as (8) and (9), one can also derive that Theorem 8: Let be defined by (15). Then, ular bent if and only if

where

We can discuss according to Lemma 3 as follows. 1) , and : In this case, by Lemma 3, one has

is reg-

for . Until now, for any odd prime , the only known result for is the case for , which is recently obtained in [19]. For the general case, new bent functions of form (15) can be obtained if are determined. Thus, in what follows, we only consider . Note that in this case, and the case for lie in the same cyclotomic coset modulo . if . Moreover, one has that and Then, let . , we can determine the bentness For this special case of of based on the coefficients instead of for that given as in Theorem 8.

sin cos since sin and , where is defined as in Lemma 3, sin and cos denote the sine and cosine functions, respectively. , and : Again by Lemma 3, one gets 2)

and then derives that if and only if sin . , and : Similar as 2), one can easily 3) conclude that if and only if sin cos . 4) , and : For this case, Lemma 3 implies

i.e., Therefore, we can obtain the following result.

if and only if .

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Theorem 9: Let be defined by (16), , , be defined as in Lemma 3, and be given by (2). Then, is regular bent if and only if (17) holds, where (17) is shown at the bottom of the page. Observe that when runs through , so does . Then, for , one can have any given

Remark 9: From this Corollary, one can deduce that if both the pairs and such that in Corol. Otherwise, one gets lary 5 is bent, then one has cos sin sin cos sin sin a contradiction. Then, if and only if

cos which leads to is bent for both the pairs and

, and

cos

(18) denotes the conjugate of where hand, by Lemma 3, one has if if

. On the other

(19)

.

By (18) and (19), the characterizations of the bentness of can be simplified in some cases. Note that the function defined by (16) is the ones in. For this case, one has vestigated in [19] if one takes . Then by Theorem 9, (18), and (19), we have be defined by (16) with , Corollary 5: Let , , be defined as in Lemma 3, and be given by (2). Then, is regular bent if and only if cos is equal to sin cos

sin

cos

which is consistent with the results given in [19, Th. 1]. Moreover, for the case of , we improve the results given in [19, Th. 2] by pointing out that can be bent only . when For another special case, if one takes , then a class of -ary binomial functions over can be obtained as follows:

if if

.

In particular, if , then is regular bent if and only if and . Proof: The first assertion is due to Theorem 9. For the case , to complete the proof, we need to prove that cannot be bent if . For , is regular bent if and only if cos sin sin cos . This together with (18) and (19) implies sin , i.e., , which contradicts with . This completes the proof.

(20) where is a positive integer with . By Theorem 9, we can obtain the following result. ,

be defined by (20) with Corollary 6: Let , , be defined as in Lemma 3, and be given by (2). Then, is regular bent if and only if is equal to sin

sin

sin sin

if

,

if

,

if if

, ,

(21) .

, then is regular bent if and In particular, if only if . Proof: The first assertion is due to Theorem 9, and the second one can be easily obtained from (18) and (19). This completes the proof. defined by (20) and CorolNote that from the function with lary 6, we can also derive that is regular bent if and only if . However, it had been proved that holds only for [14], [20]. Moreover, when , set ; then and the function defined by (20) for is . This implies that if the pair such that is bent, then so is the pair . For the case

if if

,

if if

, ,

(17)

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

, this symmetric property does not hold any more. But in this case, for any , by , , which implies either both one can derive that and belong to or both not belong to since is a square element. Further, one has if . Then by , we have Corollary 6, let , Corollary 7: Let , , and be given by (2). Then, is not bent. Proof: Since , then by Corollary 6, one has that is bent if and only if . This is impossible for [20]. Thus, we only need to prove is not bent for . When , Garaschuk and Lisonêk [10] had proved that is even if and only if or is a and , i.e., is even if and square in only if or since is a square in if and only if is a square in . However, for , one has since . This together with (21) implies ,a contradiction with . This completes the proof. , , and ; then Example 6: Let and . Let be a primitive element in the finite field and be defined as (2) over . We consider the function of form (20) in the finite field . 1) Let and ; then , , and . By a Magma program, one can know , , , , , and . By (21), one has sin sin

, which shows that sin

sin

. ; then , , and a Magma program, one can have and

2) Let

. By , , , , and . By Lemma 3 and (21), one gets ,

sin

sin . ; then , , and . By a Magma program, one can similarly have , , , , , and sin , which imply sin . and ; then 4) Let , , and . Obviously, both and since they are nonsquares. By a Magma program, one can get and , i.e., . Computer experiments show that all these functions are regular bent, which are consistent with the results given in Corollary 6. i.e., 3) Let

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element in the finite field , be a th primitive root of unity, and be defined as (2) over . We consider the function of form (20) in the finite field . 1) Let and ; then , , and . Note that both and . By a Magma program, one has and , i.e., . 2) Let and ; then , , and . Note that both and . By a Magma program, one gets and , i.e., . By computer experiments, both of the functions are verified to be bent, which are compatible with the results given in Corollary 6. B. Second Class of -ary Bent Functions Similar to the binary case, in this section, we can also derive new -ary bent functions with multiple trace terms of the form (22) where , , , are integers, is a subset of , and is the smallest positive integer satisfying and . (I) Bent Functions With Multiple Trace Terms: : Similar to the binary case, we can get the following result. Theorem 10: Let be defined by (22) with , and . 1) If , then is regular bent if and only if and . 2) If , then is regular bent if and only if . Proof: By a direct computation, for and , by , one can get

and

Example 7: Let and

,

, and ; then . Let be a primitive

Then by Theorem 1,

Thus, one can deduce that 1) If and

The requirement 2) If

This completes the proof.

is bent if and only if

, then

is a permutation of

due to (18). , one has that

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Remark 10: Let and be two functions of the form in Theorem 10 with the pairs and , respectively. Similar conditions as in Theorem 10 can be derived such that is bent if and are suitably chosen. and ; then . Example 8: Let We consider the functions in Theorem 10 as follows. 1) If one takes , , and , then by a Magma program one can get and . Note that . Thus, we have . 2) If one takes , and , then one similarly has , , and . This implies . Computer experiments show that the functions are bent, which are consistent with the results given in Theorem 10. (II) Bent Functions With Multiple Terms: and : For this case, the function of the form (23) is considered, where and By Theorem 1,

are integers with . is bent if and only if

Notice that and one has is bent if and only if

. This implies that for . Thus,

(24) and

are permutations of if . Then, we can obtain the following result according to Lemma 3, (24), (18), and (19). Theorem 11: Let , only if sin

be defined by (23) with , and . Define . Then, is regular bent if and cos is equal to sin

if if

and , then

, then is not bent; is regular bent if and cos .

Remark 11: Here, we need to point out the following. 1) Similar as in Remark 9, if both the pairs and such that are bent, then one can claim that , i.e., is bent for both the pairs and if and only if and cos with . 2) For any other suitable subset , the characterization of the bentness of can be similarly discussed. VI. CONCLUDING REMARKS In this paper, new classes of binary bent and -ary regular bent functions, including binomials, trinomials, and functions with multiple trace terms, are obtained based on some exponential sums over finite fields. The bentness of all these functions are characterized by some exponential sums, most of which have close relations with the well known Kloosterman sums. It should be noted that some of the results obtained in this paper generalize the results in [11], [23], and [29] for and those of in [19] for , respectively. REFERENCES

Further, one can deduce that

since both

In particular, if and if and only if and

.

[1] A. Canteaut, P. Charpin, and G. Kyureghyan, “A new class of monomial bent functions,” Finite Fields Appl., vol. 14, no. 1, pp. 221–241, 2008. [2] L. Carlitz, “Explicit evaluation of certain exponential sums,” Math. Scand., vol. 44, pp. 5–16, 1979. [3] P. Charpin and G. Gong, “Hyperbent functions, Kloosterman sums and Dickson polynomials,” IEEE Trans. Inf. Theory, vol. 9, no. 54, pp. 4230–4238, Jul. 2008. [4] P. Charpin and G. Kyureghyan, “Cubic monomial bent functions: A ,” SIAM J. Discr. Math., vol. 22, no. 2, pp. 650–665, subclass of 2008. [5] P. Charpin, E. Pasalic, and C. Tavernier, “On bent and semi-bent quadratic boolean functions,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4286–4298, Dec. 2005. [6] P. Delsarte and J. M. Goethals, Irreducible Binary Cyclic Codes of Even Dimension. Chapel Hill, NC: Univ. North Carolina, May 1970, pp. 100–113. [7] J. F. Dillon, “Elementary Hadamard difference sets,” Ph.D. dissertation, Univ. Maryland, Collage Park, 1974. [8] J. F. Dillon and H. Dobbertin, “New cyclic difference sets with Singer parameters,” Finite Fields Appl., vol. 10, no. 3, pp. 342–389, 2004. [9] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke, and P. Gaborit, “Construction of bent functions via Niho power functions,” J. Comb. Theory, Ser. A, vol. 113, no. 5, pp. 779–798, 2006. [10] K. Garaschuk and P. Lisonêk, “On ternary Kloosterman sums modulo 12,” Finite Fields Appl., vol. 14, pp. 1083–1090, 2008. [11] F. Gologlu, “Almost bent and almost perfect nonlinear functions, exponential sums, geometries and sequences,” Ph.D. dissertation, Univ. Magdeburg, Magdeburg, Germany, 2009. [12] T. Helleseth, “Some results about the cross-correlation function between two maximal linear sequences,” Discrete Math., vol. 16, no. 3, pp. 209–232, 1976. [13] T. Helleseth, H. D. L. Hollmann, A. Kholosha, Z. Wang, and Q. Xiang, “Proofs of two conjectures on ternary weakly regular bent functions,” IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 5272–5283, Nov. 2009. [14] T. Helleseth and A. Kholosha, “Monomial and quadratic bent functions over the finite field of odd characteristic,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2018–2032, May 2006. [15] T. Helleseth and A. Kholosha, “New binomial bent functions over the finite fields of odd characteristic,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4646–4652, Sep. 2010. [16] T. Helleseth and A. Kholosha, “On generalized bent functions,” in Proc. IEEE Inf. Theory Appl. Workshop, 2010, pp. 1–6.

LI et al.: SEVERAL NEW CLASSES OF BENT FUNCTIONS FROM DILLON EXPONENTS

[17] T. Helleseth and A. Kholosha, “Sequences, bent functions and Jacobsthal sums,” Lecture Notes Comput. Sci., vol. 6338, pp. 416–429, 2010. [18] X. D. Hou, “ -ary and -ary versions of certain results about bent functions and resilient functions,” Finite Fields Appl., vol. 10, pp. 566–582, 2004. [19] W. J. Jia, X. Y. Zeng, T. Helleseth, and C. L. Li, “A class of binomial bent functions over the finite fields of odd characteristic,” IEEE Trans. Inf. Theory, vol. 58, no. 9, pp. 6054–6063, Sep. 2012. [20] K. P. Kononen, M. J. Rinta-aho, and K. O. Väänänen, “On integer values of Kloosterman sums,” IEEE Trans. Inf. Theory, vol. 56, no. 8, pp. 4011–4013, Aug. 2010. [21] P. V. Kumar, R. A. Scholtz, and L. R. Welch, “Generalized bent functions and their properties,” J. Combin. Theory Ser. A, vol. 40, pp. 90–107, 1985. [22] N. G. Leander, “Monomial bent functions,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 738–743, Feb. 2006. [23] S. Mesnager, “Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials,” IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 5996–6009, Nov. 2011. [24] S. Mesnager, “Semibent functions from Dillon and Niho exponents, Kloosterman sums, and Dickson polynomials,” IEEE Trans. Inf. Theory, vol. 57, no. 11, pp. 7443–7458, Nov. 2011. [25] S. Mesnager and J. Flori, A note on hyper-bent functions via Dillon-like exponents [Online]. Available: eprint.iacr.org/2012/033.pdf [26] R. Lidl, G. L. Mullen, and G. Turnwald, Dickson Polynomials, ser. Pitman Monographs in Pure and Applied Mathematics. Reading, MA: Addison-Wesley, 1993, vol. 65. [27] R. Lidl and H. Niederreiter, “Finite fields,” in Encyclopedia of Mathematics and Its Applications. Reading, MA: Addison-Wesley, 1983, vol. 20. [28] O. S. Rothaus, “On bent functions,” J. Combin. Theory Ser. A, vol. 20, no. 3, pp. 300–305, 1976. [29] B. Wang, C. Tang, Y. Qi, Y. Yang, and M. Xu, A new class of hyperbent Boolean functions in binomial forms [Online]. Available: http:// arxiv.org/pdf/1112.0062.pdf [30] N. Y. Yu and G. Gong, “Constructions of quadratic bent functions in polynomial forms,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3291–3299, Jul. 2006. [31] D. B. Zheng, X. Y. Zeng, and L. Hu, “A family of -ary binomial bent functions,” IEICE Trans. Fund., vol. 94-A, no. 9, pp. 1868–1872, 2011.

Nian Li received the B.S. and M.S. degrees in mathematics from Hubei University, Wuhan, China, in 2006 and 2009, respectively. He is currently working toward the Ph.D. degree at the Southwest Jiaotong University, Chengdu, China, and currently, he is a visiting Ph.D. student (Sept. 2011–Aug. 2013) in the Department of Informatics, University of Bergen, Norway. His research interests include sequence design and coding theory.

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Tor Helleseth (M’89–SM’96–F’97) received the Cand. Real. and Dr. Philos. degrees in mathematics from the University of Bergen, Bergen, Norway, in 1971 and 1979, respectively. From 1973 to 1980, he was a Research Assistant at the Department of Mathematics, University of Bergen. From 1981 to 1984, he was at the Chief Headquarters of Defense in Norway. Since 1984, he has been a Professor in the Department of Informatics at the University of Bergen. During the academic years 1977–1978 and 1992–1993, he was on sabbatical leave at the University of Southern California, Los Angeles, and during 1979–1980, he was a Research Fellow at the Eindhoven University of Technology, Eindhoven, The Netherlands. His research interests include coding theory and cryptology. Prof. Helleseth served as an Associate Editor for Coding Theory for the IEEE TRANSACTIONS ON INFORMATION THEORY from 1991 to 1993. He was Program Chairman for Eurocrypt’93 and for the Information Theory Workshop in 1997 in Longyearbyen, Norway. He was a Program Co-Chairman for SETA04 in Seoul, Korea, and SETA06 in Beijing, China. He was also a Program Co-Chairman for the IEEE Information Theory Workshop in Solstrand, Norway in 2007. During 2007–2009 he served on the Board of Governors for the IEEE Information Theory Society. In 1997 he was elected an IEEE Fellow for his contributions to coding theory and cryptography. In 2004 he was elected a member of Det Norske Videnskaps-Akademi.

Xiaohu Tang (M’04) received the B.S. degree in applied mathematics from the Northwest Polytechnic University, Xi’an, China, the M.S. degree in applied mathematics from the Sichuan University, Chengdu, China, and the Ph.D. degree in electronic engineering from the Southwest Jiaotong University, Chengdu, China, in 1992, 1995, and 2001 respectively. From 2003 to 2004, he was a postdoctoral member in the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology. From 2007 to 2008, he was a visiting professor at University of Ulm, Germany. Since 2001, he has been in the Institute of Mobile Communications, Southwest Jiaotong University, where he is currently a professor. His research interests include sequence design, coding theory and cryptography. Dr. Tang was the recipient of the National excellent Doctoral Dissertation award in 2003 (China), the Humboldt Research Fellowship in 2007 (Germany).

Alexander Kholosha received the PhD. degree in Mathematics from the Eindhoven University of Technology, Eindhoven, the Netherlands, in 2003. He is a Researcher at the Department of Informatics, University of Bergen, Bergen, Norway. His research interests lie in the area of cryptology in general and he is in particularly interested in cryptographic properties of Boolean and pseudo-Boolean functions, sequences, key-stream generation for stream ciphers and cryptanalysis. Dr. Kholosha was a Program Co-Chairman for the International Workshop on Coding and Cryptography 2009 in Ullensvang, Norway.